Squeezed states of light

and their applications in laser interferometers

Roman Schnabel

Institut fur Laserphysik Zentrum fur Optische QuantentechnologienUniversitat Hamburg Luruper Chaussee 149 22761 Hamburg Germany

Abstract

According to quantum theory the energy exchange between physical systemsis quantized As a direct consequence measurement sensitivities are funda-mentally limited by quantization noise or just lsquoquantum noisersquo in short Fur-thermore Heisenbergrsquos Uncertainty Principle demands measurement back-action for some observables of a system if they are measured repeatedly Inboth respects squeezed states are of high interest since they show a lsquosqueezedrsquouncertainty which can be used to improve the sensitivity of measurementdevices beyond the usual quantum noise limits including those impactedby quantum back-action noise Squeezed states of light can be producedwith nonlinear optics and a large variety of proof-of-principle experimentswere performed in past decades As an actual application squeezed lighthas now been used for several years to improve the measurement sensitivityof GEO 600 ndash a laser interferometer built for the detection of gravitationalwaves Given this success squeezed light is likely to significantly contributeto the new field of gravitational-wave astronomy This Review revisits theconcept of squeezed states and two-mode squeezed states of light with a fo-cus on experimental observations The distinct properties of squeezed statesdisplayed in quadrature phase-space as well as in the photon number rep-resentation are described The role of the lightrsquos quantum noise in laserinterferometers is summarized and the actual application of squeezed statesin these measurement devices is reviewed

Keywords

Preprint submitted to Physics Reports December 19 2017

arX

iv1

611

0398

6v3

[qu

ant-

ph]

17

Dec

201

7

Contents

1 Introduction 4

2 Observations on light fields in squeezed states 921 Definition of a lsquosingle modersquo 1022 Observations on squeezed states using a single PIN photo-diode 1123 Observations on squeezed states using a balanced homodyne

detector 1324 Observations on two-mode squeezed states using balanced ho-

modyne detectors 1725 Observations using photon counters 1926 Conclusions 21

3 Theoretical description of squeezed states 2331 The quadrature amplitude operators 2332 Phase space representations of squeezed states 2733 Covariance matrix representation of (single-party) squeezed

states 3234 Phase space representation of two-mode (bi-partite) squeezed

states 3335 Covariance matrix representation of bi-partite squeezed states 3436 Photon numbers of squeezed states 36

4 Squeezed-light generation 3941 Overview 3942 Degenerate type I optical-parametric amplification (OPA) 4043 Cavity-enhanced OPA 4344 The generation of squeezed light for laser interferometry 48

441 High squeeze factors ndash minimizing decoherence 50442 Squeezing in the gravitational-wave (GW) detection band 52443 The first squeezed-light source for GW detection 54444 Generation of two-mode (bi-partite) squeezing 55

45 Conclusions 56

5 Quantum noise in laser interferometers 5751 Interferometric measurements 5752 Quantum measurement noise and shot noise 58

2

53 Quantum back-action and quantum radiation pressure noise 6554 Interferometer total quantum noise and the standard quantum

limit 6755 Squeezed light for surpassing the standard quantum limit 6956 Optomechanically induced (ponderomotive) squeezing 7557 Conclusions 77

6 The first application of squeezed light in an operating gravita-tional-wave detector 7761 Gravitational waves 7862 Interferometric detection of gravitational waves 7963 Squeezed-light enhancement of the gravitational-wave detector

GEO 600 8264 Are squeezed states the optimal nonclassical resource in

gravitational-wave detectors 8565 Conclusions 88

7 The application of 2-mode-squeezed light in laser interfero-meters 8871 Quantum Dense Metrology 8872 Conclusions 92

8 Summary and Outlook 93

3

1 Introduction

Laser interferometers are used to monitor small changes in refractiveindices rotations or surface displacements such as mechanical vibrationsThey transfer a differential phase change between two light beams into achanging power of the output light which is photo-electrically detected forexample by a photo diode The light is produced in a lasing process thatusually aims for a coherent (Glauber) state In practice laser light is oftenin a mixture of coherent states producing excess noise in the interferometricmeasurement But even if the laser light is in a (pure) coherent state its de-tection is associated with noise usually called lsquoshot-noisersquo This arises fromthe quantisation of the electro-magnetic field which for a coherent stateresults in Poissonian counting statistics of mutually independent photons

1000099009800 10100 10200

Photon number n

0000

0002

0004

0006

0008

0010

0012

Pro

babi

lity Squeezing

Poisson distribution

Figure 1 Poissonian and squeezed photon statistics ndash The upper boundary of eacharea represents the probability distribution of detected photon number n when perform-ing a large number of measurements on an ensemble of identical states having an averagephoton number of n = |α|2 = 10000 where α is the coherent field excitation or lsquodis-placementrsquo The broader curve shows the lsquoPoissonianrsquo distribution which describes thecounting statistic of mutually independent particles ie those of the coherent state Dueto the large value of α the distribution is almost Gaussian with a standard deviation ofplusmnradicn The narrow curve corresponds to the equally displaced 10 dB squeezed state which

obviously has a lsquosub-Poissonianrsquo photon statistic Note that squeezed states with smallor even without any coherent excitation (squeezed vacuum states) exhibit quite differentphoton statistics ndash see Fig 13 for example

4

If the coherent state is highly excited and thus the average number ofphotons n per detection interval is large the Poissonian distribution canbe approximated by a Gaussian distribution with a standard deviation ofplusmnradicn During the past decades squeezed states of light have attracted a lot

of attention because they can exhibit less quantum noise than a coherent stateof the same coherent excitation ie they can show sub-Poissonian countingstatistic see Fig 1

θ

Shot noise Squeezed noise

(a) (b)

Brightlaser input

Squeezedvacuum input

Faradayrotator

Photo diode

Michelsoninterferometer

Signal5050

(i)Ph

oto

curr

ent [

rel

units

]

Time [ms]

(ii)

0 5 10

Figure 2 Squeezed-light enhanced Michelson interferometer ndash (a) In addition tothe conventional operation of a Michelson laser interferometer with bright coherent lighta broadband squeezed-vacuum field is injected into the signal output port and overlappedwith the bright interferometer mode The interferometer is operated close to a dark fringesuch that most of the bright coherent light as well as most of the squeezed vacuum areback-reflected from the Michelson interferometer respectively Due to interference withthe broadband squeezed vacuum the interferometerrsquos output light on the photo diodeshows reduced variance in the photon number statistic as shown in Fig 1 Overlappingthe two light fields is possible with theoretically zero loss by the combination of a Faradayrotator and a polarizing beam splitter (PBS) A signal is produced by modulating therelative arm length (b) Simulated data for photo diode measurements Without squeezing(i) the signal of the laser interferometer is not visible With squeezing (ii) the shot noiseis reduced and here a sinusoidal signal visible

Squeezed states belong to the class of lsquonon-classicalrsquo states which areconsidered to be at the heart of quantum mechanics These states are de-fined as those that cannot be described as a mixture of coherent states Inthis case their Glauber-Sudarshan P -functions [Sudarshan (1963) Glauber(1963)] do not correspond to (classical) probability density functions ie theyare not positive-valued functions As a lsquoclassicalrsquo example the P -function ofa coherent state corresponds to a δ-function

5

But the question remains what property of coherent states justifies thename lsquoclassicalrsquo even though coherent states are quantum states and showquantum uncertainties My answer to this question is the following Allexperiments which only involve coherent states and mixtures of them allowfor a description that uses a combination of classical pictures As we will seebelow this description swaps between two different classical pictures and isthus not truly classical but semi-classical (A more precise description of thenature of coherent states uses the term lsquosemi-classicalrsquo)

Let us consider a laser interferometer that uses light in a coherent stateFirstly the light beam is split in two halves by a beam splitter The twobeams travel along different paths and are subsequently overlapped on abeam splitter where they interfere exactly as classical waves would do Theelectric fields superimpose thereby producing the phenomenon of interfer-ence Up to this point there is no reason to argue light might be composedof particlesSecondly the new (still coherent) beams that result from the interference areabsorbed for instance by a photo-electric detector In the case of coherentstates the detection process can be perfectly described in the classical parti-cle picture in which the particles appear independently from each other in atruly random fashion yielding the aforementioned Poisson statistic Duringthe detection process no wave feature of the light is present Let us havea closer look A truly random (lsquospontaneousrsquo) event is an event that hasnot been triggered by anything in the past This allows us to make a clearcut between the first part of the experiment described by the classical wavepicture and the second part of the experiment described by the classicalparticle picture Both lsquoworldsrsquo are disconnected The subsequent applicationof two classical pictures is not truly classical but lsquosemi-classicalrsquo It is in-deed the observation that the photons occur individually with truly randomstatistics that allows this semi-classical description In the case of a mixtureof coherent states the photon statistics are super-Poissonian which can beunderstood as a mixture of different Poissonian distributions In the caseof a slowly changing coherent state the mean value n depends on time Inall these cases the semi-classical description is appropriate Let me pointout that in this very reasonable description photons do not exist before theyare detected eg absorbed Further note that the famous double-slit exper-iment with coherent states also allows for the same semi-classical description

For squeezed states [Yuen (1976) Walls (1983)] the situation is different

6

As before the interference can be fully described by the classical wave pic-ture The result of the detection process however is different from that ofmutually independent random events It is also different from any super-Poissonian statistics that could be produced by mixing an arbitrary num-ber of different andor time-dependent Poissonian distributions Insteadthe squeezed probability distribution in Fig 1 suggests that the probabil-ity of detecting a photon decreases with the more photons that are alreadydetected in the same time interval over which a single measurement is in-tegrated From this observation one must conclude that the photons donot individually appear in a random fashion upon detection There mustbe lsquoquantumrsquo correlations between the photons These correlations mustexisted before detection since there is no interaction between the photonsduring their detection Pre-existing correlations between detected photonsseem to imply that the photons themselves existed before detection ie attimes when interference occurred In a semi-classical description howeverphotons are classical particles and cannot interfere for instance on a beamsplitter At this point the semi-classical picture breaks down Squeezedstates are therefor lsquononclassicalrsquoThe failure of the semi-classical model described above generally certifiesnonclassicality

Squeezed states are usually not characterized by counting their pho-tons but by measuring canonical continuous-variable phase-space observ-ables Measurements are performed as usual on an ensemble of identicalstates and quasi-probability density functions are calculated from the dataThe Glauber-Sudarshan P -function is the quasi-probability density distribu-tion over coherent states If the P -function of a state is entirely positivethe state is a coherent state or a (classical) mixture of coherent states Thestate is considered as semi-classical If the P -function is not a positive-valuedfunction the state cannot be expressed as a (classical) mixture of coherentstates and is thus nonclassical [Gerry and Knight (2005) Vogel and Welsch(2006)] A non-positive-valued P -function is the sufficient and necessary con-dition for the failure of the semi-classical model The Wigner function is thequasi-probability phase-space representation over the canonical continuous-variable phase-space observables themselves [Gerry and Knight (2005)] TheWigner functions of squeezed states are entirely positive Although subject todiscussion this fact does not mean that squeezed states are less nonclassicalthan Fock states or cat states which not only have a nonclassical P -function

7

but also a partially negative Wigner function (A cat state is a quantum su-perposition of two macroscopically distinct states [Monroe (2002)] referringto Schrodingerrsquos-cat gedanken experiment [Schrodinger (1935)]) In practicesqueezed states can even be regarded as superior nonclassical states becausethey represent the only nonclassical state that has been produced in a steadystate fashionIn almost all experiments so far the generation of Fock states and cat statesinvolves a probabilistic event such as the detection of a photon in anotherbeam path to herald these states In fact squeezed states provide the non-classical resource for the probabilistic preparation of Fock states as well as catstates But only the squeezed states themselves show a nonclassical effect in astationary way Limited only by the time duration and the frequency span ofthe mode that is in a squeezed state the squeezing effect can be continuouslyobserved independently of the time when the measurement is performed andalso independently of the measurement integration time This fact is of greatimportance for applications of squeezed states in measurement devices sincea squeezed-light-enhanced measurement remains unconditional and the ef-fective measurement time is not reduced

In past decades squeezed states of light were used in many proof-of-principle experiments to research their potential for improving the sensitivityof laser interferometers [Grangier et al (1987) Xiao et al (1987) McKen-zie et al (2002) Vahlbruch et al (2005) Goda et al (2008) Taylor et al(2013)] or the performance of imaging beyond the shot-noise limit [Lugiatoet al (2002) Treps et al (2003)] both accompanied by a huge number oftheoretical works Potential applications in secure optical communication(quantum key distribution) were also proposed and proof-of-principle ex-periments demonstrated [Ralph (1999) Furrer et al (2012) Gehring et al(2015)] This review restricts itself to the improvement of laser interfero-meters since only here has the application of squeezed light gone beyondproof-of-principle The gravitational-wave detector (GWD) GEO 600 hasoperated with squeezed light now for more than seven years starting in 2010[Abadie (2011) Grote et al (2013)] GEO 600 is a 600 m long Michelsonlaser interferometer built for the detection of gravitational waves Thesewaves are audio-band and sub-audio-band changes of space-time curvatureoriginating from cosmic events such as the merger of neutron stars or blackholes as detected recently [Abbott (2016)] In GWDs such as GEO 600 [Doo-ley et al (2016)] Advanced LIGO [Aasi (2015)] Advanced Virgo [Acernese

8

(2015)] and KAGRA [Aso et al (2013)] conventional laser technology hasbeen pushed to extremes over the past decades Noise spectral densities nor-malized to space-time strain of less than 10minus23 Hzminus12 have been measured[Abbott (2016)] Progress will continue and based on the successful appli-cation in GEO 600 squeezed light is now widely accepted to provide a newadditional technology to contribute to the new field of gravitational-waveastronomy It was also successfully tested in one of the LIGO detectors in2013 [LSC (2013)] and is an integral part of the European design study forthe 10 km Einstein-Telescope [Punturo et al (2010)]

GEO 600 has already taken several years of lsquosqueezedrsquo observational datawhich has increased its sensitivity at signal frequencies above 500 Hz Withthe implementation of a squeezed light source in GEO 600 the applicationof nonclassical states in metrology has been pushed beyond merely proof-of-principle

lsquoTwo-mode squeezed statesrsquo show a squeezed uncertainty in at least onejoint continuous variable of two subsystems lsquoArsquo and lsquoBrsquo Examples of jointvariables are differences and sums of phase-space observables of A and BTwo-mode squeezed states not only belong to the class of nonclassical statesbut due to their bi-partite character also to the class of lsquoinseparablersquo orlsquoentangledrsquo states They are the ideal states to demonstrate the Einstein-Podolsky-Rosen paradox [Einstein et al (1935)] as first achieved in [Ouet al (1992)] Apart from fundamental research on quantum mechanicsrecent proof-of-principle experiments demonstrated their usefulness in inter-ferometric measurements that go beyond the application of simple squeezedstates [Steinlechner et al (2013) Ast et al (2016)] This experiment is thefinal topic of this review

2 Observations on light fields in squeezed states

Generally there are two different kinds of observables that can be subjectof a measurement performed on a quantum system The first kind is associ-ated with the systemrsquos wave property In optics it corresponds to the electricfield strength at a given phase angle ϑ The according (dimensionless) opera-tors are called the quadrature amplitudes Xϑ and have a continuous spectrumof eigenvalues Quadrature amplitudes are measured in very good approx-imation with a balanced homodyne detector using the interference with a

9

bright local oscillator beam see Fig 3 (a) In practice any measurement ofXϑ integrates over some sideband (Fourier) spectrum within the angular fre-quencies Ω plusmn ∆Ω2 The sideband information always needs to be quotedA straight forward but rather untypical way is by adding subscripts whichleads to Xϑ

Ω∆Ω The classical analogue of the quadrature amplitude operator

XϑΩ∆Ω is the modulation depth of the optical field at modulation phase angle

ϑ and at angular modulation frequency Ω measured over the band ∆Ω lt 2ΩThe uncertainties of the statersquos quadrature amplitudes at different phases ϑare limited by a Heisenberg uncertainty relation see section 3 The secondkind of measurement is associated with the systemrsquos particle property andis given by the photon number operator n associated with a measuring timeinterval ∆t Its precise measurement requires a photon counter ideally withsingle photon resolution The measurement result obviously has a discretespectrum Continuous as well as discrete observables are usually subject toquantum uncertainties and thus quantum noiseUsually the measurementrsquos integration time and frequency band actuallydefine the physical system that is characterized In quantum optics experi-ments the interrogated physical system is called a lsquomodersquo

21 Definition of a lsquosingle modersquo

Let us define a light field or generally any quantum system to be asingle mode if it corresponds to the lsquosmallest entity of a waversquo In this caseits spectral and temporal distributions as well as waist size and divergenceare at their Fourier limits and all other properties such as optical axis waistposition and polarization are well defined For instance a linearly polarizedlongitudinal resonance of an optical standing-wave cavity defines such a singlemode if the cavity finesse is high and transversal modes are non-degenerateThe complete photo-electrical detection of a cavity mode however is notstraight forward Most quantum optical experiments are instead performedon propagating light In this case single modes are defined by spatial filtersand by temporal-spectral measurement windows both being at the Fourierlimit Examples for single modes are a laser pulse and a spectraltemporalcutout from a continuous observation of a quasi-monochromatic continuous-wave light beam in the spatial TEM00 mode both at the Fourier limits

In classical physics the only remaining free parameter of a given singlemode is its excitation energy In quantum physics the situation is differentFor a given energy a single mode can be in many different quantum states

10

which differ in their quantum statistics Examples are coherent states num-ber (Fock) states and squeezed states

22 Observations on squeezed states using a single PIN photo-diode

An ideal PIN photo-diode absorbs the full energy of a light mode andproduces one photo electron for every absorbed photon energy It uses theinternal photo-electric effect inside a semiconductor such as silicon or In-GaAs In contrast to avalanche photo-diodes PIN photo-diodes operatewith unity gain lsquoPINrsquo stands for lsquopositiversquo lsquointrinsicrsquo and lsquonegativersquo and isdescribing the doping of the semiconductor layers A PIN photo-diode is op-timally suited for the continuous monitoring of a rather bright light field ofup to several tens of milliwatts An example is the photo-diode in the outputport of a gravitational-wave detector as shown in Fig 2 (a) The prominentwavelength of 1064 nm which is emitted by NdYAG lasers has an opticalfrequency of ν = ω(2π) = 282 middot 1014 Hz The period of the field oscillationis a few femtoseconds and cannot be directly resolved with photo-electric de-tectors However variations of the electric field around the averaged opticalfield oscillation on longer time-scales can be resolved Applying an electronicbandpass filter at the sideband angular frequency Ω plusmn ∆Ω2 to the photovoltage provides information about the lsquodepth of the lightrsquos amplitude mod-ulationrsquo which is also called the lsquoamplitude of the amplitude quadraturersquo Itcan also slowly vary in time and reads

Xϑ=0

Ω∆Ω(t) equiv XΩ∆Ω(t) equiv X (1)

The subscript is usually skipped as it is done with the time dependenceas indicated on the right Applying the electronic bandpass filter in factdefines the mode of the light being detected The structure of the defini-tion in Eq (1) forms the basis of interferometric signals and quantum noisealso in the semi-classical case of coherent states Lets take an example Inthe recent observation of gravitational waves [Fig 1 bottom row in Abbott(2016)] the time-frequency representation of the gravitational-wave signalcorresponded to the amplitude quadrature amplitude XΩ∆Ω(t) of the inter-ferometer output light Note that a larger value of ∆Ω allows for changes ofthe quadrature amplitude on shorter time scales

If the light fieldrsquos lsquomodulation modersquo does not contain any quanta simplybecause there are no photons that have a frequency difference of plusmnΩ with

11

respect to the carrier it is in its ground state In this case lsquovacuum noisersquois observed which originates from the ground state uncertainty Since thevacuum noise only becomes measurable as a beat with a bright light field itcan also be seen as the carrierrsquos band-path filtered shot noise A modulationmode in a displaced vacuum state (a coherent state) corresponds to nonzerocoherent modulationThe measured level of the vacuum noise generally depends on the power ofthe bright carrier light and on the electronic amplification In any case itprovides the reference for certifying lsquosqueezingrsquo Observations using a singlePIN photo-diode require an independent measurement to quantify vacuumnoise A necessary condition is that attenuating the total fieldrsquos light powerresults in the same attenuation of the measured XΩ∆Ω values If they showa stronger attenuation a coherent modulation or thermal noise might bepresent If they show a weaker attenuation the photo-diode and its electron-ics might be saturated

Fig 2 (b) illustrates how a broadband squeezed field improves the mea-surement of an amplitude modulation in time domain based on a PIN photo-diode Shown is a simulated time sequence of XΩ∆Ω-data sampled from thephotoelectric voltage In this simulation all sideband frequencies from zero(DC) to the cutoff frequency of the detector electronics (Ωcut) are included(Ω = ∆Ω2 = Ωcut2) No additional band pass filter is applied making it amaximally broadband detection Although the data in Fig 2 (bi) contains aclassical amplitude modulation of the detected light this signal is not visibledue to random noise here representing shot noise Fig 2 (bii) shows thesame situation but with shot noise that is squeezed over the full detectionband The quantum uncertainty of the modulation depth is squeezed andthe classical signal becomes visible

It needs to be noted that a single PIN photo-diode can only measure theamplitude of the amplitude quadrature XΩ∆Ω(t) but not the non-commutingobservable the lsquoamplitude of the phase quadraturersquo

Xϑ=90

Ω∆Ω (t) equiv YΩ∆Ω(t) equiv Y (2)

For values that are small compared to the field strength of the bright fieldthe quantity Y approximately describes the bright fieldrsquos lsquophase modulationdepthrsquo

12

23 Observations on squeezed states using a balanced homodyne detector

-72

-69

-66

-63(i)

(ii)

(iii)

(a)

LO

(Squeezed)signal input

Phaseshifter

PD1

PD2

5050

01 0200 03

-75

-60

-57Balanced homodyne detector

(b)

Time [s]

Noi

se p

ower

[dB

m]

Figure 3 Balanced homodyne detection (BHD) ndash (a) Setup The quadrature atchoosable angle ϑ of the signal field is measured by overlapping the latter with a localoscillator (LO) field of the same mode parameters on a balanced beam splitter and record-ing the difference voltage from two PIN photo-diodes as shown In order to meet theBHD approximation the LO needs to be much more intense than the signal field A closeto perfect mode overlap between LO and signal input field is crucial For a non-perfectoverlap the detector measures the input state with unwanted contributions of the vacuumstate (b) Noise power measurements (i) on an electronically amplified and band-passfiltered quadrature amplitude of the vacuum field (Xvac

Ω∆Ω) (signal input blocked) (ii) on

a squeezed quadrature (XsqzΩ∆Ω) of a squeezed vacuum state (ϑ = 0) and (iii) on re-

spective quadratures of the same squeezed state where the phase angle ϑ was continuouslyshifted by changing the optical path length of the LO The measurement data shows about5 dB of squeezing and was first published in [Chelkowski et al (2007)] Ω2π = 5 MHz∆Ω2π = 100 kHz

In contrast to a single PIN photo diode a balanced homodyne detector(BHD) is suitable to measure the quantum statistic of all types of modu-lations ie for all angles ϑ Such a detector consists of two identical PINphoto-diodes a balanced beam splitter and an external homodyne local os-cillator field that is much brighter than the signal beam and that has anadjustable phase The signal beam corresponds to the squeezed field whichin many experiments is in a squeezed vacuum field having an optical powerthat usually corresponds to just a few photons per mode The two beams areoverlapped on the balanced beam splitter with close to perfect mode match-ing and the two interference outputs are focussed onto the photo diodes seeFig 3 (left) The electric output signal of the BHD is the difference of the

13

photo diode voltages The LO takes over the role of the carrier light fieldbut with the possibility to choose the phase shift ϑ This way eigenvaluesof X Y or Xϑ can be measured where the latter is given by the followinglinear combination of the first two

Xϑ(t) = cos(ϑ) X(t) + sin(ϑ) Y(t) (3)

If the modulation depths of signal and local oscillator beams are weak com-pared to their coherent amplitudes |α| and |αLO| the output voltage of aBHD corresponds to eigenvalues of the following operator

V (t) prop 2cos(ϑ) |αLO||α|+ |αLO| Xϑ(t) + |α| XϑLO(t) (4)

The lsquohomodyne approximationrsquo further involves |αLO| |α| such that theterm on the right can be neglected even if the local oscillator shows someclassical quadrature excitation The output voltage of a BHD is usually spec-trally analysed or at least spectrally filtered which removes the DC part infull analogy to a single photo diode (see previous subsection) Sampling thefiltered voltage provides eigenvalues proportional to the generalized quadra-ture amplitude in Eq (3)

V BHDΩ∆Ω(t) prop |αLO| Xϑ

Ω∆Ω(t) (5)

Fig 3 (a) shows the setup of a balanced homodyne detector for the char-acterization of squeezed states Setting ϑ = 0 eigenvalues of the ampli-tude modulation depths can be sampled from the photo voltage accordingto Eq (5) Setting ϑ = 90 eigenvalues of the phase modulation depths aremeasured The datarsquos expectation values 〈Xϑ〉 provide the coherent displace-ment of the squeezed state The datarsquos variances

∆2Xϑ equiv 〈(Xϑ)2〉 minus 〈Xϑ〉2 (6)

provide the statersquos (quantum) noise A pure squeezed state as well as asqueezed state that experienced photon loss have Gaussian quantum statisticsand are thus fully described by the expectation values and variances (first andsecond moments) of two orthogonal quadratures but only if one quadraturereflects the lowest quadrature variance

14

In most experiments with squeezed light the photo electric voltage ac-cording to Eq (5) is not sampled with a data aquisition system but the signalis directly fed into a spectrum analyser measuring the noise power of the volt-age If the expectation value 〈Xϑ〉 is zero the noise power is proportionalto the variance ∆2Xϑ in Eq (6) The reference for quantifying the squeezefactor is measured by blocking the (squeezed) signal field in Fig 3 (a) Themeasured vacuum noise level corresponds to the LOrsquos (electronically ampli-fied) shot noise level

Traces (ii) and (iii) in Fig 3 (b) show measured noise powers of the mod-ulation mode (Ω2π = 5 MHz ∆Ω2π = 100 kHz) being in a squeezed vac-uum state (i) is proportional to the variance of the ground state uncertainty∆2Xvac

Ω∆Ω (ii) is proportional to the quantum noise variance of the squeezed

quadrature amplitude ∆2XsqzΩ∆Ω (iii) is proportional to the quantum noise

variance of the quadrature amplitude with scanned phase ∆2XΩ∆Ω(ϑ(t))

To fully characterize a quantum state ie to do quantum state tomogra-phy [Vogel and Risken (1989)] a BHD is a prerequisite But also interfero-metric measurements with balanced homodyne detectors instead of singlePIN photo-diodes have several advantages A correctly implemented BHDreadily provides the vacuum noise level when the signal beam is blockedWith a BHD the optimum operating point of the interferometer is preciselyat a dark fringe If a perfect dark fringe can practically be achieved ampli-tude noise of the laser does not couple into the signal port If the interfero-meter has balanced arm length also frequency noise of the laser then does notcouple into the signal port Some quantum non-demolition schemes with theprospect of evading quantum radiation pressure noise require the detection ofa non-canonical quadrature angle [Jaekel and Reynaud (1990) Kimble et al(2001)] Here the adjustable phase of a BHD provides a straight forwardapproach The experimental exploration of BHDs for gravitational-wave de-tectors only has started recently [Steinlechner et al (2015)]

A light field can be analysed with respect to many different modulationfrequencies Ω The result constitutes a spectrum [Breitenbach et al (1998)]where in principle every modulation mode can be in a different quantumstate Fig 4 shows spectra of squeezed states from 5 MHz to 100 MHz with∆Ω2π = 1 MHz The lower curve shows the spectrum of the most strongly

15

-14-12-10-8-6- 4-20 2 4 6 8

10 12 14 16 18

6 8 10 20 40 60 80 1005

Squeezed noise

Anti-squeezed noise

Vacuum noise

Frequency [MHz]

Nor

mal

ized

noi

se v

aria

nce

[dB

]

Figure 4 Spectrum of quadrature amplitude variances ndash Shown are the quantumnoise properties of a large number of modulation modes having a resolution bandwidth of∆Ω(2π) = 1 MHz For all traces the balanced homodyne detector output was analysedwith a spectrum analyser Squeezing of XΩ∆Ω (bottom trace) and anti-squeezing of

YΩ∆Ω (top trace) versus f = Ω(2π) are shown relative to the vacuum noise variance Thespectrum below 5 MHz is not shown since it contained less squeezing due to laser relaxationoscillation in the carrier field as well as disturbances from back-scattered light [Vahlbruchet al (2007)] Disturbances at frequencies above 70 MHz originated from relatively largedetector dark noise which was subtracted from all traces shown here The thin linerepresents a theoretical model that takes into account for the linewidth of the squeezingcavity The data was first presented in in Ref [Mehmet et al (2010)]

squeezed variances in this case the variances of XΩ∆Ω The upper spectrum

shows the variance in the orthogonal quadrature amplitude (YΩ∆Ω) Allvariances are normalized to those of the corresponding vacuum state Thesqueeze factor reduces towards higher frequencies due to the linewidth ofthe squeezing cavity The anti-squeezing is always higher than the absolutevalue of the squeezing due to Heisenbergrsquos uncertainty relation and due to thepresence of optical loss The curves do not represent pure squeezed states butmixed squeezed states with a significant contribution from vacuum statesdue to optical loss Pure squeezed states can only be produced by making

16

the influence of all decoherence processes negligibleThe choice of the resolution bandwidth (RBW ∆Ω) during data taking

and processing defines the spectral-temporal modulation modes includingtheir number within the detected spectrum For any setting of the RBWthe quantum mechanical properties of the quadrature amplitudes XΩ∆Ω and

YΩ∆Ω [Caves (1985)] fully correspond to those introduced for quadratures instandard text books and which are reviewed in Sec 3

24 Observations on two-mode squeezed states using balanced homodyne de-tectors

Two-mode squeezed states are composed of two subsystems lsquoArsquo and lsquoBrsquoand are bi-partite entangled states with a Gaussian quantum statistic Toavoid conflicts with different usage of the term lsquomodersquo they can synony-mously be named lsquobipartite Gaussian entangled statesrsquo or lsquobipartite squeezedstatesrsquo which will be mainly used in this Review In the same way multi-partite Gaussian entangled states correspond to multi-partite squeezed states

The measurement observables that prove or disprove the bi-partite squeez-ing property are XA

Ω∆Ω minus XBΩ∆Ω and Y A

Ω∆Ω + Y BΩ∆Ω where the minus and

plus signs may be swapped Bi-partite squeezed states are precisely thosestates that were discussed by Einstein Podolsky and Rosen (EPR) in theirseminal paper [Einstein et al (1935)] Fig 5 shows a measurement resulton bi-partite squeezed light [Eberle et al (2013)] The variances of bothjoined observables are squeezed as shown in the two lower traces They wererecorded consecutively by adding or subtracting the outputs of two balancedhomodyne detectors But by interfering the subsystems on a beam splitterone could even measure both joined observables simultaneously This pos-sibility is correctly described in quantum theory since their commutator iszero

The so-called EPR paradox arises as follows If we either measure XAΩ∆Ω

and XBΩ∆Ω or Y A

Ω∆Ω and Y BΩ∆Ω it is obvious from the data in Fig 5 that

we can always predict the measurement result at subsystem lsquoBrsquo when know-ing the result at subsystem lsquoArsquo This seems to suggest that both quantitiesat lsquoBrsquo are precisely defined simultaneously before the measurement on lsquoArsquowhich contradicts the rigorous (and correct) interpretation of their non-zerocommutator that they are not precisely defined simultaneously

To solve this paradox EPR conjectured that the wavefunction as definedby quantum theory does not provide the full information This led to a

17

discussion of whether hidden variables existed that needed to be included ina complete theory of quantum mechanics (see also Bell [Bell (1966)]) Theexperimentally observed violation of Bellrsquos inequality [Bell (1964) Aspectet al (1981) Giustina et al (2013) Hensen et al (2015)] however ruled outthe existence of (local) hidden variables

Based on that the EPR paradox needs to be solved in a different wayContrary to what EPR assumed it is in fact possible to predict the valueof an arbitrary observable of a physical system A with certainty via a mea-surement on system B although this observable was not defined before themeasurement Without any interaction a measurement on subsystem lsquoArsquo notonly creates lsquorealityrsquo of eg XA

Ω∆Ω simultaneously lsquorealityrsquo is also created

regarding the observable XBΩ∆Ω describing subsystem lsquoBrsquo Here the term

lsquorealityrsquo has the meaning as defined by EPR [Einstein et al (1935)] Simi-larly the detection of one photon of a two photon entangled number statenot only produces the reality of this photon but also that of a second oneA discussion of Einstein-Podolsky-Rosen entanglement can also be found in[Schnabel (2015)] Note that the EPR paradox can also be described aslsquoquantum steeringrsquo [Schrodinger (1935) Cavalcanti et al (2009) Handchenet al (2012)] It should also be mentioned that two-mode squeezing beingdetected with BHDs and not with photon counters cannot be used to violatea Bell inequality The latter topic is outside the scope of this Review

Bi-partite squeezed states were first characterized with balanced homo-dyne detectors by the group of J Kimble in 1992 [Ou et al (1992)] Gener-ally the EPR paradox becomes more pronounced the stronger the bi-partitesqueezing is A measure of the strength of EPR entanglement was introducedby M Reid [Reid and Walls (1985)] According to this measure the resultin Fig 5 can be quantified to ε2 = 00309 where the critical value is one Itcorresponds to the strongest Gaussian EPR entangled state generated so far

For a long time it looked like that two-mode squeezed states are notuseful for laser interferometers The reason for that belief was that a laserinterferometer as any other measurement device too is built to measureone observable It seems to be ideal already if the quantum noise in thissingle observable is squeezed The increased quantum noise in the orthog-onal observable is not harmful in this case and squeezing in two differentobservables useless Only recently realistic scenarios were discussed in whichtwo-mode squeezing in fact does improve the performance of a laser inter-

18

-12

-10

-8

-6

-4

-2

0

0 1 2 3 4 5 6 7 8 9 10

100

dB

109

dB

Δ (XAvac + X B

vac ) = Δ (Y

Δ (XA + X B)

Avacminus Y B

vac )2 2

2

Δ (YAminus Y B)2

Nor

mal

ized

noi

se v

aria

nce

[dB

]

Times [s]

Figure 5 Two-mode squeezing measurement ndash For this measurement the outputs oftwo balanced homodyne detectors are added or subtracted and the variances (noise pow-ers) of the results recorded The upper trace was measured with modes lsquoArsquo and lsquoBrsquo beingin their ground states This measurement served as a reference level Strong two-modesqueezing was observed as shown by the lower two traces The sideband frequency wasΩ(2π) = 8 MHz and the resolution bandwidth was ∆Ω(2π) = 200 kHz The measure-ment results were first published in Ref [Eberle et al (2013)]

ferometer [Steinlechner et al (2013)] The proof-of-principle experiment isreviewed in Sec 7

25 Observations using photon counters

Alternatively to field quadratures an optical mode in a squeezed statecan also be characterized at least partly by detecting its photon numberdistribution For a pure squeezed vacuum state such a measurement wouldreveal the existence of solely even photon numbers including a large prob-ability for zero photons The average photon numbers of squeezed vacuumstates with feasible squeeze factors are very small of the order of one persecond and bandwidth in hertz see Fig 13 (a) ndash (c) A distribution with closeto zero probability of odd photon numbers however has not been measuredso far The reason is the lack of ideal photon counters First of all theefficiency of these detectors ie their probability of converting one photoninto one click and no photon into no click must be almost perfect lsquoLostrsquophotons as well as dark counts wash out the oddeven oscillations Further-more most detectors available can only distinguish between zero and one

19

photon This problem can be solved by distributing the squeezed mode ontoa large number of single photon detectors using an array of beam splitterssuch that all paths have a low probability of carrying more than one pho-ton Photon number measurements on squeezed vacuum states neverthelessplay an extremely important role in quantum optics When the squeezingstrength is very low the probability of detecting more than 2 photons can beneglected and the detection of a photon heralds the existence of a second one

0

300

600

900

1200

-40 -20 0 20 400

5

10

15

20

25

Two-

fold

coi

ncid

ence

s [1

(4s

)](a) (b)

Non-degeneratetwo-mode squeezing

5050

APDA1

APDA2

APDHerald

Photon coincidencedetection

Delay [ns]

Thr

ee-f

old

coin

cide

nces

[1

(4s)

]1500

Figure 6 Coincidence clicks from non-degenerate photon pairs ndash The first suchexperiment was reported in Ref [Hong and Mandel (1986)] (a) shows a setup with threeavalanche photo-diodes (APDs) for proving the successful heralding of a single photonnumber state (b) Histograms of the two-fold coincidence detections at APDHerald andAPDA1 (red) and at APDHerald and APDA2 (yellow) with theoretical models (solid lines)If the two-mode squeezing just carried one photon in each spatial subsystem the three-foldcoincidence detection should be zero Indeed the according histogram (grey points righty axis) shows only a few events These are produced by false (dark) counts of the APDsThe delay for the three-fold coincidences is defined as the time between counts at lsquoA1rsquo andlsquoA2rsquo given that the trigger APDHerald detected a photon (within a 100 ns time window)The data was taken on photons that were up-converted from 1550 nm to 532 nm and itwas first published in Ref [Baune et al (2014)]

If a mode of light is always excited by either zero or two photons lsquocondi-tionalrsquo or lsquoheraldedrsquo one-photon Fock states can be realized (Measurementson an ensemble of the n-photon Fock state would always produce the mea-surement result n ie Fock states have a zero photon number uncertaintyThey are also called lsquonumber statesrsquo) The above concept of producing aone-photon Fock state obviously requires the deterministic and balanced dis-tribution of the down-converted signal and idler fields into two different paths

20

In order to achieve this the signal and idler fields need to be non-degenerateUsually a mode in a squeezed state is composed of degenerate signal andidler fields and this degeneracy thus needs to be removed Possible waysare producing the down-converted fields at well separated wavelengths [Vil-lar et al (2005) Su et al (2006) Li et al (2010) Samblowski et al (2011)]separating the upper and lower sidebands belonging to an ordinary squeezedmode by frequency filters [Schori et al (2002) Hage et al (2010)] and us-ing spatial filters [Hong et al (1987)] A frequently used approach is usingtype II parametric down-conversion where the photons within a pair are al-ways orthogonally polarized [Ou et al (1992) Kiess et al (1993) Kwiat et al(1995)]The list of experiments with conditional or heralded photon number states islong They showed for instance nonclassical g(2)-functions [Hong et al (1987)]and violations of Bell inequalities [Weihs et al (1998)] Fig (6) shows a re-sult from a more recent experiment in which a bipartite-squeezed state withsubsystems at 1550 nm and 810 nm was produced the subsystem at 1550 nmsubsequently up-converted to 532 nm and the lsquoquantum non-Gaussianityrsquo ofheralded up-converted single photons demonstrated [Baune et al (2014)]Squeezed states are also the resource for the conditional generation of super-positions of coherent states [Ourjoumtsev et al (2006) Neergaard-Nielsenet al (2006)] and so-called N00N-states [Afek et al (2010)]

The generation of nonclassical states mentioned in the paragraph aboveis not stationary but relies on a probabilistic trigger event The produc-tion of squeezed states themselves usually happens in a stationary fashionThis distinction has an important consequence for applications of nonclassi-cal states in measurement devices Only (stationary) squeezed states allowfor a continuous improvement of a measurement Avoiding any loss of mea-suring time is generally of high relevance for the detection of short-livedsignals with unknown arrival time as well as for the detection of long-livedquasi-monochromatic signals since the signal-to-noise-ratio (SN) improveswith measuring time

26 Conclusions

The detection of squeezed light produces measurement results that canbe considered as remarkable Let us focus on experiments where a mode in abright coherent state is overlapped with a mode in a squeezed vacuum state

21

as shown in Figs (1) and (3) In both setups the squeezed vacuum field caneasily be blocked which allows us to compare the measurement results ona bright coherent state with and without the interference with the squeezedvacuum state Without squeezing the photo-electric detectors measure alarge number of photon events with a large quantization noise (shot noise)The large noise reflects the fact that all photon events were independent fromeach other as shown in Fig 2 (bi) With squeezing the photo-electric de-tectors again measure a large number of photon events with an expectationvalue that is even slightly higher but nevertheless the quantization noise ofall detected photons is significantly reduced Fig 2 (bii)Based on the discussion of EPR entanglement in Subsec 24 the photo-electric detection of the output light of a squeezing-enhanced laser inter-ferometer (with αlowastα1) produces the reality of photons This way we cankeep the lsquowave picturersquo in which no photons exist when light travels alongthe interferometer arms and when it interferes at the beam splitter Whenthe energy of the beam is elevating electrons to the conductance band ofthe photo-diodersquos semi-conductor n photon events simultaneously appearwithin the measuring interval with probability P (n) What conclusion hasto be drawn if the probabilities resemble a sub-poissonian statistic ndash Theoccurrence of photon events is still truly random but in this case not forindividual photons The occurrence of photons is correlated in such a waythat the probability of detecting an additional photon in the same time in-terval reduces the larger the number of already detected photons is Whatfollows from the discussion of EPR entanglement for a photon counting ex-periment with pure squeezed vacuum and ideal photon counters Here theprobabilistic detection of one photon entails the detection of a second onewith certainty With some smaller probability a third photon is detectedwhich entails the detection of a fourth photon with certainty and so on

If a photon of a mode that was not interrogated by the environment beforeis absorbed its reality is created in this very moment If the photon belongsto a squeezed state this process instantaneously influences the probability ofother photons becoming reality

Of course a more general statement can be made based on the insightthat interaction with the environment creates the reality of any kind ofquanta including electrons atoms and molecules

22

3 Theoretical description of squeezed states

31 The quadrature amplitude operators

Consider a single mode of light at optical frequency ω Its Hamiltonoperator reads

Hω = ~ω(n+

1

2

)= ~ω

(adaggerωaω +

1

2

)= ~ω

(X2ω + Y 2

ω

) (7)

where n is the photon number operator and aω and adaggerω are the annihilationand creation operators which obey the commutation rule

[aω a

daggerω

]= 1 The

operator aω has a complex-valued dimensionless eigenvalue spectrum andcorresponds to the complex amplitude αω in classical optics Xω and Yω arethe hermitian amplitude and phase quadrature operators The eigenvaluesof the quadrature operators are also dimensionless and proportional to theelectric fields at the oscillationrsquos antinode and at the oscillationrsquos node Inthe above equation they are defined such that their variances are ∆2Xω =∆2Yω = 14 if the oscillator is in its ground state ie if 〈n〉 = 0

Although Eq (7) simply describes the energy of an harmonic oscillator itis the essence of quantum theory since it mathematically describes the wave-particle dualism Whereas the eigenvalues of n have a discrete spectrum theeigenvalues of Xω and Yω have a continuous spectrum In classical opticsthe phase quadrature is zero In quantum optics its expectation value is alsozero but its uncertainty contributes to the overall energy

Eq (7) describes a cavity mode as well as a section that is cut from apropagating quasi-monochromatic light beam The latter example is of highrelevance in actual experiments By setting the sectionrsquos time window ie themeasuring time interval the time-frequency (lsquomodulationrsquo) mode is defined

The quadrature operators introduced in Eq (7) and displayed in Fig 7 donot correspond to lsquoXrsquo and lsquoY rsquo that are of relevance in laser interferometry andin optical communication and which were already discussed in Subsec 22and 23 The optical frequency of visible and near-infrared light is far too highto be transferred to an oscillation of photoelectric voltage Quite general alaser interferometer targets signals at audio or radio band frequencies Ωi ωSuch a measurement is achieved as stated before by decomposing the photo-electric voltage from the photo diode at the interferometer output into asingle-sided spectrum (positive frequencies only) of intervals of Ωplusmn∆Ω2

23

(a)

(b)

(c)

(d)

2π 4π

2π 4π

2π 4π

2π 4π

Figure 7 Phase spaces and electric field oscillations of monochromatic lightndash Top Left Monochromatic light in a coherent state is represented by a phasor (whitearrow) including its quantum uncertainty (white dashed circle and fuzzy area) located inthe phase-space spanned by the quadratures Xω and Yω When the phase space rotateswith optical frequency ω2π the projection of the quantum phasor onto a fixed (vertical)axis corresponds to the electric field E(t) as shown on the right side (a) Weakly displacedcoherent state (b) Corresponding amplitude squeezed state The electric field uncertaintyaround the zero average field region is anti-squeezed (c) Vacuum state at the same opticalfrequency (d) Corresponding squeezed vacuum state The meaning of the uncertaintycould be carved out by supplementing them with monochromatic waves all having theoptical frequency ω2π Changing amplitudes then display amplitude quadrature noiseChanging shifts along the time axis model the electric field uncertainty at the expectedzero crossing They are not implemented in the graphics here however since any of thosewaves does not exist due to Heisenbergrsquos uncertainty relation

24

The signals as well as the quantum uncertainties carried by a beam oflight are thus described by a spectrum of pairs of non-commuting quadratureoperators Mathematically every such operator is defined by an integral overthe Fourier components within the bandwidth The spectral weighting of theFourier components is called the lsquowindow functionrsquo By going to sidebandintervals a spectrum of a new type of optical mode is defined which describesthe modulation of the electric field in the respective frequency interval Ω plusmn∆Ω2 In this Review we call it a lsquomodulation modersquo

The quadrature operators that are defined around a modulation frequencyΩ with a bandwidth of ∆Ω are the quadrature amplitude operators thatare relevant in laser interferometry Whenever they are not related to aspecific band we use the short form XΩ∆Ω(t) equiv X and YΩ∆Ω(t) equiv Y cfEqs (1) and (2) These operators can slowly vary with time where thetime dependence is limited by ∆Ω (The time dependence is not due toquantum uncertainty which usually is time independent but for instancedue to the time dependence of the signal eg a passing gravitational wave)Let us consider now a pair of quadrature operators for a particular sidebandΩ plusmn ∆Ω2 The Hamilton operator of the corresponding modulation modeis found by switching to the frame rotating at optical frequency ω Thetransition is done by applying the unitary transformation U = exp(iωadaggerat)generating a new Hamiltonian H = U daggerHωU minus i~UpartU daggerpartt The Hamiltonianof the modulation mode reads

H = ~Ω

(nΩ +

1

2

)= ~Ω

(adaggera+

1

2

)= ~Ω

(X2 + Y 2

) (8)

where nΩ is the (occupation) number operator for the modulation modeand a and adagger its annihilation and creation operators The commutation rule[a adagger

]= 1 is unchanged X and Y are the amplitude and phase quadra-

ture amplitude operators respectively They correspond to the depth of theamplitude modulation and for weak excitations to the depth of the phasemodulation respectively They are the conventional hermitian field operatorsin experimental quantum optics Note that modulation modes at angularfrequency Ω can be described by a superposition of three optical frequenciesa carrier at ω an upper sideband at ω + Ω and a lower sideband at ω minus ΩThe quantum mechanical description of modulation states in connection tooptical carrier and upper and lower sidebands is known as the lsquoTwo-PhotonFormalismrsquo [Caves and Schumaker (1985) Schumaker and Caves (1985)]

The quadrature amplitude operators in Eq (8) are again defined such

25

that the variances of the uncertainty of a modulation field in its ground stateor in a coherent state are

∆2Xvac = ∆2Yvac = 14 (9)

Generally quadrature operators X and Y as defined in Eqs (7) and (8) arethe real and imaginary parts of the annihilation operator

a = X + iY hArr adagger = X minus iY (10)

hArr X =1

2

(a+ adagger

) Y =

1

2i

(aminus adagger

) (11)

They satisfy the commutation relation[X Y

]=i

2 (12)

and their variances are limited by a Heisenberg uncertainty relation of thefollowing form

∆2X∆2Y ge 1

16 (13)

A quantum state is called a ldquosqueezed staterdquo [Bachor and Ralph (2004)]if ∆2Xϑ lt 14 for an arbitrary field quadrature Xϑ = X cosϑ + Y sinϑ seeEq (3) The angle of the lowest variance below 14 is called the squeeze angleθ The largest factor by which the variance is below 14 is called the squeezefactor often given on a decibel (dB) scale using the following transformation

minus 10 middot log10

(∆2Xθ

∆2Xvac

) (14)

The squeeze factor can also be described by the squeeze parameter r

eminus2r =∆2Xθ

∆2Xvac

(15)

When a squeezed state experiences optical loss it remains squeezed butthe squeeze factor is reduced Also the statersquos purity is reduced ie theproduct of the quadrature uncertainties increases above the minimum valueOptical loss corresponds to mixing the state with the vacuum state Let∆2Xϑ be the variance of a quadrature amplitude ∆2Xvac the variance of the

26

(quadrature angle independent) ground state uncertainty and (1 minus η2) therelative energy loss Then the resulting quadrature variance reads

∆2Xprime

ϑ = η2∆2Xϑ + (1minus η2)∆2Xvac (16)

To maximize the benefit from squeezed states in applications stronglysqueezed states need to be generated and optical loss minimized Opticalloss occurs due to absorption and scattering in the optical components inthe path of the squeezed beam including the squeezing resonator itself anddue to non-perfect matching to the interferometer mode non-perfect inter-ference contrast of the interferometer and non-perfect quantum efficiency ofthe photo diodes The sum of all losses including those outside the inter-ferometer need to be less then 10 to allow a nonclassical quantum noisesuppression of a factor of 10 in power ie 10 dB

32 Phase space representations of squeezed states

The Wigner function ndash The properties of squeezed states are nicely dis-played by the Wigner function W (X Y ) [Wigner (1932)] An example interms of a squeezed vacuum state is shown in Fig 8 It is a quasi-probabilitydistribution which contains the statersquos full information including its quan-tum statistic There are two ways how a Wigner function provides a sufficientcriterion for nonclassicality First by containing negative values second byfeatures that have a smaller (squeezed) width compared with the Wignerfunction of the ground state Integrating the Wigner function over Y pro-vides the probability density of measurement results ie of the eigenvaluesof the observable X and vice versa

infinintminusinfin

W (X Y )dY = p(X)

infinintminusinfin

W (X Y )dX = p(Y ) (17)

where p(X) and p(Y ) are the observed probability distributions also exem-plarily shown in Fig 8

The ground state coherent states as well as (quadrature) squeezed stateshave quadrature eigenvalue probability densities that are Gaussian TheirWigner functions are also Gaussian and thus entirely positive Wigner func-tions of other nonclassical states for instance Fock states exhibit negativevalues For this reason the Wigner function is called a quasi -probabilityfunction

27

X Y XY

Figure 8 Wigner function and its projections ndash Displayed is the full information ofa squeezed vacuum state The quasi probability density of the Wigner function (centre)is plotted along a third dimension and is color-coded Also shown are the actual mea-surement results from which the Wigner function is reconstructed They are representedby the squeezed and anti-squeezed Gaussian projections onto the X and Y axes Theirprobability densities p(X) and p(Y ) are given on the vertical axes The Gaussian measure-ment statistic of the first clearly shows squeezing compared to the ground state statistic(dashed) The squeeze factor is 116 dB and the squeeze parameter r = 1335 [Mehmetet al (2010)]

Fig 9 shows the Wigner functions for (a) a coherent state (b) the ground(vacuum) state (c) a displaced squeezed state and (d) a squeezed vacuumstate All Wigner functions describe a modulation of the carrier light atsideband frequency Ω integrated over the frequency interval ∆Ω The car-rier light is not part of these Wigner functions The displacement in (a)represents a classical amplitude modulation (b) corresponds to the absenceof any photons with a frequency offset of plusmnΩ from the local oscillator field(c) and (d) represent states whose amplitude modulation depth is more pre-cisely defined than that of the ground state Fig 10 shows Wigner functionspectrum for a broadband squeezed vacuum field Every Wigner functiondescribes the modulation field at some modulation frequency Ωi integrated

28

X

Y

X

(a) (b)

X X

(c) (d)

Y

Y Y

Figure 9 Simplified representation of Wigner functions ndash The darker the shadedareas the larger is the phase-space quasi-probability Shown are four different (time-independent) states of a modulation mode at frequency Ω for a specific resolution bandwidth ∆Ω Panel (a) represents a coherent state the displacement (α) corresponds to aclassical amplitude modulation Panel (b) represents the ground (vacuum) state (c) adisplaced squeezed state and (d) a squeezed vacuum state both with squeeze angle θ = 0The latter is in analogy to Fig 8 Again the light field that carries the modulation is notpart of the pictures

over the resolution bandwidth (RBW) of ∆Ω

The Glauber-Sudarshan P -function ndash The P -function [Glauber (1963)Sudarshan (1963)] is calculated by de-convoluting the Wigner function fromthe ground state uncertainty [Gerry and Knight (2005)] For displaced vac-uum states (coherent states) the P -function corresponds to a displaced δ-function The mathematical expression of the P -function of a squeezed statecontains infinitely high orders of derivatives of the δ-function [Vogel andWelsch (2006)] Such a function contains negativities but cannot be dis-played It is possible however to define a phase-space quasi probabilityfunction for squeezed states that can be displayed and that does show neg-

29

0

Ω1

Ω2

Ω

Single-sided spectrum

Y

X

Figure 10 Hint of a Wigner function spectrum ndash A single-sided spectrum (positivefrequencies only) with respect to the carrier field can be used to visualize a broadbandsqueezed field Shown are two examples displaying a squeezed vacuum state at Ω1 anda displaced amplitude squeezed state at Ω2 The individual Wigner functions cover theresolution bandwidth ∆Ω gt 0 (not shown) In general the squeezing strength as well asthe squeeze angle and the displacement are a function of sideband frequency

ativities as a sufficient and necessary condition for certifying the squeezingeffect This lsquononclassicality functionrsquo is calculated by de-convoluting theWigner function from an uncertainty distribution that is steeper than theGaussian distribution A pronounced negativity of a squeezed vacuum stateof up to 69 standard deviations was found [Kiesel et al (2011)]

The double-sided phasor picture ndash This phasor picture links quantumstates of modulations with the quantum states of the contributing opticalfields [Bachor and Ralph (2004)] and is mathematically described by thetwo-photon-formalism [Caves and Schumaker (1985) Schumaker and Caves(1985)] Generally a weak amplitude or phase modulation at frequency Ω ofa carrier field at optical frequency ω can be understood as the carrierrsquos beatwith two optical frequencies at ωplusmnΩ The double-sided phasor picture is ableto display a spectrum of different and independent modulation frequenciesin the rotating frame of the carrier field The carrier light field is time-independent but the upper and lower sidebands are not They rotate withplusmnΩi(2π) respectively around the frequency axis

30

ω0

Yω

Xω

Upper sideband

Lower sideband

ω0+Ω1

ω0+Ω2

ω0-Ω2

ω0-Ω1

ω

Upper sideband

Lower sideband

Yω

Xω

ω0+Ω1

ω0+Ω2

ω0-Ω2

ω0-Ω1

ω

ω0

Figure 11 Double-sided phasor pictures ndash Phase spaces at optical frequency ω0 plusmnΩirotate around the frequency axis with sideband frequency plusmnΩi Its sign determines thedirection of rotation with respect to the local oscillator in the rotating frame at ω0 Apair of phase spaces need to be superposed to provide a description of a modulation fieldat |Ωi| Top Amplitude quadrature squeezed field with a coherent displacement at |Ω2|The displacement corresponds to a classical amplitude modulation The uncertaintiesof all optical frequencies are circular and larger than that of the ground state (dashed)Quantum correlations are indicated by additional symbols More details are given in themain text Bottom Corresponding spectrum of (displaced) vacuum states which do nothave any quantum correlations

31

Fig 11 shows such a double-sided phase space picture where the carrierrsquosmodulation at Ω1 is in a squeezed vacuum state and where the modulationat Ω2 is in a displaced squeezed state The picture shows how a classicalamplitude modulation as well as the quantum statistic of a modulation fieldis decomposed into contributions from upper and lower sidebands For asqueezed modulation field the upper and lower sidebands show no squeezedbut circular thermally excited quantum uncertainties The uncertainties of apair of sidebands however show correlations as well as anti-correlations InFig 11 these (anti-) correlations are marked with times and + for the modulationfrequency Ω1 and with N and bull for the modulation frequency Ω2

33 Covariance matrix representation of (single-party) squeezed states

Since squeezed states have a Gaussian quantum statistic four numbersare sufficient for their full description These numbers are the second mo-ment of the quadrature amplitude showing the strongest squeezing and thesecond moment of its orthogonal quadrature amplitude as well as their firstmoments describing the displacement These four numbers are sufficientto calculate the Wigner function shown in Fig 8 In general the quadra-ture of strongest squeezing is not perfectly aligned with one of the axesof the measurementrsquos coordinate system The so-called covariance matrix(VXXVXY VY XVY Y ) [Simon et al (1994)] accounts for phase space rotationsand enables the calculation of how these states evolve within an interfero-metric arrangement Their components are normalized to the vacuum noisevariance ∆2Xvac = ∆2Yvac and read

VXY =

langXY + Y X

rangminus 2

langXranglang

Yrang

2∆2Xvac

(18)

The following examples represent the ground state a pure 10 dB am-plitude quadrature squeezed state and a pure 10 dB squeezed state with asqueeze angle of 45

Vvac =

(1 00 1

) V0

01 =

(01 00 10

) V45

01 =

(505 495495 505

) (19)

with V4501 =RT

45V001R45 where Rα = (cosα minussinα sinα cosα) is the rota-

tion matrix

32

34 Phase space representation of two-mode (bi-partite) squeezed states

XA

YA

XB

YB

Figure 12 Bi-partite squeezed vacuum state ndash Shown is a Wigner-function-basedphase space representation in close analogy to the single party version in Fig 9 (d) Thepicture describes a single modulation at frequency Ω with bandwidth ∆Ω Measurementsat party A and B reveal local Wigner functions that correspond to thermal states sincethe uncertainties (indicated by the color and by the large circles) are symmetric and largerthan that of the ground state (indicated by the small dashed circles) The uncertaintieshowever show correlations and anti-correlations here indicated by + and times respectivelyThe strength of these (anti-)correlations are indicated by ellipses Bi-partite squeezingie entanglement is present if the short axes of the ellipses are shorter than the diameterof the ground state uncertainty The picture in fact represents Einstein-Podolsky-Rosenentanglement [Einstein et al (1935)] From a measurement of XA or YA the correspondingmeasurement result at party B can always be inferred with an uncertainty that is smallerthan the ground state uncertainty

A bi-partite state enables a measurement on subsystem A and simulta-neous a measurement on subsystem B For a large number of simultaneousensemble measurements of the same quadrature amplitude Xϑ the followingtwo joint quadrature variance can be calculated

∆2(XAϑ plusmn XB

ϑ ) (20)

A state that is symmetrically shared between two parties (A and B) is calleda two-mode squeezed state if the variances of joint quadrature measurementsfulfill the following inequality [Duan et al (2000)] ie

∆2(XA minus XB)

∆2(XAvac minus XB

vac)+

∆2(Y A + Y B)

∆2(Y Avac + Y B

vac)lt 2 (21)

33

with ∆2(XAvac plusmn XB

vac) = ∆2(Y Avac plusmn Y B

vac) = 2∆2Xvac A lsquotwo-mode squeezedstatersquo reveals entanglement in the second moments of the measurement statis-tics It is thus a lsquobi-partite Gaussian entangled statersquo

Fig 12 displays a (pure) bi-partite squeezed vacuum state distributedbetween A and B The state shows full symmetry regarding its subsystemsat the two sites The large circles and the colored area represent Wignerfunctions of the subsystems Measurements of the quadrature amplitudesXA YA XB and YB show identical variances and the correlations and anti-correlations have identical strength since ∆2(XA minus XB) = ∆2(YA + YB) lt12 for our normalization of quadrature amplitudes having a ground statevariance of 14

Generally a symmetric bi-partite squeezed state fulfills another quantita-tive (Gaussian) entanglement criterion if less than 50 of the vacuum state issymmetrically mixed into the initially pure state Bi-partite squeezed statesare always entangled but in this case they are even Einstein-Podolsky-Rosen(EPR) entangled [Reid (1989)] allowing the demonstration of the quantumsteering effect [Einstein et al (1935) Schrodinger (1935) Reid (1989) Cav-alcanti et al (2009)] The first such experiment was performed by Ou etal [Ou et al (1992)] using type II parametric down-conversion (PDC) Laterexperiments produced bi-partite squeezed vacuum states by overlapping twosqueezed vacuum states each produced with type I PDC on a balanced beamsplitter and used the entangled output for the demonstration of quantumteleportation [Furusawa et al (1998) Bowen et al (2003ca)] The criterionin Eq (21) and the EPR criterion from [Reid (1989)] was experimentallycompared in Ref [Bowen et al (2003b)] The steering effect in asymmet-ric bi-partite squeezed states were recently experimentally characterized inRef [Handchen et al (2012)]

Fig 12 shows features similar to those in the top part of Fig 11 Thisis not a coincidence and shows that a bi-partite squeezed state can also begenerated by spatially splitting the upper and lower sideband of a (single-party) squeezed state This was first experimentally demonstrated by thegroup of E Polzik [Schori et al (2002)] and later used for EPR multiplex-ing of a single longitudinal mode of a squeezing resonator [Hage et al (2010)]

35 Covariance matrix representation of bi-partite squeezed states

Also the full information of bi-partite states including the entanglementcan be cast by the covariance matrix [Simon et al (1994)] which can be used

34

to calculate the propagation of these states in laser interferometers Againall variances are normalized to the vacuum noise variance in full analogy toEq (18) The generic bi-partite covariance matrix has dimension 4times4 andreads

Vbp =

VXAXA

VXAYA VXAXBVXAYB

VYAXAVYAYA VYAXB

VYAYBVXBXA

VXBYA VXBXBVXBYB

VYBXAVYBYA VYBXB

VYBYB

with

VXAYB =

langXAYB + YBXA

rangminus 2

langXA

ranglangYB

rang2∆2Xvac

ϑ

(22)

Due to the symmetry in Eq (22) the 4times4 covariance matrix is fully spec-ified by just ten independent coefficients If the phase spaces at A and Bare aligned along the strongest correlations and anti-correlations the ma-trix components referring to different quadrature amplitudes eg VXAYA arezero Such entangled states can be produced by overlapping two squeezedfields with a squeeze angle difference of 90 on a balanced beam splitter

A symmetric bi-partite squeezed vacuum state which is also called anlsquoS-classrsquo [DiGuglielmo et al (2007)] bi-partite squeezed vacuum state shows(anti-)correlations in two joint quadratures as defined in Eq (21) For a puresuch state of 10 dB squeezing the covariance matrix reads

Vbp10|10 =

505 0 495 0

0 505 0 minus495495 0 505 0

0 minus495 0 505

The following covariance matrix describes a so-called lsquoV-classrsquo 10 dB bi-partite squeezed vacuum state Here only one joint quadrature shows 10 dBsqueezing whereas the orthogonal joint quadrature shows vacuum noise Thestate is obtained by overlapping one 10 dB squeezed state with a vacuumstate on a balanced beam splitter

35

Vbp0|10 =

055 0 045 0

0 55 0 minus45045 0 055 0

0 minus45 0 55

The first measurement of all elements of such a covariance matrix was achievedin [DiGuglielmo et al (2007)]

36 Photon numbers of squeezed states

In contrast to the ground state squeezed vacuum states do have photonexcitations As said earlier quantum theory links the wave and the particlepictures Indeed the squeeze factor of a modulation mode is directly con-nected to a certain photon number excitation Squeezed states of light areproduced via spontaneous photon pair generation eg by parametric down-conversion The following operator S is called the lsquosqueeze operatorrsquo [Gerryand Knight (2005)] It creates and annihilates photon pairs

|r θ〉 = S(r θ) |0〉 (23)

where |r θ〉 is a squeezed vacuum state with squeeze parameter r and squeezeangle θ and |0〉 is the vacuum state The definition of the squeeze operatoris

S(r θ) = exp

[1

2

(reminusiθa2 minus reiθadagger 2

)] (24)

The following shows that this definition indeed results in a state with squeezedquadrature amplitude variances Lets set θ = 0

〈0| Sdagger(r 0) X S(r 0) |0〉 =1

2〈0| Sdagger(r 0)

(a+ adagger

)S(r 0) |0〉 (25)

〈0| Sdagger(r 0) Y S(r 0) |0〉 =i

2〈0| Sdagger(r 0)

(aminus adagger

)S(r 0) |0〉 (26)

Using the Baker-Hausdorff formula we get

Sdagger(r θ) a S(r θ) = a cosh r minus adaggereiθsinh r (27)

Sdagger(r θ) adagger S(r θ) = adaggercosh r minus aeminusiθsinh r (28)

36

Since 〈0| X |0〉 = 〈0| Y |0〉 = 0 also Eqs (25) and (26) are zero To finallycalculate the variances we need

〈0| Sdagger(r 0) X2 S(r 0) |0〉 =1

4〈0| Sdagger(r 0)

(a2 + adaggera+ aadagger + adagger2

)S(r 0) |0〉

〈0| Sdagger(r 0) Y 2 S(r 0) |0〉 = minus1

4〈0| Sdagger(r 0)

(a2 minus adaggeraminus aadagger + adagger2

)S(r 0) |0〉

Given that SSdagger is the identity and using again Eqs (27) and (28) we get theexpected variances

∆2X =1

4

[cosh2 r minus 2cosh r sinh r + sinh2 r

]=

1

4eminus2r

∆2Y =1

4

[cosh2 r + 2cosh r sinh r + sinh2 r

]=

1

4e2r

Since the squeeze operator can only create and annihilate photon pairsa squeezed vacuum state without photon loss must correspond to an evennumber of photons But not only photon loss also a coherent displacementleads to flattening out the odd-even oscillations The probability of detectingN photons in a pure displaced squeezed state are derived for instance in[Gerry and Knight (2005)] and read

P (N) = |〈N |α r θ〉|2 =(05 tanh r)N

N cosh rexp

[minus|α|2 minus 1

2(αlowast2eiθ + α2eminusiθ)tanh r

]times∣∣∣HN

[(αcosh r + αlowasteiθsinh r)

radiceiθsinh(2r)

]∣∣∣2 (29)

where HN is the N th Hermite polynomialFig 13 shows the photon number distributions for 5 different pure squeezed

states according to Eq (29) Panels (a) to (c) show squeezed vacuum stateswith 43 dB 86 dB and 172 dB of squeezing Panel (d) shows the moregeneral case of a squeezed state with a coherent displacement α 6= 0 Dueto θ = 0 the state is amplitude quadrature squeezed Panel (e) refers to thecorresponding phase quadrature squeezed state For comparison panel (f)shows the photon number distribution of the coherent state with the samedisplacement

The panels in Fig 13 represent the diagonal elements of the statersquos den-sity matrix in number basis Only the latter also contains the coherences

37

002

004

006

000 000

002

004

006

008

403020100 Photon number n

403020100Photon number n

Pro

babi

lity

403020100000

005

010

015

100 2 4 6 8100 2 4 6 800

04

06

02

100 2 4 6 800

03

02

01Pro

babi

lity

Pro

babi

lity

08

00

04

06

02

08

(a) (b)

(c) (d)

(e) (f)

Figure 13 Photon number distributions ndash All panels represent pure states (a) 43 dBsqueezed vacuum state (r = 05 α = 0) (b) 86 dB squeezed vacuum state (r = 1 α = 0)(c) 172 dB squeezed vacuum state (r = 2 α = 0) (d) displaced 86 dB squeezed state(r = 1 θ = 0 α = 4) (e) displaced 86 dB squeezed state (r = 1 θ = π2 α = 4) (f)Coherent state (r = 0 α = 4) The average photon numbers are about 027 14 131174 174 and exactly 16 see Eq (30)

between photon numbers [Gerry and Knight (2005)] Figures as shown heregenerally do not give full descriptions of the states

38

A squeezed vacuum state (r 6= 0) always has a non-zero photon numberand can not be the ground state The average photon number of a puresqueezed vacuum state can be calculated using Eq (8) With the maximallysqueezed quadrature variance ∆2Xsqz the average photon number is given by

n = 〈n〉|α=0θr〉 = ∆2Xsqz +(∆2Xsqz)

minus1

16minus 1

2=eminus2r

4+e2r

4minus 1

2 (30)

with the vacuum noise variance normalized to one quarter A coherent dis-placement further adds |α|2 photons on average

4 Squeezed-light generation

41 Overview

Squeezed light was first produced in 1985 by Slusher et al using four-wave-mixing in sodium atoms in an optical cavity [Slusher et al (1985)]Shortly after squeezed light also was generated by four-wave-mixing in an op-tical fibre [Shelby et al (1986)] and by degenerate parametric down-conversion(PDC) in a 2nd-order nonlinear crystal placed in an optical cavity [Wu et al(1986)] The pumped cavity was operated below its oscillation thresholdie the parametric gain did not fully compensate the round trip losses whichis also called lsquocavity-enhanced optical-parametric amplification (OPA)rsquo

The early day experiments achieved squeeze factors of a few percent up toabout 3 dB Today squeeze factors of more than 10 dB are directly observedin several experiments [Vahlbruch et al (2008) Eberle et al (2010) Stefszkyet al (2012) Vahlbruch et al (2016)] All of them are based on cavity-enhanced OPA (below threshold) The parametrically amplified mode isdegenerate ie signal and idler modes are identical In particular the down-conversion process is of lsquotype Irsquo which means that the amplified mode has awell-defined polarization Squeezed states can also be generated above oscil-lation threshold In Refs [Villar et al (2006) Jing et al (2006)] bi-partitesqueezing was generated with above-threshold PDC Both experiments usedtype II PDC which provides orthogonally polarized signal and idler fieldsType II PDC below threshold was also used to generate squeezed and bi-partite squeezed fields [Grangier et al (1987) Ou et al (1992)] All these

39

experiments were performed in the continuous-wave regime which is also thefocus of this Review Squeezed states of modulations of trains of laser pulseshowever have been also generated since the 1980s using either PDC or theoptical Kerr effect [Slusher et al (1987) Bergman and Haus (1991) Our-joumtsev et al (2006) Dong et al (2008)] For an overview of the develop-ments in squeezed-light generation in the continuous-wave as well as pulsedregime see Ref [Bachor and Ralph (2004)] Squeezed-light generation inopto-mechanical setups [Aspelmeyer et al (2014)] which use the intensitydependent phase shift from radiation pressure was discussed in Refs [Paceet al (1993) Rehbein et al (2005) Corbitt et al (2006)] and recently ex-perimentally achieved by several groups [Brooks et al (2012) Safavi-Naeiniet al (2013) Purdy et al (2013)]

42 Degenerate type I optical-parametric amplification (OPA)

This section provides a graphical description of how degenerate type IOPAPDC turns a vacuum state into a squeezed vacuum state and a coher-ent state into a displaced squeezed state The process requires a bright pumpfield and a 2nd-order nonlinear crystal For simplicity we set all nonlinearitiesabove 2nd-order to zero

Let us consider a short segment of the second-order nonlinear crystalpumped with light of optical frequency 2ν All other modes that enter thecrystal shall not contain any photons ie are in their vacuum states Ofthese the only mode of interest is that at optical frequency ν which spatiallyoverlaps with the pump mode Fig 14 shows the total electric field of theoptical input E in and the 2nd-order nonlinear dielectric polarisation of thecrystal P The latter is proportional to the total electric field of the outputEout The pump field at 2ν periodically drives the vacuum field at ν betweenregions of low and high polarisation This process transforms the vacuumstate into a squeezed vacuum state in the output [Bauchrowitz et al (2013)]The output further contains the hardly depleted pump field and frequencydoubled parts of the pump field at 4ν It is again emphasized that Fig 14displays OPA in a small segment of the crystal In reality the nonlinear effectaccumulates over the crystal length or even over several passages since thecrystal is usually put into an optical resonator A noticeable effect is achievedif all infinitesimal contributions constructively interfere This is achieved incase of phase matching ie if the wave fronts of the modes at 2ν and νpropagate with the same speed and thus do not run out of phase Note that

40

t

t

Squeezing

Figure 14 Optical parametric amplification of a vacuum state ndash The upper leftcorner shows the crystal polarization P(E) = ε0

(χ(1)E + χ(2)E2

) ie the separation of

charge carriers by the electric component of an optical field E The graph illustrates howan input quantum field (from below) is projected into an output quantum field (towardsthe right) In the example shown here the input field is composed of a classical pumpfield E in

2ν at frequency 2ν and zero-point fluctuations E inν of a field at frequency ν cf

Fig 7(c) The superposition E in of these two fields is transferred into a time-dependentdielectric polarization that is the source of (and thus directly proportional to) the elec-tric component of the output field Eout The quantum uncertainty of the output fieldshows a phase-dependent (parametric) amplification at frequency 2ν Spectral decompo-sition of the output field Eout reveals coherent amplitudes at frequencies 2ν and 4ν and asqueezed vacuum state Eout

sqzν cf to Fig 7(d) The concept of this figure was published inRef [Bauchrowitz et al (2013)]

41

t

t Amplitudequadraturesqueezing

Figure 15 Optical parametric amplification of a coherent state ndash The pictureshows how a displaced vacuum state is transformed into a displaced squeezed state Thepumprsquos electric field is maximal when the input field at fundamental frequency ν shows azero crossing The phase relation described results in an output state that is amplitudequadrature squeezed If the pump field at the input in phase was shifted by half of itswavelength the squeezed output field were a phase quadrature squeezed The squeezinggeneration displayed here corresponds to the transition from Fig 7(a) to Fig 7(b) but withan additional reduction of the displacement The concept of this figure was published inRef [Bauchrowitz et al (2013)]

in actual squeezing experiments the 4ν component is usually suppressed byphase miss-matching

Fig 15 shows the same process but now for an input field at frequencyν in a coherent state In this case the relative phase between the two inputstates ∆ϕ = ϕ2ν minus 2ϕν is relevant In Fig 15 the relative phase is set suchthat the expectation value of the field at frequency ν is zero when the pump

42

(a) (b)

(c) (d)

X

Y

X

YΔϕ = ndash50deg

X

Y

Δϕ = 0deg

X

Y

Δϕ = 90deg

Figure 16 Phase-space illustration of degenerate OPA ndash The (displaced) dashedcircle in each diagram represents the uncertainty of the initial state at optical frequencyν The (displaced) shaded area represents the state after degenerate optical parametricamplification All quantum uncertainties shown correspond to pure states The boldgreen arrow describes the bright second-harmonic pump field whose uncertainty can beneglected The phase between the 2nd harmonic pump and the initial state (∆ϕ = ϕ2ν minus2ϕν) determines the result of the parametric amplification

field reaches its maximum (∆ϕ = 90) The output at the fundamentalfrequency is then an amplitude squeezed state with a deamplified coherentamplitude

Fig 16 summarizes the squeezing operation on the vacuum state as wellas on displaced vacuum states for different phase relations ∆ϕ between thetwo input fields

43 Cavity-enhanced OPA

Placing the nonlinear crystal inside a cavity can greatly enhance thedown-conversion efficiency but not only that A cavity introduces a thresholdfor the pump power above which the parametric gain is infinite just limited

43

by the finite pump power In this case the vacuum uncertainty of the inputfield at frequency ν is amplified to a bright laser field at frequency ν Thedevice is then called an optical-parametric oscillator (OPO) For the gener-ation of squeezed states however the pump power is usually kept (slightly)below threshold Due to nonzero optical loss there exists a pump powersmaller than the threshold above which the tiny improvement of squeezingis not noticeable anymore Getting the pump power closer to the thresholdcould even reduce the observed squeeze factor if a fluctuating squeeze an-gle projects anti-squeezing into the observed quadrature amplitude [Franzenet al (2006) Suzuki et al (2006) Dwyer et al (2013)] The cavity has an-other important purpose It confines the transverse spatial mode usuallyto TEM00 This mode confinement is crucial for any efficient application ofthe squeezed state in laser interferometry since it allows the suppression ofanti-squeezing from other transversal modes The squeezing process requiresa nonlinear material that should show negligible absorption at both opticalfrequencies involved in particular at the wavelength of the squeezed modeIn Refs [Vahlbruch et al (2008) Mehmet et al (2009)] 10 dB and 116 dBof squeezing were achieved using MgOLiNbO3 The highest squeeze factorstoday are produced in (quasi phase matched) periodically poled KTP [Eberleet al (2010) Mehmet et al (2011) Stefszky et al (2012) Vahlbruch et al(2016)]

The optical cavity that is built around the nonlinear crystal is vital forsqueezed-light generation and it deserves a detailed consideration Gener-ally the mode propagating away from a cavity is the result of interferenceat the cavity coupling mirror One contribution is given by the intra-cavityfield attenuated by the amplitude transmission coefficient t of the outcouplingmirror The second contribution is given by the outside field that is reflectedby the same mirror with amplitude reflectivity r =

radic1minus t2 and spatially

overlapped with the first Also the mode from a squeezing resonator is suchan interference product

The impedance matched resonator

Let us consider first an empty optically stable and loss-less Fabry-Perotresonator built from two identical mirrors each with amplitude reflectivityr = r1 = r2 lt 1 A propagating field be perfectly mode-matched to one ofthe cavity resonances In this setup the resonator shows zero reflection andthe resonator is said to be impedance matched (for all such input fields)

44

|α〉

r1α

r12

= +

|α〉 r22 = r1

2

minusr1α equiv t1 αcav = 1minus r12 minusα sdot r1

1minus r12

0

XY

Figure 17 Empty impedance-matched resonator ndash Mode-matched and resonant lightthat enters the cavity from the left here displayed by a quantum phasor for a coherentstate is fully transmitted including its quantum uncertainty The back-reflected statedestructively interferes with itself for all frequencies well within the cavity linewidth The180 phase shift of the transmitted cavity field amplitude minusr1α is a necessary conditionin order to fulfill energy conservation on cavity resonance Impedance matching is forinstance achieved for a resonant cavity with matched mirror reflectivities (r2

2 = r21) and

zero optical loss The complex amplitude of the field traveling towards left inside the cavityis readily deduced from the figure and reads αcav = minusαr1(1minus r2

1)minus12 (highlighted by thedashed circle) In the displayed setup zero field uncertainties are reflected however alsothe vacuum state that enters the cavity from the right is fully transmitted (not shown)

r1α

XY

r12 r2

2 =1

= + |α〉

minus 1+ r1( )α equiv t1 αcav = 1minus r12 minus 1+ r1( )α

1minus r12

|α〉

Figure 18 Empty maximally overcoupled resonator ndash Maximal overcoupling isachieved for a resonant cavity with a perfect end-mirror reflectivity (r2

2 = 1) and zerooptical loss For a given input-mirror reflectivity r2

1 the intracavity light power is maximalMode-matched and resonating light entering the resonator from the left is fully reflectedThe complex amplitude of the field traveling towards left inside the cavity is readilydeduced from energy conservation to αcav = minusα(1 + r1)(1 minus r2

1)minus12 (dashed circle) Inthis setup no uncertainty from the right couples to the left side of the cavity

45

Obviously the interference described in the previous paragraph is fullydestructive The same resonator also shows zero reflection of the input fieldrsquosquantum uncertainty since the interference happens between parts of thesame quantum state The mode propagating away from such a resonatorhowever is not in a nonclassical but in a vacuum state because the vacuumstate that enters the cavity through the opposite site is also fully transmittedThe interference at the coupling mirror of an impedance matched resonatoris displayed in Fig 17

r1α

r12

+

=

OPA PDC

Xcavg r1(1+r1) Xcav

Δ2Xg 0 Δ2Yg

XY

r22 =1

|α〉

Figure 19 Squeezing resonator ndash Shown is the interference at the zero-loss squeezingresonator operated at threshold The lower line represents the perfectly squeezed modepropagating away from the cavity towards the left The parametric gain medium inside thecavity deamplifies the X quadrature of the cavity mode (Xcav) by the factor r1(1 + r1)which is the ratio of the intra-cavity field amplitudes of the two previous figures The Xquadrature of the field that is back-reflected towards the left destructively interferes withitself similar to the situation of the impedance matched cavity in Fig 17 The parametricpower gain per resonator round-trip (G) needs to mimic the effect of an end mirror withreflectivity r2

2 = r21 For this reason the deamplification of Xcav corresponds to a round-

trip deamplification factor of r1 equivradic

1G The round-trip amplification factor for Ycav

then is 1r1 equivradicG which exactly compensates for the outcoupling and thus determines

the parametric oscillation threshold (threshold for bright lasing) The variances of thequantum uncertainties ∆2Ycav and ∆2Y are thus infinite In this setup no field uncertaintyfrom the right couples to the left of the cavity and a perfectly X-quadrature-squeezed fieldoutside the squeezing resonator is produced

46

The perfectly over-coupled single-ended resonator

We now increase the reflectivity of the far mirror lsquo2rsquo to being perfect (r2 = 1)This way the counter-propagating vacuum state can not enter the cavityAgain a propagating field be perfectly mode-matched through mirror lsquo1rsquo toone of the cavity resonances For frequencies well inside the cavity linewidththe situation is displayed in Fig 18 The setup protects the left side of thecavity against vacuum fluctuations entering through mirror lsquo2rsquo but of coursedoes not squeeze quantum noise The intra-cavity built-up factor is too highfor achieving destructive interference below the vacuum uncertainty on theleft side of the resonator

The impedanced-matched single-ended squeezing resonator

Building on the two previous concepts the straight forward approach now isto start from the perfectly over-coupled single-ended resonator and insert anattenuator into the cavity that does not couple the cavity mode to any bathbut still results in a roundtrip efficiency of precisely r1(lt 1) in amplitudeOptical loss is not appropriate since it increases the coupling of the cavitymode to a thermal bath neither would any phase-insensitive atenuator beappropriate It is easy to show that a phase-insensitive attenuator adds ad-ditional uncertainty since otherwise the commutation relation [a adagger] = 1 isviolated The amplification process that matches our requirement is OPA Toachieve infinite squeezing in X on cavity resonance a second-order nonlinearcrystal needs to be put into the cavity and pumped such that the intra-cavityamplitude quadrature is attenuated by the factor (1 + r1)r1 (on cavity reso-nance) with respect to the empty cavity This factor is readily deduced fromFigs 17 and 18 Due to the symmetry in parametric amplification the intra-cavity phase quadrature is then amplified by (1 + r1)r1 and the round-tripgain has a value of 1r1 in amplitude In this situation not only infinitesqueezing but also the (laser) threshold of the resonator is achieved sincethe round-trip gain of the intra-cavity phase quadrature equals its roundtriploss here fully given by the incoupling mirror

The physical descriptions in Figs 17 to 19 are fully consistent with ob-servations in squeezing experiments The consideration above in particularshows that the intra-cavity field shows a finite squeezing strength while theexternal field shows infinite squeezing The strongest intra-cavity squeezefactor possible is (1 + r1)2r2

1 In the high reflectivity limit this factor corre-

47

sponds to 6 dB Averaged over the full cavity mode the squeeze factor of thecavity mode is in this limit even limited to 3 dB [Walls and Milburn (2008)]Higher intra-cavity squeeze factors are possible for lower mirror reflectivities

44 The generation of squeezed light for laser interferometry

With the insights gained in the previous subsection we now turn to ac-tual experiments The application of squeezed states in laser interferometrycertainly requires large squeeze factors (idealy accompanied with the highestpossible purity) to maximize the impact in terms of sensitivity improvementIn cavity-enhanced OPA the highest parametric gain is achieved on cavityresonance ie at zero sideband frequency But this is not the main reasonwhy this Subsection focusses on the generation of squeezed states at lowsideband frequencies The application of squeezed states in a laser inter-ferometer requires that their sideband frequencies cover the devicersquos signalband Ground-based gravitational wave (GW) detectors have a detectionband from about 10 Hz to 10 kHz frequencies which can be considered aslsquolowrsquo compared to typical frequencies in quantum optics experiments

Squeezing at MHz sideband frequencies is easier to observe than at acous-tic frequencies because the latter are often polluted with excess noise fromlight beams that serve as control beams [Bowen et al (2002) McKenzie et al(2004)] and parasitic interferences from back-scattered light [Vahlbruch et al(2007)] Furthermore the observation of squeezing at low sideband frequen-cies requires a more stable setup since larger measuring times are necessaryThe observation of strong squeezing at MHz frequencies however alreadysets an upper limit to the optical loss of the setup At least the same squeezefactor can be observed at lower frequencies

There are two different main topologies for squeezing resonators TheFabry-Perot-type standing-wave resonator consists of a minimum number ofmirror surfaces and has the advantage of being compact and thus robustagainst mechanical vibrations Usually one or even two mirror coatings aredirectly placed on the spherical and polished surfaces of the nonlinear crys-tal itself [Wu et al (1986) Grangier et al (1987) Breitenbach et al (1998)Vahlbruch et al (2008) Eberle et al (2010) Vahlbruch et al (2016)] TheBowtie traveling-wave resonator has the advantage of providing a separa-tely accessible counter propagating mode for cavity length control [Ou et al(1992) Takeno et al (2007)] It shows no direct back-reflection of incoupledlight which helps reducing parasitic interferences [Stefszky et al (2012)]

48

(a)

(b)

(c)

(d)Squeezingresonator

To inter-ferometer

OPA

SHG

BHD

Mode cleaner

LO

Laser

DBS

DBS

Figure 20 Generation of squeezed light ndash (a) Example of a 2nd-order nonlinearcrystal for the squeezed-light generation at 1064 nm Shown is a bi-convex 65 mm long7MgOLiNbO3 crystal whose polished surfaces also carry the mirror coatings of the res-onator The crystal thus realizes a monolithic squeezing resonator as it was used forthe first demonstration of 10 dB squeezing [Vahlbruch et al (2008)] (b) Optical con-figuration of a half-monolithic (hemilithic) standing-wave squeezing resonator Here thecavity length can be adjusted by displacing the coupling mirror The crystal surface insidethe cavity is anti-reflection coated The photograph shows a 10 mm long PPKTP crystalsqueezing resonator as used for the GEO 600 squeezed-light source [Abadie (2011)] (c)Mechanically stable housing of a standing-wave squeezing resonator The crystalrsquos temper-ature is stabilized at its phase matching condition using Peltier elements (d) Schematicfor the squeezed-light generation After spatial filtering of continuous-wave laser lighttwo hemilithic standing-wave resonators are employed The first generates second har-monic pump light (SHG) The second (OPA) generates a squeezed vacuum field at theinitial wavelength The squeezed states are observed by a balanced homodyne detector(BHD) or alternatively sent and mode-matched to the optical mode of an interferometerbeforehand LO local oscillator DBS dichroic beam splitter

49

Fig 20 (a) and (b) show photographs of typical nonlinear crystals used forsqueezed-light generation at near infra-red wavelengths The crystals shownhere form a monolithic standing-wave squeezing resonator (a) or are partof a half-monolithic standing-wave squeezing cavity (c) shows a tempera-ture stabilized and mechanically stable housing of the squeezing resonator(d) shows a schematic of a full setup for the generation of squeezed vacuumstates of light for an application in a laser interferometer The only brightinput required for the squeezing resonator (OPA) is the second-harmonicpump field The resonator mode at fundamental frequency is thus initiallynot excited by photons ie it is in its ground state characterized by vacuumfluctuations due to the zero point energy see Fig 7 (c) [Gerry and Knight(2005)] The pump field spontaneously decays in the degenerate pair of sig-nal and idler fields The combined down-converted field leaving the resonatorexhibits quantum correlations which give rise to a squeezed photon countingnoise when overlapped with a bright coherent local oscillator beam Thedetection is done either in a balanced homodyne detector (BHD) or with asingle photo diode The squeeze factor increases the closer the pump powerof the squeezing resonator gets to the oscillation threshold and the lower theoptical loss on down-converted photon pairs is

441 High squeeze factors ndash minimizing decoherence

Squeezed states of light have significant impact on the sensitivity of laserinterferometers if large squeeze factors can be produced Squeezing of 3 dBimproves the signal-normalized quantum-noise spectral density by a factorof 2 This factor corresponds to doubling the (coherent state) light powercirculating inside the interferometer Squeezing of 10 dB corresponds to aten-fold power increase The experimentally demonstrated squeeze factorswere considerably improved in recent years [Takeno et al (2007) Vahlbruchet al (2008) Polzik (2008) Eberle et al (2010) Stefszky et al (2012)]culminating in a value of as large as 150 dB [Vahlbruch et al (2016)] Thisvalue corresponds to the same reduction of signal-normalized quantum noisethat is achieved by increasing the light power by a factor of 32 (At this pointit is already noted that squeezing the quantum noise can simultaneouslyreduce quantum measurement noise (shot noise) as well as quantum backaction noise (radiation pressure noise) This is not possible with scaling thelight power of coherent states see Subsec 55)

50

Ideally a parametric squeezed-light source can produce an infinite squeez-ing level see Fig 19 fundamentally just limited by the energy provided bythe pump field In practice the limit is set by decoherence mechanismsThe by far most important one is optical loss Optical loss occurs duringsqueezed-light generation its propagation through the interferometric setupincluding imperfect mode matchings and finally the photo-electric detec-tion Also detector dark noise [Schneider et al (1998)] phase noise [Takenoet al (2007)] and excess noise [Bowen et al (2002)] impair the observablesqueezing strength

Optical loss is usually understood as coupling the squeezed mode to a zerotemperature bath ie overlapping it with a vacuum mode For any amountof loss the resulting state is still squeezed But to be able to directly observesay 10 dB of squeezing the total loss on the state needs to be less than 10in this example cf Eq (16) To minimize optical loss the nonlinear crys-tal as well as lenses and beam splitters in the interferometric path need toshow very low absorption and scattering at the wavelength of the squeezedlight PPKTP shows absorption of about 10minus4cm and below at near-infraredwavelengths Low OH content fused silica is a suitable material for all otheroptics Absorptions of less than 10minus6cm were measured [Hild (2007)] Coat-ings on crystal surfaces and on all other optical components should also showlowest optical loss Total loss of the 10minus6 level are available today Superpol-ished surfaces which show roughnesses with less than 1 A root mean square(integrated over spatial scales from approximately 1 micron to 100 microns)and thus very low scattering are necessary to achieve these low numbersMinimizing the total number of optical components is essential From thisperspective a monolithic squeezing resonator as shown in Fig 20 (a) is theoptimum choice The squeezed mode needs to be matched to the mode ofthe laser interferometer or to the mode of the balanced homodyne detectorVisibilities of up to 998 have been achieved [Eberle et al (2010)] whichcorresponds to a loss of about 04 Of great importance also is the quantumefficiency of the photo-diodes used for detecting the squeezed field (togetherwith the interferometric signal) Recently a quantum efficiency of photo-diodes in a squeezing experiment of (995plusmn 05) was measured [Vahlbruchet al (2016)] To minimize photon loss the photo-diodes had no protectionwindow an anti-reflection coating on the semi-conductor material and theremaining reflection was re-focussed with an external mirror

Also the dark-noise spectral density of the detection electronics reducesthe observable squeezing and needs to be as low as possible Similar to optical

51

noise it also provides a contribution to the observed variance The dark noiseof the detection electronics needs to be much lower than the detected photoncounting noise In [Vahlbruch et al (2016)] it was 28 dB below shot noisebut still reduced the observable squeeze factor from 153 dB to 150 dB

Excess noise emerges if the squeezed mode couples to a nonzero tem-perature bath or to a mode whose excitation is strongly fluctuating (Thecoupling process can always be understood as a beam splitter coupling andis physically described by overlapping electric fields Coupling to a zero tem-perature bath leads to Eq (16)) The captured excess noise variance thenneeds to be added to the initial squeezing variance which deteriorates theobserved squeezing stronger than just mixing in the vacuum mode Excessnoise is less likely to occur at MHz frequencies but can be significant ataudio-band sideband frequencies and below and is thus a serious issue ingravitational-wave detectors [Chua et al (2014)] The reason for that isthat acoustically or thermally excited motions of surfaces produce frequencyshifts of back-scattered light mainly at these low frequencies [Vahlbruch et al(2007)]

Phase noise corresponds to stochastic phase fluctuations between thesqueezed field and the local oscillator within the measuring time It cor-responds to mixing the squeezed mode with itself with a fluctuating squeezeangle [Suzuki et al (2006) Franzen et al (2006)] Phase noise in squeez-ing experiments typically is less of an issue than optical loss [Dwyer et al(2013) Oelker et al (2016) Vahlbruch et al (2016)] The setuprsquos phasenoise can be reduced by making the squeezing resonator more compact andthus mechanically more stable against acoustic and thermal fluctuations ofthe environment and by improving the quality of the servo loops for cavitylength and propagation length controls Operating a squeezed-light resonatorin vacuum might also be beneficial The ability to run a high performancesqueezed-light generator in vacuum was demonstrated in [Wade et al (2015)]

442 Squeezing in the gravitational-wave (GW) detection band

High squeeze factors have been first demonstrated at sideband frequen-cies of a few MHz and above where excess noise is generally negligible whenworking with visible or near-infra-red light Today we know that extendingthe squeezing spectrum towards the audio-band and even below is technicallynot always easy but straight forward once a high squeeze factor is achieved

52

at MHz frequencies In most squeezing experiments the main laser light

Figure 21 Photograph of the GEO 600 squeezed-light source ndash The breadboarddimensions are 135 cm times 113 cm The squeezing resonator is high-lighted by the white ar-row and is set up as a standing-wave hemilithic cavity containing a plano-convex PPKTPcrystal of about 10 mm length (see also Fig 20 b) It is pumped with continuous-wave532 nm light that is produced by frequency doubling of light (at angular frequency ω)from a commercial NdYAG laser Two more laser fields at about 1064 nm having fre-quency offsets of more than 10 MHz with respect to ω(2π) serve as optical control fieldsBoth fields are mode-matched and injected into the squeezing resonator together with thesecond-harmonic pump field

at the squeezing wavelength is accompanied by significant noise up to thelaser relaxation oscillation For this reason laser control fields at the opticalcarrier-frequency in the optical path of the squeezed mode need to be avoided[Bowen et al (2002) Schnabel et al (2004) McKenzie et al (2004)] and thesqueezing resonator length and the optical path stabilized by other means[McKenzie et al (2005) Vahlbruch et al (2006)] Furthermore and mostimportantly excess noise due to back-scattering is an issue Back-scattering(also called lsquoparasitic interferencesrsquo) is produced if DC light scatters out ofthe optical path hits a vibrating surface and re-scatters back into the opticalpath [Vahlbruch et al (2007)] Significant back-scattering can be produced

53

in interferometers for the detection of gravitational waves since light powersof several hundreds of kilowatts are used Even back-scattering from the mil-liwatt local oscillator of balanced homodyne detectors is an issue at acousticsideband frequencies and below The recipe for avoiding parasitic interfer-ences turns out to be threefold (i) avoiding scattering by using ultra-cleansuperpolished optics with close to perfect anti-reflex coatings (ii) avoidingback-scattering by carefully blocking all residual (faint) light fields and (iii)reduce the vibrationally and thermally excited motion of all mechanical andoptical parts that could potentially act as a re-scattering surface with re-spect to the optical path [Vahlbruch et al (2007) McKenzie et al (2007)]The insights described above led to the first demonstration of audio-bandsqueezing at frequencies down to 200 Hz [McKenzie et al (2004)] and laterto the first demonstration of squeezing over the full gravitational-wave de-tection band even from as low as 1 Hz [Vahlbruch et al (2007)] Whilea standing-wave squeezing resonator [Ou et al (1992)] can be built in avery compact way that is rather insensitive against mechanical vibrations[Chelkowski et al (2007)] a traveling-wave bow-tie squeezing resonator [Wuet al (1986)] is more tolerant against back-scattered light [Chua et al (2011)]The strongest squeezing in the audio-band of up to 116 dB was reported inRef [Stefszky et al (2012)]

443 The first squeezed-light source for GW detection

The first squeezed-light source for the continuous operation in GW de-tectors had been designed and completed between 2008 and 2010 [Vahlbruch(2008) Vahlbruch et al (2010)] Since then this source has been producingsqueezed vacuum states in a fully phase controlled way using co-propagatingfrequency-shifted bright control beams [Vahlbruch et al (2006)] as an inte-gral part of the GW detector GEO 600 The source is a turn-key device witha fully automated re-lock system [Vahlbruch et al (2010) Khalaidovski et al(2012)] Re-locking is required if the temperature of the environment changessignificantly which drives the actuators outside their dynamic ranges

Up to 9 dB of squeezing over the entire GW detection band was observedusing a balanced homodyne detector (BHD) located close to the squeezingresonator The squeeze factor has been limited by optical loss due to ab-sorption in the PPKTP crystal transmission of the back-surface and thenon-perfect AR-coating of the crystalsrsquos intra-cavity surface The adjustableair gap has been necessary to allow for an easy way to apply length control

54

-12

-8

-4

0

4

8

12

16

20

10 100 1k 10k

Rel

ativ

e no

ise

pow

er [

dB]

Frequency [Hz]

shot noise (a)

squeezed noise (b)

anti-squeezed noise (c)

~ 9dB

Figure 22 Broadband squeezing spectrum ndash Noise power spectra measured on theoutput of the GEO 600 squeezed-light source shown in Fig 21 with a balanced homodynedetector The traces correspond to the spectra of quadrature amplitude variances normal-ized to vacuum noise The resolution bandwidth used increases towards higher frequenciesto reduce the measurement time (a) Shot noise normalized to unity which serves as thereference level (0 dB) (b) Squeezed noise covering the complete detection band of ground-based GW detectors (c) Anti-squeezed noise Peaks at 50 Hz and 100 Hz are the electricmains frequency and its first harmonic The data was published in Ref [Vahlbruch et al(2010)]

Additional optical loss in the path to the balanced homodyne detector mainlyarose due to a Faraday isolator that eliminated parasitic interferences Fi-nally the mode missmatch to the BHD as well as its non-perfect quantumefficiency provided additional loss Inferring the squeeze factor without theBHD detection loss more than 10 dB of squeezing are provided by the sourceSince 2010 it has been used in basically all observational runs of the GEO 600GW detector see Section 6

444 Generation of two-mode (bi-partite) squeezing

lsquoTwo-mode squeezed lightrsquo or lsquobi-partite squeezed lightrsquo is light that allowsfor joint measurements at two locations A and B These joint quadraturemeasurements reveal correlations and anti-correlations with a remaining un-certainty smaller than the ground-state uncertainty which certifies the pres-

55

ence of entanglement cf Subsec 34 Bi-partite squeezed light has been gen-erated by type I and by type II parametric down-conversion In case of type Ithe squeezed fields from two squeezing resonators as described in Subsec 44are overlapped on a balanced beam splitter with a 90 phase shift The twooutput fields together represent the entangled mode [Furusawa et al (1998)Bowen et al (2003c) Eberle et al (2013)] In case of type II signal and idlerfields are non-degenerate regarding polarisation and a single cavity contain-ing an appropriate crystal and a polarising beam splitter are sufficient for theproduction of bi-partite squeezing Also in this case the measurements ofthe quadrature amplitudes of signal and idler fields show large uncertaintiestogether with bi-partite correlations and anti-correlations that are strongerthan the ground state uncertainty of individual subsystems [Ou et al (1992)Villar et al (2006) Jing et al (2006)]

To date the strongest entanglement of bi-partite squeezed light has beenproduced based on type I parametric down-conversion [Eberle et al (2013)]The requirements of producing strong entanglement are identical to those ofproducing strong squeezing outlined above The strength of bi-partite en-tanglement can be given in decibels in full analogy to the squeeze factorPractically the strength of bi-partite squeezing is always somewhat smallerthan that of single party squeezing since it requires an additional mode-matching that results in additional optical loss

45 Conclusions

The first observation of squeezed light was achieved in 1985 [Slusher et al(1985)] Shortly after cavity-enhanced optical parametric amplification forsqueezed-light generation was demonstrated [Wu et al (1986)] which todayenables the observation of up to 15 dB of squeezing [Vahlbruch et al (2016)]Quite generally the maximum squeezing level that is observed does not de-pend on the strength of the optical nonlinearity Squeezing cavities can easilybe operated at their oscillation threshold where they should provide infinitesqueezing if decoherence is zero The main limiting factor is optical lossincluding that of the photo-electric detection

Dedicated experimental research and development towards a squeezed-light source for applications in gravitational-wave detectors can be tracedback to 2002 [McKenzie et al (2002) Bowen et al (2002)] Since then a sur-prising amount of progress has been made culminating in the first squeezed-light source specifically built for the integration into a gravitational-wave

56

detector For the future squeeze factors above 15 dB will certainly be possi-ble by further reducing optical loss This statement is supported by the highdegree of matching between experimental data and a theoretical loss modelas presented in Fig 3 of [Vahlbruch et al (2016)]

5 Quantum noise in laser interferometers

51 Interferometric measurements

The purpose of a laser interferometer is the precise measurement of smallchanges of an optical path length with respect to a reference path Forthis the interferometer transfers the change of the phase difference betweentwo light fields into an amplitude quadrature change of the interferometerrsquosoutput light The latter can easily be detected by a single photo diode Ofgeneral interest are differential changes of the optical path length that aremuch smaller than the laser wavelength ie differential phase changes thatare much smaller than 2π In this case the differential phase change can bedescribed in very good approximation as a differential change of the phasequadrature instead

In order to transfer the phase quadrature signal with minimum loss ahigh interference contrast at the interferometerrsquos beam splitter is essentialAdditionally instrumental noise in terms of unwanted excitations of the out-putrsquos amplitude quadrature needs to be reduced to a minimum Noise arisesdue to power fluctuations of the input laser light back-scattered laser lightinside the interferometer thermally driven displacements of mirror surfacesand in many more ways The important measure of the sensitivity of aninterferometer obviously is its signal-to-noise-ratio The most useful measureis given in terms of the noise spectral density S(f=Ω2π) that is normalizedto the physical unit of the signal S(f) is in fact a lsquonoise-to-signal-ratiorsquoand can be seen as the signal-normalized variance of the photo diode out-put decomposed into spectral components versus sideband frequency f withthe resolution bandwidth of 1 Hz As an example S(100 Hz) = 10minus39 m2Hzmeans that the instrumental noise in the one hertz band around 100 Hz equalsa signal that would be produced if the mirror of one interferometer arm os-cillates with an amplitude of just

radic10 middot 10minus20 m in the very same band Such

small spectral densities are achieved by gravitational-wave detectors [Abbott(2016)]

57

52 Quantum measurement noise and shot noise

The most fundamental noise source in laser interferometers is due to thequantum noise of light which is in fact two-fold [Caves et al (1980)] Firstof all there is lsquoquantum measurement noisersquo which arises in the processof photo-electric detection For coherent states the quantum measurementnoise is the lsquophoton counting noisersquo of mutually independent photons andusually simply called lsquoshot-noisersquo Fig 2 (b i) shows a time series of suchnoise hiding the actual signal The frequency components of the shot noiseare well described by the quantum uncertainty of the output fieldrsquos amplitudequadratures XΩ∆Ω see Subsec 22 (Recall this quantity corresponds to the

differential phase quadrature YΩ∆Ω of the light beams in the interferometerarms) The photon counting noise has a white Fourier spectrum howeverthe lsquoshot noisersquo of an interferometer is usually normalized to the signal whosetransfer function is usually not white for instance due to the presence of armcavities or a signal-recycling cavity

All current and planned gravitational-wave detectors are Michelson-typelaser interferometers with operating points very close to a dark fringe Thelight power in the output port is just a couple of tens of mW which canbe handled by a single photo diode In this configuration the signal-to-shot-noise-ratio is actually maximized which can be shown in three steps [Bachorand Ralph (2004)] For the first step we use plane waves to describe theelectric field in the output port of a Michelson interferometer For perfectinterference contrast at the balanced beam splitter ie for perfect modematching and for defining φ = 0 as the dark port condition we get

Eout(t φ) =1

2E0 sin(ωt+ φ)minus 1

2E0 sin(ωt) (31)

where E0 is the amplitude of the total internal field whose two parts hasaccumulated a differential phase It directly follows for the squared fields

E2out(t φ) =

(sin

φ

2

)2

(E0 cos(ωt+ φ2))2 (32)

We now turn to a light beam with a localized transversal mode that can befocussed onto a photo-electric detector The photo diode has perfect quantumefficiency ie the rate of photo electrons is not only proportional to the rateof output field photons but also has a unity slope efficiency Since the optical

58

frequency is too high to be resolved we consider the averaged light power

P out(φ) =

(sin

φ

2

)2

P (33)

The next step is a formulation of the signal being the derivative of detectedphoton number versus phase Let n be the average value of the photonnumber per measuring time interval Eq (33) can then be rewritten as

nout(φ) =

(sin

φ

2

)2

n (34)

rArr dnout(φ) = n sinφ

2cos

φ

2dφ (35)

The final step is the calculation of the signal-to-shot-noise-ratio Shot noiserefers to coherent states which have a standard deviation of the photonnumber of σ(n) =

radicn

dnout(φ)

σ(nout)=n sinφ

2cosφ

2dφ

radicn sinφ

2

(36)

and find for a signal-to-noise ratio of unity for coherent states and for anon-zero but still small phase difference ∆φCoh 2π

1 =radicn cos

φ

2∆φCoh with φ 6= 0 (37)

In this equation the smallest measurable phase difference is given for φrarr 0

∆φCohmin =

1radicn (38)

This is the well-known shot-noise limit of high-precision phase sensing ∆φCohmin

is the smallest phase shift that can be measured with a signal-to-noise ra-tio of one when using n mutually independent photons per measuring time(those of a coherent state) when the loss of photons is assumed to be zeroThe typical purpose of a laser interferometer is the continuous sensing (moni-toring) of a continuously changing phase An illustrative example is the phasesignal produced by the black hole merger measured by Advanced LIGO onSept 14 in 2015 (Fig 1 in [Abbott (2016)]) The measuring interval should

59

be short to be able to resolve the time-evolution of the signal Generally themeasurement of an arbitrary signal that lasts for a finite time thus needs tobe understood as l subsequent measurement intervals using n photons eachIt can be shown that Eq (38) is also valid for interferometers operated athalf fringe ie when each output port contains the same light power In thiscase photo diodes need to be placed in both output ports and the actualsignal is provided by their difference voltage

Due to its importance the shot-noise limit deserves some remarksThe phase φ in Eq (31) is the phase difference of two mode-matched fieldsand might be accumulated by a single pass along the length L such as ina Mach-Zehnder interferometer or in a double pass such as in a (simple)Michelson interferometer or in four passes as realized in a Michelson inter-ferometer with folded arms [Grote (2005)] The shot-noise limit in Eq (38)and its scaling therefore holds independent of the number of passes Theclaim in Ref [Higgins et al (2007)] that the scaling according to Eq (38) canbe surpassed by multiple passes is not justifiedThe fact that Eq (38) is derived by approaching φ rarr 0 correctly describesthe actual operation point of gravitational-wave detectors which is close tobut not exactly at a dark port In practice a tiny offset from dark port ischosen at which the shot noise is well above the photo diodersquos electronic darknoiseEq (38) solely depends on the number of quanta but not on the lightrsquos wave-length λ Of course the shot-noise limit for the change of an optical pathlength ∆L does depend on the wavelength and ∆φCoh

min needs to be replacedby ∆φCoh

min = 2π∆LCohminλ

Finally an essential result of the shot-noise limit is that the ideal precisemeasurement should use lsquoas much quanta as possible per measuring inter-valrsquo which translates to lsquoas much light power in the interferometer armsas possiblersquo Eq (38) is indeed the one and only reason why gravitational-wave detectors use high power lasers power-recycling and arm resonatorsExtending the measuring time for a given light power can also improve thesensitivity but only if the signal repeats ie is periodic Let us assume thatone period of the signal is resolved by l intervals using n photons each Inthis case repeating the overall measurement k times improves Eq (38) by1radick The fundamental statement of Eq (38) however does not change

since the actual photon number n may then simply incorporate the factor k

60

For a given average photon number the shot-noise limit in Eq (38) canonly be surpassed by using photons that are quantum correlated ie by usingnonclassical states of light How is the shot-noise limit surpassed with thehelp of squeezed states A Michelson interferometer that is operated closeto a dark fringe acts like an almost perfect mirror for both input ports Allthe input light is back-reflected towards the laser source This also accountsfor the quantum uncertainty of the input light The quantum uncertaintythat impinges onto the photo diode thus (mainly) enters the interferometerthrough its (almost) dark port An interferometer that uses displaced co-herent states entering from one port can thus be improved by replacing theordinary vacuum entering the signal output port by a squeezed vacuum stateThis was the proposal by CM Caves in 1981 [Caves (1981)] which is labeledhere with lsquoCSVrsquo If the differential phase quadrature of the interferometer issqueezed Eq (38) then within the limit of large coherent state displacementα sinh2r improves to

∆φCSVmin asymp

eminusrradicn (39)

(The above expression is an approximation since the squeezing operationproduces a small number of photons that are not accounted for here) Ofcourse the mode of the squeezed vacuum needs to be precisely matchedto the mode of the interferometer The first experimental demonstrations ofsqueezed phase measurements used a Mach-Zehnder [Xiao et al (1987)] and apolarization interferometer [Grangier et al (1987)] Fig 2 shows how spatialdegeneracy between an externally generated squeezed mode and the signalmode in a Michelson interferometer is achieved using a polarizing beam split-ter and a Faraday rotator Again the limit in Eq (39) can only be achievedif optical loss is zero Optical loss not only reduces the signal but here alsoreduces the squeeze parameter see Eqs (15) and (16)

Let us consider an example The sensitivity of a laser interferometer thatuses coherent states with an excitation of 1023 photons per second can beimproved by a factor of

radic10 by either adding 09 middot 1024 photons per second

or by adding about just 2 photons per second and bandwidth in hertz thatbelong to the 10 dB squeezed vacuum confer Eq (30) Since the full sig-nal band of ground-based GW detectors covers sideband frequencies up to10 kHz just 2 middot 104 photons per second are necessary At a wavelength ofλ = 1064 nm these values correspond to a power increase by 168 kW and

61

37 fW respectively

The question arises whether a scaling of the sensitivity better than propradic1n is possible It was theoretically shown that in principle the scaling

can indeed considerably be improved yielding the so-called Heisenberg limitor Heisenberg scaling [Bondurant and Shapiro (1984) Yurke et al (1986)Braunstein (1992) Holland and Burnett (1993)]

∆φHLmin prop

1

n (40)

The Heisenberg scaling requires nonclassical states that have a certain num-ber of quanta similar to Fock states ie n = n The theoretically optimalstates describe a superposition of n (N) indistinguishable photons in oneinterferometer arm while having zero (0) photons in the second arm and viceversa and were named lsquoN00Nrsquo-states [Dowling (2008)] A specific propertyof these states is lsquosuper-resolutionrsquo The output ports of the interferometershow an n-times faster oscillation of the interference fringes when changingthe phase between the two interferometer arms Super-resolution correspondsto an n-times improved signal transfer function and was demonstrated forinstance in Refs [Rarity et al (1990) Kuzmich and Mandel (1998) Mitchellet al (2004) Afek et al (2010)] The presence of this nonclassical phe-nomenon however does not prove a sensitivity better than the semi-classicalbound according to Eq (38) Sensitivity is rather related to the signal-to-noise-ratio and needs to take into account all imperfections as well as theprobability of a successful detection of the sensing state [Thomas-Peter et al(2011)] All experiments so far used post-selection on particular measure-ment outcomes and neglected the typically large probability that nothingwas detected

Super-resolution was demonstrated with up to n = 5 [Afek et al (2010)]In addition to the fact that super-resolution does not prove a sensitivity bet-ter than the semi-classical bound photon numbers in state-of-the-art super-resolution experiments are extremely small compared to the photon numberof about 1023 (within a measuring interval of one second) of coherent statesused in Ref [Abbott (2016)] and of about 1022 using coherent states plussqueezed vacuum states used in Ref [Abadie (2011)]( which did prove a sen-sitivity better than the semi-classical bound)

Another interesting and related question is what the smallest phase is

62

that can be estimated in a single measurement again using a given numberof quanta Taking into account that no prior information about the phaseshift exists still a scaling proportional to 1n is possible In the limit of largen however an additional factor of π is required in the nominator of Eq (40)[Sanders and Milburn (1995) Berry and Wiseman (2000)] yielding

∆φHLmin asymp πn (41)

The state that can actually achieve this bound is different from the N00Nstate and was found in [Summy and Pegg (1990) Luis and Perina (1996)Berry and Wiseman (2000)] A N00N state is not the optimum state forphase estimation (via a single measurement) since it only provides one bit ofinformation A recent review on generell aspects on phase measurements isgiven by Ref [Demkowicz-Dobrzanski et al (2015)]

It is important to note that Eqs (38-40) do not consider photon lossExperiments that demonstrated super-resolution and aimed for achievingthe scaling in Eq (40) were conditioned on zero photon loss Let η gt 0 bethe average efficiency of detecting (all) photons Eq (38) then reads

∆φηCohmin =

radic1

η n (42)

Eq (39) turns into

∆φηCSVmin asymp

radicηeminus2r + 1minus η

η n(43)

and Eq (40) turns into [Demkowicz-Dobrzanski et al (2012 2013)]

∆φηmin =

radic1minus ηη n

for 0 lt η lt 1 (44)

For non-zero photon loss most interestingly the ultimate sensitivity of aphase measurement for a given photon number also shows a 1

radicn -scaling

The difference between the CSV strategy of using bright coherent states incombination with squeezed vacuum states which is bounded by Eq (43) andthe strategy of using the optimal nonclassical state which is bounded byEq (44) is marginal in practice [Demkowicz-Dobrzanski et al (2013)] Forgravitational-wave detectors and for any other laser interferometer using in-tense light there is no need for an alternative to the CSV strategy

63

We now turn back to the shot noise according to Eq (38) Generally noisecan be decomposed into its spectral contributions For a simple Michelsoninterferometer without arm resonators and without a signal-recycling cavitythe square-root of the single-sided shot-noise spectral density normalized tothe differential arm length change x in units of m

radicHz is given by [Saulson

(1994)] radicSMI

SNx =

radic~c2

2ωPprop 1radic

P (45)

where ω is the optical angular frequency of the quasi-monochromatic carrierlight and P the total light power in both arms including the built-ups fromcavities In combination with a squeezed vacuum whose relative phase gen-erates squeezing of the output lightrsquos amplitude quadrature the right handside reduces according to the factor eminusr Note that the single-sided spectraldensity is only defined for positive sideband frequencies and thus twice aslarge as the double-sided spectral density

The spectral density of the measurement of a GW induced strain is givenby the same expression but normalized to h = xL (If the gravitational waveis oriented in an optimal way with respect to the Michelson interferometerone arm is squeezed while the other is expanded by the same amount of ∆L =x2 and h then corresponds to the actual gravitational-wave amplitude)The square-root of the single-sided shot-noise spectral density normalized tostrain in units 1

radicHz is given by

radicSMI

SNh =

radic~c2

2L2ωP (46)

Equations (45) and (46) show that the smallest measurable signal (corre-sponding to unity signal-to-shot-noisendashratio) is inversely proportional to thesquare root of the laser power and has a white spectrum for sideband frequen-cies much smaller than the carrier frequency see horizontal line in Fig 23

All first- and second-generation GW detectors use power-recycling andadditional cavities to improve their sensitivities Fabry-Perot arm resonatorsdo not only increase the light power but additionally also increase the signalfor signal frequencies inside the resonator linewidth For lossless Fabry-Perot

64

arm resonators the spectral densities in Equations (45) and (46) need to bemultiplied by the following factor [Kimble et al (2001)]

HFP =

radicL2(γ2

FP + Ω2)

c2 (47)

where γFP = cTFP(4L) is the Fabry-Perot arm resonatorrsquos half bandwidthand TFP is the light power transmission of the input mirror The end mirrorsare assumed to have perfect reflectivity A similar expression can be derivedfor describing the improvement due to signal-recycling [Buonanno and Chen(2001)]

In summary shorter laser wavelengths higher light powers and squeezingof the amplitude quadrature of the interferometer output reduce shot noise ina broadband way ie for all signal frequencies Fabry-Perot arm resonatorsas well as signal-recycling provide improvements mainly for frequencies insidethe resonator linewidths

53 Quantum back-action and quantum radiation pressure noise

In laser interferometers quantum back-action noise results from the un-certainty of the lightrsquos radiation pressure force on the interferometer mirrorsand is also called lsquo(quantum) radiation pressure noisersquo (RPN) Its origin isthe quantum uncertainty of the differential amplitude quadrature XΩ∆Ω ofthe fields in the interferometer arms It results in an uncertain momentumtransfer to the mirrors and thus in an position uncertainty of the mirrors atfuture times with respect to their differential mode of motion [Caves et al(1980)] The physical mechanism of radiation pressure corresponds to anintensity dependent phase shift [Pace et al (1993)]The higher the light power in the arms of a laser interferometer the loweris its shot-noise spectral density see Eq (45) Unfortunately the spectraldensity of quantum back-action noise increases with light power The single-sided force noise spectral density reads

radicSRPNF =

radic8~ωPc2

(48)

Whereas the force noise of the quantum radiation pressure has a white spec-trum the RPN does not since the mirrorrsquos reaction to external periodicforces depends on frequency The link between the Fourier component of an

65

external force F (Ω) and the Fourier component of the displacement x(Ω) isgiven be the mechanical susceptibility HM It reads for an harmonic oscillatorwith mass M

HM(Ω) =1

M | minus Ω2 + Ω2M + iΩΩMQ|

(49)

where ΩM is the oscillatorrsquos resonance frequency and Q its quality factorThe square root of the single-sided spectral density of the RPN normalized

to the displacement of an harmonic oscillator with mass M is then given by

radicSRPNx = HM(Ω)

radic8~ωPc2

(50)

In GW detectors the test mass mirrors are suspended as pendula with highmechanical Q-factors and their centre of mass motion corresponds to that ofa harmonic oscillator The resonance frequencies of the pendula are lowerthan the detection band of interest The mechanical susceptibility is thereforeoften approximated for the so-called free-mass regime as H fm

M (Ω) = (mΩ2)minus1The square root of the single-sided spectral density of the RPN normalizedto differential displacement of two mirrors with each of mass M in a simpleMichelson interferometer is given by [Saulson (1994)]radic

SfmMIRPNx =

radic2~ωPc2m2Ω4

propradicP (Ω ΩM) (51)

where m = M2 is the mirrorsrsquo reduced mass In case of a simple Michel-son interferometer that is enhanced with arm cavities the spectral density inEq (51) needs to be multiplied with the expression given in Eq (47) In com-bination with a squeezed vacuum whose relative phase generates squeezing ofthe output lightrsquos phase quadrature the right hand side reduces according tothe factor eminusr Note if the radiation pressure noise is squeezed the shot noisemust be anti-squeezed or vice versa The radiation pressure noise calibratedto strain of space time is given by the right side of Eq (51) divided by theinterferometer arm length LIn summary heavier masses longer laser wavelengths lower light powersand squeezing of the amplitude quadrature in the interferometer arms re-duce radiation pressure noise

66

1 10 100 1000Frequency [Hz]

Radiation pressure noise

Shot noise

SQL

10-21

10-18

10-15

10-12

radicSx

[mradic

Hz]

ndashndash

ndashndash

Figure 23 Displacement-normalized quantum noise spectral densities ndash Consid-ered is a simple Michelson interferometer with neither arm cavities nor signal recyclingThe two end mirrors (m = 100 g) of the interferometer arms are suspended as pendulahaving a resonance frequency of ΩM2π = 1 Hz and a Q-factor of 107 The interferometeruses quasi-monochromatic light (in coherent states) with a total power of 4 kW Opticalloss and the offset from a dark output fringe is assumed to be negligible Wavelengthλ = 1550 nm The standard quantum limit (SQL) corresponds to the lowest noise achiev-able at a given sideband frequency when varying the light power without using quantumcorrelations

54 Interferometer total quantum noise and the standard quantum limit

Both shot noise and radiation pressure noise contribute to the total quan-tum noise of a given interferometer If they are not quantum correlatedwhich is the case for a conventional Michelson interferometer when detectingthe output lightrsquos amplitude quadrature their variances add up (The resultis not shown in Fig 23) It can easily be deduced from the previous sectionsthat changing the laser power will shift the two quantum noise contributionsHowever the total quantum noise never goes below the standard quantumlimit (SQL) [Braginsky and Manukin (1967)]

Let us consider Fig 23 for sideband frequencies much greater than thependulum resonance Here the test mass mirrors react as free masses whenexerted to external forces The SQL in this free-mass regime is calculated

67

by minimizing the sum of the squares of Eqs (45) and (51) [Saulson (1994)]

Sfmtotx =

~c2

2ω

[1

P+

4ω2

c4m2Ω4P

] (52)

Its derivative reads

dSfmtotx

dP=minus1

P 2+

4ω2

c4m2Ω4 (53)

Setting the above equation to zero provides the optimum laser power versussideband frequency in order to achieve the lowest total quantum noise

P fmopt =

c2mΩ2

2ω (54)

Inserting the optimal light power into Eq (57) provides the square root ofthe single-sided noise spectral density of the free-mass SQL in m

radicHzradic

SfmSQLx =

radic2~mΩ2

(55)

Again m is the reduced mass and dividing by the interferometer arm lengthL yields normalization to the GW-induced strain h Eq (55) shows that theSQL falls off with sideband frequency The corresponding equation for aMichelson interferometer that uses arm cavities readsradic

SfmFPSQLx =

radic~

mΩ2

(1

HFP

+HFP

) (56)

with HFP according to Eq (47)Using the expression for the SQL the square root of the total quantum

noise spectral density of a Michelson interferometer in the free-mass approx-imation can be written asradic

SfmFPtotx =

radicSfmFP

SQLx

2

[1

k+ k

] (57)

with the radiation pressure coupling parameter

k(Ω) =2ωP

mc2Ω2 (58)

68

For a fixed light power and fixed reduced mass of the mirrors the quantumnoise limited interferometer reaches the SQL when k = 1 which is realizedat the angular sideband frequency ΩSQL =

radic2ωP(mc2)

Note that neither squeezing the phase quadrature nor squeezing the am-plitude quadrature of the interferometer light leads to sub-SQL performance[Caves (1981)] also confer Ref [Schnabel (2005)] Fig 3 (left) As we willsee in the next sections the standard quantum limit can be surpassed if shotand radiation pressure noise are correlated Then the total quantum noise isnot given by the sum of the variances ie the sum of the squares in Eqs (45)and (51)

55 Squeezed light for surpassing the standard quantum limit

A measurement with sensitivity better than the standard quantum limit(SQL) is also called a lsquoquantum non-demolition (QND)rsquo measurement [Bra-ginsky and Khalili (1995 1996) Kimble et al (2001)] Several QND tech-niques for laser interferometers were proposed in recent decades [Jaekel andReynaud (1990) Kimble et al (2001) Purdue and Chen (2002) Chen (2003)McClelland et al (2011) Danilishin and Khalili (2012) Graf et al (2014)]What they have all in common is they exploit quantum correlations betweenobservable uncertainties

Arguably the most extensive way of introducing quantum correlationsand surpassing the SQL is the injection of squeezed states of light [Jaekeland Reynaud (1990)] If the squeezed quadrature angle of the injected statesis neither 0 nor 90 the quantum uncertainties of the amplitude and phasequadrature amplitudes that describe the differential field in the two interfe-rometer arms become correlatedLet us consider a very simplified setup that just consists of a quasi-mono-chromatic light field that is back-reflected from a quasi-free mirror Thelight power and the mass be such that reflected light in a coherent stateresults in a measurement of the mirror position with a noise spectral den-sity at the SQL at sideband angular frequency ΩSQL At this frequencyquantum measurement noise and back-action noise are of the same sizeie the uncertainty in X produces an equally large additional uncertaintyin Y Upon reflection the quadrature amplitude variances change from∆2X = ∆2Y = 14 to 2∆2X = ∆2Y = 12 This result corresponds tothe situation in Fig 23 at the crossing frequency of shot noise and radiation

69

pressure noise The coupling of the uncertainty variances can be describedby the matrix K = (1 minusk 0 1) where k = 1 at the SQL If the modulationstate at ΩSQL is the ground state its variances are transferred according to

KT

(1 00 1

)K =

(1 0minus1 1

)(1 00 1

)(1 minus10 1

)=

(1 minus1minus1 2

) (59)

In accordance with Fig 23 the variance of YΩSQLis twice as large as the vac-

uum noise varianceNow let the quantum noise of the light field be 10 dB squeezed at 45

(Eq (19)) The projection of the quantum uncertainty onto the X-observableproduces the radiation pressure noise by being transferred with the couplingfactor k = 1 at the SQL into the Y -observable in fact with negative signsince a larger value of X produces a larger optical path length and thus aretardation of the phase Due to the squeezing at 45 the initial uncertaintyin Y cancels with the additional uncertainty that originates from the onein X The following calculation shows that the strength of the cancellationcorresponds to the initial squeezing strength Upon reflection the quantumuncertainties transform in the following way(

1 0minus1 1

)(505 495495 505

)(1 minus10 1

)=

(505 minus01minus01 02

) (60)

The state of light after reflection has a squeezed phase quadrature amplitudeThe improvement in comparison to Eq (59) is exactly 10 dB The quantumnoise improvement corresponds to the input squeeze factor and is also a mea-sure by what factor the SQL is surpassed Squeezed vacuum injection thusallows surpassing the SQL upon measuring the conventional Y -quadrature(which is realized by a single photo diode in the interferometerrsquos output port)as first realized by [Unruh (1983) Yuen (1983) Jaekel and Reynaud (1990)]

In the example above the input squeeze angle is optimized for a sin-gle sideband frequency Injecting a broadband squeezed vacuum field withfrequency-independent squeeze angle of 45 would result in a rather bad inter-ferometer quantum-noise performance at frequencies far smaller or largerthan ΩSQL Fig 24 shows the quantum-noise performance if the input fieldhas squeeze angles that are optimized for every k(Ω) as given in Eq (58)Injected squeezing can thus lead to a broadband sub-SQL performance ifthe quantum measurement noise (shot noise) and the quantum back-actionnoise (radiation pressure noise) are correlated in an optimal way Due to the

70

10 50 100 500 1000 5000Frequency [Hz]

Dis

plac

emen

t noi

se sp

ectra

l den

sity

[mradic

Hz]

10-20

10-19

10-18

Total quantum noise 0dB

Total quantum noise -10dB

SQL

Y

X

Y

X

Y

X

Figure 24 Surpassing the SQL with squeezed-light injection ndash At shot-noise lim-ited sideband frequencies squeezing of the Y -quadrature amplitude improves the noisespectral density of the interferometer At radiation-pressure-noise limited sideband fre-quencies squeezing of the X-quadrature amplitude improves the noise spectral density ofthe interferometer If both kinds of quantum noise contribute equally (at the SQL markedwith a dot) a squeeze angle of 45 results in surpassing the SQL by the full squeeze factorsee Eq (60) In the graph here the squeeze angle is optimized for all frequencies result-ing in a broadband quantum noise reduction [Jaekel and Reynaud (1990)] Measurementsensitivities beyond the SQL (shaded area) are in the so-called quantum non-demolition(QND) regime [Kimble et al (2001)] Dashed horizontal lines represent the (squeezed)shot noise Dashed straight lines with negative slope represent the (squeezed) radiationpressure noise The calculations use 10 dB of squeezing a conventional Michelson inter-ferometer with neither arm resonators nor signal recycling a light power at the beamsplitter of 1 MW at a wavelength of λ = 1550 nm and mirror masses of 1 kg

correlation shot noise and radiation pressure noise can be squeezed simulta-neously

71

Light with a frequency-dependent squeeze angle

The discovery that shot noise and radiation pressure noise can be squeezedsimultaneously and thus a broadband reduction of quantum noise beyond theSQL be achieved required the insight that the spectral analysis of light definesa spectrum of many lsquosideband modulation modesrsquo that all can be in differ-ent quantum states An ordinary squeezing resonator which is on resonancefor light at twice the pump wavelength produces a spectrum of modulationmodes that all have the same squeeze angle A frequency-dependent squeezeangle can be introduced by reflecting such a field from a detuned single-ended filter cavity which was suggested by Kimble and coworkers [Kimbleet al (2001)] They showed that the optimal frequency dependence thatleads to the broadband improvement shown in Fig 24 can be realized byusing altogether two filter cavities as shown in Fig 25 Motivated by this re-sult research and development on filter cavities for optimizing the frequencydependence of broadband squeezed fields has been very active in recent years[Corbitt et al (2004) Chelkowski et al (2005) Dwyer et al (2013) Kweeet al (2014) Straniero et al (2015) Oelker et al (2016)]

Photo diode Squeezed vacuum

Faraday Rotator

Coherent light

YΩΔΩ

Figure 25 Frequency dependent squeezing injection ndash A broadband squeezed fieldwith a frequency-dependent squeeze angle that is optimal for gravitational-wave detectorsis produced by reflecting off an ordinary broadband squeezed field from two detuned opticalfilters [Kimble et al (2001)]

A light field with a frequency-dependent squeeze angle was first demon-strated in Ref [Chelkowski et al (2005)] see Figs 26 and 27 The experi-

72

Detuned filter cavityLockingphoto diode

Isolator

EOM R asymp 1

SHG

EOM

DBSOPA

LaserIsolator

Mode cleaner

LO

Homodyne detector

Figure 26 Generation of a frequency-dependent squeezing ndash A frequency-dependent orientation of the squeeze ellipse was first demonstrated in Ref [Chelkowskiet al (2005)] Initially a conventional spectrum of squeezed vacuum states of light was gen-erated in a squeezing resonator (lsquoOPArsquo) The squeezed vacuum was transmitted throughan optical isolator to a detuned filter cavity After reflection the squeezed vacuum stateswere absorbed in a balanced homodyne detector (BHD) The phase of the BHDrsquos localoscillator (LO) was changed for quantum state tomography of the squeezed states in diffe-rent regions of the spectrum The result showed a frequency-dependent orientation of thesqueeze ellipse see Fig 27 SHG second harmonic generation EOM electro-optical mod-ulator for applying phase modulation sidebands for cavity length control DBS dichroicbeam splitter R mirror reflectivity λ4 quarter wave plate for turning linear polarizedlight into circular polarised light and vice versa

ment consisted of a standing-wave squeezing resonator which produced ans-polarized broadband amplitude quadrature squeezed field accompanied bya dim continuous-wave DC control field with a wavelength of λ = 2πcω =1064 nm The squeeze bandwidth covered sideband frequencies up to aboutΩ(2π) = 30 MHz which corresponded to the linewidth of the squeezing res-onator The optical cavity for producing the frequency dependence of thesqueeze angle was a standing-wave cavity composed of a plane incouplingmirror of reflectivity r1 =

radic097 and a concave end mirror of reflectivity

r2 =radic

09995 The cavity length was L = 50 cm resulting in a linewidthof 147 MHz The squeezed field first passed a Faraday isolator to preventinterference effects between the filter cavity and the squeezing resonator A

73

Phase quadrature

min

max

-2 -1 0 1 2

141 MHz

-2

-1

0

1

2

Am

plit

ud

e q

uad

ratu

re

Figure 27 Frequency-dependent squeezing ndash Picture top right Reconstructed con-tour plot of the Wigner function of the sideband modulation at Ω(2π) = 141 MHz afterreflection from a 1515 MHz detuned filter cavity The state shows quantum correlationsbetween phase and amplitude quadratures ie squeezing at an angle of here about 40The white circle visualizes the standard deviation of the vacuum state uncertainty Thewhite ellipse represents the standard deviation of the squeezed uncertainty Small picturesMeasurement results on the same continuous-wave laser beam at various sideband frequen-cies around 15 MHz For each tomographic picture noise histograms of 100 equidistantquadrature angles were measured In each case the laser beam was phase locked to a ref-erence beam and the quadrature angle stably controlled and stepwise rotated The phasereference was given by a phase modulation at 198 MHz [Chelkowski et al (2005)] Thepicture was first published in Ref [Schnabel (2005)] (copyright ccopy2007 by Imperial CollegePress)

λ4-waveplate turned the s-polarized field into a circularly polarized beamwhich was then mode matched into the detuned cavity The retro-reflectedfield was analyzed by a balanced homodyne detector (BHD) for quantumstate tomography The filter cavity was electro-optically controlled to be de-tuned by 1515 MHz with respect to the DC control field The cavity length

74

control was achieved by the Pound-Drever-Hall (PDH) locking techniqueutilizing a circularly polarized laser beam that carried 15 MHz phase mod-ulation sidebands and was coupled into the filter cavity from the back Theoutput voltage of the BHD was characterized by a spectrum analyser as wellas used to perform quantum state tomography In the latter case the BHDoutput voltage was mixed down with an electronic local oscillator at differentradio-frequencies around 15 MHz and low-pass filtered to set the resolutionbandwidth to ∆Ω(2π) = 100 kHz The final electric signal corresponds to atime series of quadrature amplitude measurements XθΩi∆Ω Quantum statetomography is a method to reconstruct the phase space quasi-probability dis-tribution (Wigner function) of quadrature amplitudes from sets of measuredXθΩ∆Ω distributions when varying the angle θ [U Leonhardt (1997)] Forevery sideband frequency Ωi 100000 quadrature values were measured di-vided up on 100 equidistant quadrature angles Each quadrature angle wasstably controlled with a precision of plusmn1 Fig 27 shows the reconstructedWigner functions which were all measured on the same laser beam but atdifferent sideband frequencies For these measurements the detuned filtercavity was locked to the lower sideband at minus1515 MHz The result clearlyshows the frequency-dependent orientation of the squeeze ellipse In a morerecent experiment a frequency-dependent squeeze angle was also realized inthe kHz regime [Oelker et al (2016)]

56 Optomechanically induced (ponderomotive) squeezing

The radiation pressure of light when acting on a movable mirror re-sults in an intensity dependent phase shift [Pace et al (1993)] The cou-pling produces a so-called lsquoponderomotive effectrsquo [Braginsky and Manukin(1967)] which is of third order optical nonlinearity and which transforms abright coherent state inside an interferometer into a squeezed state of light[Vyatchanin and Matsko (1993)] This type of squeezed-light generationis usually called lsquoponderomotive squeezingrsquo or lsquooptomechanical squeezingrsquoConsequently even if no squeezed field is injected into the interferometercorrelations between the quadrature amplitudes are generated that allow forsurpassing the SQLPonderomotive squeezing as produced by the interferometer itself can onlybe exploited for evading back-action (radiation pressure noise) It can notbe used to squeeze the interferometer shot noise This is why ponderomotivesqueezing is fundamentally less extensive than injecting externally producedsqueezed states of light [Corbitt et al (2006)] suggested an external pon-

75

deromotive squeezing source for gravitational-wave detectors In this casedue to its external generation also the interferometerrsquos shot noise can besqueezed Recently ponderomotive squeezing was observed for the first time[Brooks et al (2012) Purdy et al (2013)] The achieved squeeze factors aremuch smaller than those produced by optical-parametric down-conversion[Vahlbruch et al (2016)]

Let us have a look again at Eq (59) Rotating the covariance matrix onthe right by arctan(minus

radic54minus 12) asymp minus58 indeed reveals squeezing

(cos 58 minussin 58

sin 58 cos 58

)(1 minus1minus1 2

)(cos 58 sin 58

minussin 58 cos 58

)asymp(

262 00 038

) (61)

The vacuum-noise normalized variance of 038 corresponds to about 42 dBof ponderomotive squeezing This is the general value that is produced atthe angular sideband frequency ΩSQL At higher frequencies the squeezefactor gets smaller at lower frequencies higher The squeezing strength of42 dB can be observed if the photo diode in the interferometer output portis replaced by a balanced homodyne detector using a local oscillator phase ofabout minus58 It can be shown however that the optimal signal-to-quantum-noise-ratio at the SQL is achieved for a local oscillator phase of exactly 45At this angle back-action is fully evaded

Full evasion of radiation pressure noise at all frequencies requires an op-timized frequency dependence of the relative local oscillator phase This canbe achieved by reflecting off the interferometer output field from two detunedfilter cavities [Kimble et al (2001)] The scheme was called lsquovariational out-putrsquo In the case of zero optical loss this scheme can fully evade radiationpressure noise just leaving the shot noise as the only quantum noise contri-bution

The variational-output scheme can be used to enhance the frequency-dependent squeezed input scheme The right site of Eq (60) shows thatthe output statersquos squeezing is not optimally detected in the Y -quadratureRather than with a single photo diode the detection should be done witha balanced homodyne detector with optimized phase of its local oscillatorIn this case the output lightrsquos quantum noise is solely given by squeezedshot noise The total quantum noise in Fig 24 would then be given by thelowest (dashed) horizontal line This combined scheme was called lsquosqueezed

76

variationalrsquo [Kimble et al (2001)] It can be realized by reflecting off theinterferometer output light from in total two optical filter cavities placed infront of the balanced homodyne detector

57 Conclusions

The highest quantum-noise-limited sensitivities of high-precision laserinterferometers are achieved by employing a large number of quanta tomaximize the signal strength in combination with strongly squeezed statesto minimize the quantum noise From this perspective it is clear that thequantum-noise-limited sensitivity of future gravitational-wave detectors willbe further improved ndash by increasing the light power and the squeeze factorTo be able to do so the optical loss in these devices needs to be reduced

In principle the optical loss in laser interferometers can be made smallbut never zero Recent theoretical research has shown that for any non-zero loss the sensitivity scales proportional to 1

radicn at best where n is the

average photon number per measurement This scaling is efficiently achievedby combining strongly displaced coherent states with squeezed vacuum statesof light

If a repeated measurement is not only limited by quantum measurementnoise but also by quantum back-action noise squeezed states of light can beused to simultaneously reduce both ie in the case of an interferometer shotnoise and radiation pressure noise

6 The first application of squeezed light in an operating gravita-tional-wave detector

Squeezed states of light have been successfully used to improve the sensi-tivity of the gravitational-wave detector GEO 600 from 2010 up to the pointwhen this Review was written [Abadie (2011) Grote et al (2013)] Afterdecades of proof-of-principle experiments [Xiao et al (1987) Grangier et al(1987) McKenzie et al (2002 2004) Vahlbruch et al (2005 2006 20072008) Goda et al (2008)] the implementation of a squeezed-light source inGEO 600 has resulted in the first sensitivity improvement beyond shot noiseof a measurement device that targets new observations in nature The im-plementation of squeezed states in GEO 600 was not done to provide anotherproof-of-principle demonstration but was realized because it offered a rela-tively cheap way of further improving the measurement sensitivity Of course

77

the sensitivity of GEO 600 can also be further increased by purely classicalmeans however the implementation of arm resonators to enable higher lightpowers without increasing the thermal load on the beam splitter or eventhe realization of longer interferometer arms are much more expensive Inthis respect the sensitivity improvement of GEO 600 with squeezed light canarguably be regarded as the first lsquotruersquo application that developed out of thefield of lsquononclassical (quantum) metrologyrsquo (Note that the term rsquoquantummetrologyrsquo is currently defined in different ways [Giovannetti et al (2006)Gobel and Siegner (2015)] and the term rsquononclassicalrsquo referring to a non-classical P-function gives a distinct description)

61 Gravitational waves

Einsteinrsquos General Theory of Relativity [Einstein (1916)] or simply lsquoGen-eral Relativityrsquo (GR) predicts that accelerating mass distributions producegravitational radiation analogous to electromagnetic radiation from accel-erating charges Experimental evidence of their existence is given by theobservation of the slow spiraling together of two neutron stars caused bythe loss of orbital energy to gravitational waves The inspiral rate exactlymatches the predictions of Einsteinrsquos theory [Weisberg and Taylor (2005)]Recently Advanced LIGO observed gravitational waves for the first time[Abbott (2016)] thereby giving the go-ahead for gravitational-wave astron-omy The gravitational-wave source was the final inspiraling and the mergerof two black holes 13 billion light years away from earth

Gravitational-waves evolve in the far field of the source propagate withthe speed of light and are measurable on earth with laser interferometersFig 28 displays a gravitational wave propagating along a certain directionGravitational waves are dynamical changes of space-time They are transver-sal and quadrupolar in nature and have two polarization states

A variety of known astrophysical and cosmological sources are predictedto emit gravitational radiation that should reach the Earth with a measur-able strength [Sathyaprakash and Schutz (2009)] The first gravitational waveevent detected was produced by two black holes of 36 and 29 solar massesDuring the final 02 seconds of their inspiraling they produced a peak gravi-tational strain in our solar system of 10minus21 covering frequencies up to 250 Hz[Abbott (2016)] Other predicted sources are mergers of neutron stars supernovae and background signals from the Big Bang According to GR GWsfrom complex astrophysical sources carry a plethora of information that willhave a major impact on gravitational physics astrophysics and cosmology

78

L +Δ L

Binary system

fBS

Figure 28 Space-time oscillation ndash Gravitational waves are dynamical deformations ofspace-time that form in the plane perpendicular to the direction of wave propagation Asa result distances between free-falling test masses in a transverse plane will change witha strain h = ∆LL For black hole or neutron star binary systems with orbital frequencyfBS distances will oscillate at frequency fGW = 2fBS The wavelength of this oscillation isgiven by λGW = cfGW where c is the speed of light The wave of orthogonal polarizationwith respect to the one shown is rotated by 45 around the propagation axis

62 Interferometric detection of gravitational waves

Current gravitational wave detectors are kilometre-scale laser interfero-meters [Dooley et al (2016) Aasi (2015) Acernese (2015) Aso et al (2013)]Continuous-wave laser light is split into two beams traveling in orthogonaldirections Both beams are reflected back towards the central beam splitterwhere they interfere Gravitational waves change the optical path lengthdifference and thus the light power directed towards the photo-diode thatis positioned in the signal output port of the beam splitter A gravitationalwave at frequency fGW = ΩGW(2π) reveals itself as a light-power modula-tion at the same frequency The spectral decomposition of the output signalis described by a spectrum of the quadrature amplitude YΩ∆Ω introduced inSec 3 It corresponds to the amplitude quadrature amplitude of the outputlight and relates to the differential phase quadrature of the interferometerarms

79

Photo diode

Squeezed vacuum

Faraday Rotator

Signal

Quantum noise

Coherent light XΩΔΩ

YΩΔΩ

(a) (b)

YΩΔΩFigure 29 Squeezed-light-enhanced interferometric measurement ndash (a) Michelsoninterferometer with arm cavities power recycling and signal recycling (see main text forexplanation) The interferometer is operated close to a dark fringe such that the quantumnoise entering from the dark port is back-reflected The squeezed field is mode-matched tothe signal output field (b) Phase space diagram of the gravitational-wave signal outputat sideband frequency Ω(2π) The quantum noise is squeezed below the ground stateuncertainty and thus the signal to quantum noise ratio improved

The first key ingredient of an interferometric gravitational-wave detectorare suspended heavy mirrors that can be regarded as quasi-free in the di-rection of laser light propagation thereby acting as test masses that probespacetime Being on ground current detectors are located in rather noisy en-vironments that allow the realization of undisturbed quasi-free mirrors onlyabove a sideband frequency of the order of 10 Hz Since sufficiently strongGW signals are expected up to a frequency of 10 kHz todayrsquos gravitationalwave detectors target at signals in the acoustic band from 10 Hz to 10 kHzThe quasi-free motion of the test mass mirrors in this frequency regime isachieved by suspending the mirrors as sophisticated multiple-stage pendulain vacuum chambers [Aasi (2015)] Far above the pendularsquos resonant frequen-cies which are typically around 1 Hz the centre of masses of the mirrors areisolated from vibrations of the ground and they react on frequency compo-nents of small external forces approximately as free masses The mirrors andtheir suspensions are built from materials having exquisitely high mechani-cal quality factors This helps to concentrate the thermal energy that causes

80

displacements of the mirror surface into well-defined vibrational frequencymodes At these particular very sharp frequencies no gravitational wavescan be detected

The second key ingredient of an interferometric gravitational-wave de-tector is laser light with a power of up to hundreds of kilowatts or evenmegawatts The light is quasi-monochromatic and needs to show very lowamplitude (quadrature) noise and phase (quadrature) noise at sideband fre-quencies within the detection band Low amplitude noise is necessary toprovide a shot noise limited output field It is also necessary to avoid classi-cal radiation pressure noise which becomes an issue if the light power or themirror masses in the two arms are not identical Low phase noise is requiredif the storage time of the light in the two arms is not identical This mightaccidentally occur due to different linewidths of the arm cavities or mightbe part of the interferometer design to allow for the length control schemeproposed by Schnupp [Heinzel et al (1998)] To maximize the light power in-side the interferometerrsquos cavities it should be produced in an almost perfecttransversal spatial distribution of a Gaussian TEM00 mode

Light sources of gravitational-wave detectors are ultra-stable NdYAGmaster-slave systems that provide up to 200 W of light at 1064 nm [Winkel-mann et al (2011) Kwee et al (2012)] The high power in the interferometerarms is achieved by cavity built-ups in the so-called power-recycling cavityand in the arm cavities Power recycling uses a partially reflective mirror thatis located between the light source and the interferometer beam splitter Itssurface is matched to the lightrsquos wave front and forms an optical cavity to-gether with the rest of the interferometer Since gravitational-wave detectorsare operated close to a dark fringe large power built-ups can be achievedThe highest power built-up is achieved for a mirror transmission equal to the(given) interferometer round trip loss In this case an impedance-matchedcavity is achieved The power-recycling cavity as well as the arm cavitiesare stabilized on resonance for the input light The difference between theirfunctionality is that the power-recycling cavity does not limit the detectionbandwidth of the interferometer GEO 600 as well as Advanced LIGO em-ploy a third type of cavity the so-called signal-recycling cavity Similarlyto power recycling a partially reflecting mirror that is placed between theoutput port of the beam splitter and the photodiode is used to resonantly en-hance the GW signal [Meers (1988)] The signal-recycling cavity resonantlyenhances the signal modulation fields within its linewidth without furtherenhancing the carrier light power In combination with low linewidth Fabry-

81

Perot arm resonators it can also be used to extract the signal by reducing theeffective finesse of the arm resonators for the signal sidebands This schemeis called resonant sideband extraction [Heinzel et al (1996)] The signal-recycling cavity has also been tested in a detuned setting in which just theupper or lower sideband is extracted or resonantly enhanced respectively[Heinzel et al (2002)] Current gravitational-wave detectors however usecarrier-tuned signal recycling

All these techniques are lsquoclassicalrsquo approaches for maximizing the signal-to-shot-noise ratio At frequencies above a few hundred Hertz howevershot-noise is still the limiting noise source in gravitational-wave detectorsFuture gravitational-wave detectors will therefore use even higher light pow-ers but further increasing the light power becomes more and more challeng-ing Optical materials with less light absorption need to be found to coun-teract an increasing thermal load inside the mirrors Mirror masses need tobe further increased to counteract the increasing radiation pressure noiseNonclassical approaches are superior and become more and more attractivethe farther classical approaches are pushed to the extremes Nonclassicalapproaches allow for simultaneously increasing the signal-to-shot-noise ratioand the signal-to-radiation-pressure-noise ratio without changing light poweror mirror masses see Fig 24 They also allow for a complete evasion of radia-tion pressure noise [Braginsky and Khalili (1995 1996) Kimble et al (2001)]see Subsec 56

63 Squeezed-light enhancement of the gravitational-wave detector GEO 600

In 2010 GEO 600 was equipped with the squeezed-light source shownin Fig 18 The location of the squeezed-light source close to the outputport is shown in Fig 30 It was known that GEO 600 was shot-noise limitedat sideband frequencies above about 700 Hz In this frequency regime thereplacement of the ordinary vacuum states that entered the interferometerfrom the output port by a spectrum of squeezed vacuum states was expectedto reduce the noise spectral density into the nonclassical regime It was notprecisely clear what squeezing factors could be expected since the opticalloss upon mode-matching an external field into the output port propagationalong the arms and the final photo-electric detection was not determined

Critical components were the quantum efficiency of the photo-diode aswell as the optical loss of Faraday rotator for overlapping the squeezed fieldwith the output mode Also the transversal output mode of GEO 600 was an

82

issue since it contained significant contributions from higher order modesto which a good mode matching of the squeezed mode was not possible Theimplementation of the squeezed-light source thus was accompanied with anew custom made InGaAs photo-diode with 3 mm diameter The goal was aquantum efficiency of greater 99 [Vahlbruch et al (2016)] Also the Fara-day rotator was custom-made and optimized for lowest optical loss which

Figure 30 GEO 600 ndash View into the central building of the British-German GW detectorlocated close to Hannover Germany The vacuum chambers contain the suspended beamsplitter power- and signal recycling mirrors additional input and output optics as well asmirrors to realize a double pass of the laser light through the 600 m long interferometerarms By courtesy of the AEI

83

involved a rather precise rotation of the polarisation of 45 plusmn 05 over anaperture of more than 15 mm Finally a ring cavity (output mode cleaner)was placed in front of the photo-diode which acted as a passive filter forhigher transversal modes Since GEO 600 was not limited by radiation pres-sure noise and since it used a carrier-tuned signal-recycling cavity a frequencyindependent orientation of the squeezing angle was optimum After several

100 200 300 500 700 1000 2000 3000 500010-22

10-21

10-20

Obs

erva

tory

noi

se c

alib

rate

d to

str

ain

[1radic

Hz]

10-19

10-18

10-17

Noise calibrated to test m

ass displacement [m

radicHz]

Sideband frequency f

Figure 31 Nonclassical reduction of the GEO 600 instrumental noise ndash Shownare the square roots of the noise spectral densities without squeezed vacuum states (top)and with squeezed vacuum states (bottom) [Abadie (2011)] Their calibrations [Affeldtet al (2014)] to gravitational-wave strain and differential arm length change are shownon the left and right y-axes respectively Note that both traces increase towards higherfrequencies due to the frequency-dependent signal enhancement of the signal-recyclingcavity The injection of squeezed vacuum states leads to a broadband noise reductionof up to 35 dB at shot-noise limited frequencies The spectral features are for instancecaused by excited violin modes of the mirror suspensions (600ndash700 Hz and harmonics)Data by courtesy of the AEI

months the combination of the squeezed-light source and the gravitational-wave detector succeeded The main laser of the squeezed-light source wasphase locked to the main laser of GEO 600 and a stable mode matching be-

84

tween the squeezed field and the interferometer output field was achieved[Abadie (2011)] The effect on GEO 600rsquos (strain normalized) noise spectraldensity is shown in Fig 31 At frequencies above about 700 Hz the squareroot of noise spectral density was reduced by up to one third This value cor-responds to a quantum noise variance of 045 (minus35 dB) with the shot noisevariance normalized to unity For isotropically distributed gravitational-wavesources this factor produces a detection rate increase by a factor of 153 asymp 34After its integration into GEO 600 the squeezed-light source was used in allscientific runs seeking for gravitational waves for instance in the observa-tional run S6eVSR4 that was undertaken from June 3rd to September 5th

in 2011 [Grote et al (2013)]Towards the end of 2011 right before the start of the detector upgrade

to Advanced LIGO a nonclassical sensitivity improvement was also demon-strated in one of the LIGO detectors [Aasi (2013)] In this experiment anonclassical sensitivity improvement corresponding to up to 215 dB abovefrequencies of about 150 Hz was achieved The successful test is a strongmotivation for a squeezed-light upgrade of Advanced LIGO Note that thedesign of Advanced LIGO was completed in 1999 and squeezed-light sourceswere not mature at those times

In the past years the squeezing enhanced GEO 600 detector was not onlyused for observations but also was the control of the injected squeezed modefurther improved Stabilizing the overlap between squeezed mode and brightmode of the interferometer to close to perfect is necessary to reduce theeffective optical loss and to maximize the measurable squeezing factor Re-cently superior methods for stabilizing the longitudinal phase of squeezedvacuum mode were found [Dooley et al (2015)] and the first automatic align-ment system for stabilizing and optimizing the transversal mode overlap wasdemonstrated [Schreiber et al (2016)]

64 Are squeezed states the optimal nonclassical resource ingravitational-wave detectors

For a given number of photons Eq (41) quotes the ultimately smallestphase change that can be measured with a signal-to-noise-ratio of one Thescaling with number of photons per measuring time of this Heisenberg limitseems appealing compared to the scaling achievable with coherent states orsqueezed states according to Eqs (38) and (39) respectively The Heisenberglimit however is only valid for precisely zero photon loss Since the non-classical states required to achieve Eqs (40) and (41) show an exponentially

85

increasing sensitiveness to loss when increasing the photon number the ac-tual scaling can not be deduced from Eq (40) Proposals to use Fock statesand the so-called N00N states for optimizing interferometer sensitivities [Hol-land and Burnett (1993) Dowling (1998) Mitchell et al (2004) Afek et al(2010)] are thus only applicable when the experiment is conditioned on zerophoton loss As discussed in recent publications the correct expression forthe fundamental sensitivity limit needs to consider not only the total photonnumber inside the interferometer but also the total photon loss [Dorner et al(2009) Ko lodynski and Demkowicz-Dobrzanski (2010) Knysh et al (2011)Escher et al (2011) Demkowicz-Dobrzanski et al (2012)]

Based on these earlier works Ref [Demkowicz-Dobrzanski et al (2013)]proved that the nonclassical sensitivity enhancement of GEO 600 reportedin Ref [Abadie (2011)] has been exceedingly close to fundamental quantuminterferometry bound under given energy constraints and photon loss lev-els More than that it was generally proven that the approach of com-bining displaced coherent states and squeezed vacuum states is optimal forgravitational-wave detectors

In Ref [Abadie (2011)] the gravitational-wave detector GEO 600 used aneffective number of photons per second of approximately n = 2 middot 1022 whichcorresponded to a total optical power inside the interferometer arms of P asymp37 kW at a wavelength of 1064 nm The total optical loss was 1minus η asymp 038The injected squeezing factor was eminus2r asymp 01 For these numbers the ratioof Eqs (43) and (44) is calculated to

∆φCSVmin

∆φgenmin

asymp

radicηeminus2r + 1minus η

1minus ηasymp 108 (62)

which is a good approximation within the limit of large coherent state dis-placements α sinh2r The quantum noise of GEO 600 including thesqueezed-light source was just 8 above the fundamental quantum inter-ferometry bound An increased squeezing strength of 16 dB (eminus2r asymp 0025)which is in reach would bring the approach based on coherent states andsqueezed vacuum states to within just 2 above the fundamental bound

Future GW detectors will have significantly reduced optical loss values(1 minus η) lsquoLossrsquo includes scattering and absorption at mirrors non-perfectfringe contrasts and the non-perfect quantum efficiency of the photo detec-tor Optical loss reduction is important for at least four reasons First itleads to an increased signal second it leads to a reduced quantum noise when

86

employing squeezed states third less absorption reduces the thermal load onthe test mass mirrors and fourth less scattering reduces the probability ofback-scattered light which produces disturbance signals [Billing et al (1979)Vahlbruch et al (2007) Punturo et al (2014)] The higher the finesse valuesof the arm and signal-recycling cavities are the more significant is opticalloss at mirror test masses the beam splitter and the signal-recycling mirrorThe finesse value of the power-recycling cavity and the loss of mirrors andlenses that guide the output field to the photo-diode are less critical Suitablephoto detectors of 995 quantum efficiency are available today [Vahlbruchet al (2016)] but achieving a total optical loss of 10 is still challengingThe reason for that is that first of all a measurement device aiming for bestabsolute sensitivity should use as much quanta (photons) as possible Highfinesse values for the enhancement cavities are thus essential but results inan unavoidable scaling-up of the effect of mirror losses A realistic exampleof future gravitational wave detectors thus considers η = 09 with a squeezingfactor of 20 dB (eminus2r = 001) In this case the quantum noise will be about4 above the ultimate fundamental bound for a given photon number

From Eq (62) it can be concluded that there is no need for any moresophisticated nonclassical states than squeezed states In particular non-classical states with a defined photon number such as N00N states are notrequired Within the approximation quoted this result is independent ofthe photon number This result is also independent of the physical systemused for interferometric phase estimation and can also be made for quantum-enhanced atomic clock calibration in the presence of dephasing Here the-oretical results indicate that the precision of Ramsey interferometry withspin-squeezed states is close to the optimal one in the asymptotic regime ofa large number of atoms [Huelga et al (1997) Ulam-Orgikh and Kitagawa(2001) Escher et al (2011)] as already stated in Ref [Demkowicz-Dobrzanskiet al (2013)] More sophisticated nonclassical states with fixed number ofn quanta might still be useful for the exceptional case when the absorptionof one quantum already results in zero measurement sensitivity anyway Anexample is an ensemble measurement where the absorption of a single photondemolishes the source of the phase change to be characterized A typicallyused approach of conditioning the measurement result on n clicks of n singlephoton counters conditions on precisely zero loss and is thus able to use theadvantage of Eq (40) over Eq (39)

87

65 Conclusions

Squeezed states of light will contribute to realizing gravitational-wave ob-servatories with much higher sensitivities than existing or planned ones Tobenefit from squeezed states in a most efficient way optical loss in terms ofabsorption and scattering must be minimized In particular the optical lossof mirror coatings and mirror substrates need to be minimized The rele-vant mirrors include the test masses the balanced beam splitter the signalrecyclingextraction mirror and all optical components between the latterand the photo diode Excellent spatial mode matching between the brightinterferometer field and the squeezed vacuum field is also of great impor-tance Achieving this requires further improvement of the surface figures ofall reflective optical components of the interferometer as well as improvedhomogeneity of all optical components that the light passes through

The quantum noise reduction achieved in a gravitational-wave detectoris of course always smaller than the highest squeeze factor provided by thesqueezed-light source As an example let us consider the observation of 15 dBof nonclassical noise suppression directly at the source If the squeezed fieldsenses an additional loss of 5 when propagating through the interferometerwhich is a very challenging number from todayrsquos point of view the remainingsqueezing level is about 11 dB see Eq (16)

7 The application of 2-mode-squeezed light in laser interferome-ters

71 Quantum Dense Metrology

At first glance the application of bi-partite (two-mode) squeezed statesto a device whose goal is measuring a single observable seems meaninglessSqueezing the uncertainty of that observable should be the optimum one cando This is indeed true when concerning just quantum noise but recentlyit was discovered that in the presence of classical disturbances bi-partitesqueezing can improve such measuring devices [Steinlechner et al (2013)]The concept was named quantum dense metrology (QDM) The potentialimprovement of a gravitational-wave detector with bi-partite squeezed statesis shown in Fig 32 (a) A description is given in the caption The pre-condition for a potential improvement can be best understood within a phasespace diagram Fig 32 (b) contains two different kinds of lsquosignalsrsquo The firstis the actual signal which always shows up as a phase space displacement

88

along the Y axis The second is a disturbance signal that can produce adisplacement in arbitrary direction in phase space A prominent example

Faraday Rotator

Coherent light

Photo detectors Balanced homodyne detectors

Squeezed vacuum

Squeezed vacuum

Quantum noise in

Y (A) and X (B)

XΩΔΩ(B)

YΩΔΩ(A)

(a) (b)

Disturbance projected

onto Y

and X

XΩΔΩ(B)

YΩΔΩ(A)

Example of (unknown) disturbance

Signal

Figure 32 Bi-partite-squeezed-light-enhanced measurement ndash (a) Setup for theapplication of bi-partite (two-mode) squeezed light in a laser interferometer on the basisof QDM Two squeezed vacuum fields are overlapped on a balanced beam splitter withprogrammable squeeze angles for instance with a relative angle of 90 which produces abi-partite state as shown in Fig 12 The beam splitter outputs are entangled for any rela-tive angle greater than zero One part is matched to the interferometer mode The secondpart is kept outside as a reference beam The interference of the interferometer outputand the reference beam is arranged with such a phase difference that it reproduces the twosqueezed inputs on the photo detectors The two squeezed beams are photo-electricallydetected measuring the respective squeezed quadrature (using balanced homodyne detec-tors) Both beams carry half of all interferometer induced modulations which includesignals as well as disturbances A single readout as shown in Fig 21 cannot distinguishbetween the two kinds The double readout shown here provides additional informationand allows for recognition of the disturbance [Steinlechner et al (2013)] as well as in prin-ciple a modeling of the disturbance and with a correct model an improvement of thenoise spectral density of the interferometer [Ast et al (2016)] (b) Phase space diagramdescribing phase quadrature readout A as well as amplitude quadrature readout B Bothshow squeezed quantum noise The amplitude quadrature readout does not contain anygravitational-wave signal ie any feature in this channel must be due to disturbancesThis information can be used to improve the interferometer

89

for such a disturbance is parasitic interference due to back-scattered laserlight [Vahlbruch et al (2007)] Back-scattering is a limiting noise at lowsignal frequencies of gravitational-wave detectors [Billing et al (1979) Vinetet al (1997) Hild (2007) Ottaway et al (2012) Punturo et al (2014)] Notethat all noise that couples in via unwanted motions of the test mass mirrorsso-called lsquodisplacement noisersquo always produces a phase space displacementalong the Y axis and cannot be tackled with QDM

Fig 33 shows measurement results obtained in Ref [Steinlechner et al(2013)] In a table-top experiment one part of a bi-partite squeezed state ofa continuous-wave mode at 1064 nm was mode-matched into the output portof a Michelson laser interferometer operated at its dark fringe in full analogyto Fig 32(a) A lsquosignalrsquo was produced by driving the piezo behind one of theend mirrors at a frequency of 555 MHz The lsquodisturbancersquo was introducedby re-injecting a small amount of light that leaked through the second endmirror with an additional piezo-mounted mirror The piezo was driven ata frequency of 517 MHz to produce a phase modulation An additionalDC voltage defined an arbitrary and unknown optical path length of thelight before being re-injected and as such the phase space orientation of thedisturbance signal This mechanism of a parasitic interference is realizednaturally in any interferometric device due to back-scattering of quanta frommoving surfaces in the environment

The interferometer output consisted of the signal as well as the distur-bance with a quantum uncertainty given by one subsystem of the bi-partiteentanglement It was overlapped with the second subsystem of the entan-gled state on a balanced beam splitter and the two outputs were analysedwith balanced homodyne detectors The phases of the bi-partite entangle-ment and the BHD local oscillators were controlled to resemble Fig 32(b)ie both BHDs measured a squeezed uncertainty regardless of the phase ofthe (generally unknown) disturbance

The beam splitter that combines interferometer output and the entangledreference beam unavoidably splits the signal as well as the disturbance intotwo paths For a balanced beam splitter this generally reduces the signaland disturbance power by 3 dB for both quadrature measurements Fig 33shows however that both BHDs performed about 6 dB below shot noisewhich demonstrates the usefulness of the scheme The squeeze factor can inprinciple be infinite which thus qualifies the lsquo3 dB penaltyrsquo In the abovefigure the additional information from the second BHD output was used torecognize the parasitic interference in the first BHD output providing a lsquovetorsquo

90

50 51 52 53 54 55 56 57

50 51 52 53 54 55 56 57-70

-75

-80

-85

-90

-70

-75

-80

-85

-90

Y (A) (Ω2π)Ω∆Ω

Frequency ( f )

Noi

se p

ower

(dB

m)

Noi

se p

ower

(dB

m)

X (B) (Ω2π)Ω∆Ω

~ ndash6 dB

~ ndash6 dB

Shot noise reference

Shot noise reference

Signal Projected disturbance

Projected disturbance

Vet

o

No signal

Figure 33 Bi-partite-squeezed-light-enhanced measurement ndash The result wasachieved in a table-top setup [Steinlechner et al (2013)] In the two panels the lower

(blue) traces show the squeezed quadrature noise-power spectra ∆2Y(A)Ω∆Ω(Ω2π) (top) and

∆2X(B)Ω∆Ω(Ω2π) (bottom) as simultaneously measured with balanced homodyne detectors

lsquoArsquo and lsquoBrsquo respectively The conventional Y -measurement (top) cannot distinguish be-tween signal and disturbances The additional X-measurement (bottom) does not detectany phase quadrature signal thus any feature in this measurement is a parasitic signal dueto a disturbance The respective projection onto the Y -measurement can thus be lsquovetoedrsquoIn a more sophisticated approach the X-data might be used to model and then to elim-inate the disturbance as well as its projections on both quadrature measurements Theresult is a reduced spectral density of the actual phase quadrature measurement [Ast et al(2016)] Traces shown here are slightly sloped due to the decreasing transfer functions ofthe balanced homodyne detectors The resolution bandwidth was ∆Ω(2π) = 10 kHz thevideo bandwidth was 100 Hz All traces were averaged three times

signal to trigger its removal from the data streamThe question arose whether the additional information can be used to

reduce the actual noise spectral density of the first measurement ie to re-cover signals that were buried by parasitic interferences Very recently it wasshown that this is indeed possible Ref [Ast et al (2016)] reports a table-top proof-of-principle experiment in which the additional information in the

91

QDM approach could be used for improving the sensitivity of an interfero-meter The measurement sensitivity was improved from above-shot-noise tosub-shot-noise (sub-Poissonian) performance This result was possible notbecause the way the parasitic interference arose was known but because theadditional information provided by QDM allowed for fitting a model of theexcess noise to the readout data

Quantum dense metrology (QDM) as shown in Fig 32 improves a mea-surement by simultaneously reading out two conjugate observables Bothreadout observables show a squeezed quantum noise and act as estimatorsof independent physical quantities This situation was recently described aslsquoquantum-mechanics freersquo [Tsang and Caves (2012)] QDM is based on anEinstein-Podolsky-Rosen (EPR) entangled [Einstein et al (1935)] bi-partitesystem as described in Subsec 34 EPR entanglement was previously con-sidered for the quantum-informational task of dense coding which doublesthe capacity of quantum communication channels [Bennett et al (1992)Braunstein and Kimble (2000)] The application of EPR entanglement inmetrology was first proposed by DrsquoAriano et al [DrsquoAriano et al (2001)]

72 Conclusions

A single beam that carries an optimized spectrum of squeezed vacuumstates and that is injected into the interferometerrsquos dark port provides themost efficient and practically optimal approach to reduce the quantum noisein laser interferometers by means of nonclassical states (see Section 5) Theconclusion of the section here is that two entangled beams provide a superiorapproach if the interferometerrsquos sensitivity is limited by classical noise thatis not exclusively restricted to the actual observable which is the phasequadrature amplitude Y Parasitic interferences due to laser light that isbackscattered from vibrating surfaces are an example Current gravitational-wave detectors use light fluxes of about 1024 photons per second [Abbott(2016)] Just a single photon per second and hertz which leaves the mainlight beam and is backscattered from a vibrating surface and in this way getsfrequency shifted into the detection band produces a significant disturbancesignal The lsquoquantum-dense metrologyrsquo approach might provide a powerfultechnique to tackle this problem

Very recently it turned out that QDM is not the only technique that mayexploit EPR entanglement to improve phase measurements Ref [Ma et al(2017)] proposes to use EPR entanglement to simultaneously suppress shotnoise and radiation pressure noise in a gravitational-wave detector without

92

the need for an additional filter cavity (confer subsection 55) In this caseEPR entanglement is exploited that is carried by one broadband squeezedbeam and that is present between quadrature amplitudes defined with re-spect to different optical frequencies ω and ωprime as investigated in Ref [Hageet al (2010)] Such lsquofrequency multiplexedrsquo EPR entanglement might resultin considerably lower costs of building a gravitational-wave detector with abroadband simultaneous squeezing of shot noise and radiation pressure noiseAlso this proposal does not lead to a fundamentally lower quantum noise butrather improves on classical aspects of an interferometer

8 Summary and Outlook

In many cases experiments that involve interference of quantum statescan be described in a semi-classical way This description uses the classicalwave picture for the interference part of the experiment and subsequently theclassical particle picture when the states transfer their energy to a detectoror more generally to a thermal bath This semi-classical description is notpossible when using the specific class of lsquononclassicalrsquo states Squeezed statesof light are a prominent example of these Squeezed states and other nonclas-sical states allow for observations that made Einstein Podolsky and Rosenformulate their critical and seminal paper on quantum theory [Einstein et al(1935)]

In the review here it is argued that after many successful proof-of-principle experiments with nonclassical states in the past decades the routineuse of squeezed-light in observational runs of the gravitational-wave detectorGEO 600 goes beyond proof-of-principle and is a true application of nonclassi-cal light Since 2010 the squeezed-light source has improved the measurementsensitivity of GEO 600 in basically every observational run [Abadie (2011)Grote et al (2013) Dooley et al (2016)] At quantum noise limited frequen-cies ie above a few hundreds of hertz the sensitivity has been improvedcorresponding to a squeezing strength in the noise spectral density of up to37 dB which corresponds to an increase of the average gravitational-wavedetection rate by a factor of 043minus32 = 36 This success is a strong motiva-tion to also equip the Advanced LIGO Virgo and Kagra gravitational-wavedetectors with squeezed light Similar improvement factors even down tolower signal frequencies are expected [LSC (2013)] The achievable improve-ment factors are mainly limited by the optical loss on the squeezed states

93

and much higher factors are achievable in principleUp to now squeezed states have not been used to reduce the radiation-

pressure noise in gravitational-wave detectors The reason is that so far othernoise sources are larger than radiation pressure noise and such an effect can-not be observed It is expected however that future gravitational-wavedetectors will eventually be partly limited by radiation pressure noise Fromthis point on squeezed light will be used to simultaneously reduce shot noiseand radiation pressure noiseSqueezed states are the optimum nonclassical states for gravitational-wavedetectors or more generally for all laser interferometers operating with largeaverage photon numbers per measuring interval [Demkowicz-Dobrzanski et al(2013)] In addition to using higher light powers and heavier test mass mir-rors higher squeeze factors will thus contribute to mitigate the lightrsquos quan-tum noise in laser interferometers

Two-mode (bi-partite) squeezed light has not been used in gravitational-wave detectors so far They are not capable of further reducing the quantumnoise in laser interferometers but they can be used to mitigate classicalnoise that originates from fluctuating phase space displacements A well-known such noise source is back-scattered light Proof-of-principle experi-ments were performed recently [Steinlechner et al (2013) Ast et al (2016)]This new technique could turn out to be valuable in next generations ofgravitational-wave detectors in particular in those targeting high sensitivi-ties at low sub-audio signal frequencies and using high light powers Suchan implementation in gravitational-wave detectors does not require any newtechnology Compared to a squeezed-light enhanced interferometer just asecond squeezed-light source is required

It is certainly remarkable that those quantum states that made EinsteinPodolsky and Rosen falsely think quantum theory incomplete are now ex-ploited as new technologies in measurement devices targeting new observa-tions in nature

Acknowledgements

RS thanks M Ast J Bauchrowitz C Baune S Chelkowski J DiGugliel-mo A Franzen B Hage J Harms A Khalaidovski L Kleybolte NLastzka M Mehmet S Steinlechner and H Vahlbruch for their contri-butions many fruitful discussions and their support with the figures and JFiurasek for many valuable comments on the manuscript Thanks are also

94

due to Y Chen F Khalili and H Miao for fruitful discussions within thequantum noise working group of the LIGO Scientific Collaboration (LSC)Special thanks are due to H Vahlbruch and H Grote together with theGEO 600 team for their pioneering work on the squeezed-light implementa-tion in GEO 600 RS is supported by the Deutsche Forschungsgemeinschaft(Grant No SCHN 757-6) and by the European Research Council (ERC)project lsquoMassQrsquo (Grant No 339897)

References

References

Aasi J et al Jul 2013 Enhanced sensitivity of the LIGO gravitationalwave detector by using squeezed states of light Nature Photonics 7 (8)613ndash619URL httpwwwnaturecomdoifinder101038nphoton2013177

Aasi J et al Apr 2015 Advanced LIGO Classical and Quantum Gravity32 (7) 074001URL httparxivorgabs14114547httpstacks

ioporg0264-938132i=7a=074001key=crossref

20895763c84bce3f8929251031b2475c

Abadie J et al Sep 2011 A gravitational wave observatory operatingbeyond the quantum shot-noise limit Nature Physics 7 (12) 962ndash965URL httparxivorgabs11092295httpwwwnaturecom

doifinder101038nphys2083

Abbott B P et al Feb 2016 Observation of gravitational waves from abinary black hole merger Phys Rev Lett 116 061102URL httplinkapsorgdoi101103PhysRevLett116061102

Acernese F et al 2015 Advanced virgo a second-generation interferome-tric gravitational wave detector Classical and Quantum Gravity 32 (2)024001URL httpstacksioporg0264-938132i=2a=024001

95

Contents

1 Introduction 4

2 Observations on light fields in squeezed states 921 Definition of a lsquosingle modersquo 1022 Observations on squeezed states using a single PIN photo-diode 1123 Observations on squeezed states using a balanced homodyne

detector 1324 Observations on two-mode squeezed states using balanced ho-

modyne detectors 1725 Observations using photon counters 1926 Conclusions 21

3 Theoretical description of squeezed states 2331 The quadrature amplitude operators 2332 Phase space representations of squeezed states 2733 Covariance matrix representation of (single-party) squeezed

states 3234 Phase space representation of two-mode (bi-partite) squeezed

states 3335 Covariance matrix representation of bi-partite squeezed states 3436 Photon numbers of squeezed states 36

4 Squeezed-light generation 3941 Overview 3942 Degenerate type I optical-parametric amplification (OPA) 4043 Cavity-enhanced OPA 4344 The generation of squeezed light for laser interferometry 48

441 High squeeze factors ndash minimizing decoherence 50442 Squeezing in the gravitational-wave (GW) detection band 52443 The first squeezed-light source for GW detection 54444 Generation of two-mode (bi-partite) squeezing 55

45 Conclusions 56

5 Quantum noise in laser interferometers 5751 Interferometric measurements 5752 Quantum measurement noise and shot noise 58

2

53 Quantum back-action and quantum radiation pressure noise 6554 Interferometer total quantum noise and the standard quantum

limit 6755 Squeezed light for surpassing the standard quantum limit 6956 Optomechanically induced (ponderomotive) squeezing 7557 Conclusions 77

6 The first application of squeezed light in an operating gravita-tional-wave detector 7761 Gravitational waves 7862 Interferometric detection of gravitational waves 7963 Squeezed-light enhancement of the gravitational-wave detector

GEO 600 8264 Are squeezed states the optimal nonclassical resource in

gravitational-wave detectors 8565 Conclusions 88

7 The application of 2-mode-squeezed light in laser interfero-meters 8871 Quantum Dense Metrology 8872 Conclusions 92

8 Summary and Outlook 93

3

1 Introduction

Laser interferometers are used to monitor small changes in refractiveindices rotations or surface displacements such as mechanical vibrationsThey transfer a differential phase change between two light beams into achanging power of the output light which is photo-electrically detected forexample by a photo diode The light is produced in a lasing process thatusually aims for a coherent (Glauber) state In practice laser light is oftenin a mixture of coherent states producing excess noise in the interferometricmeasurement But even if the laser light is in a (pure) coherent state its de-tection is associated with noise usually called lsquoshot-noisersquo This arises fromthe quantisation of the electro-magnetic field which for a coherent stateresults in Poissonian counting statistics of mutually independent photons

1000099009800 10100 10200

Photon number n

0000

0002

0004

0006

0008

0010

0012

Pro

babi

lity Squeezing

Poisson distribution

Figure 1 Poissonian and squeezed photon statistics ndash The upper boundary of eacharea represents the probability distribution of detected photon number n when perform-ing a large number of measurements on an ensemble of identical states having an averagephoton number of n = |α|2 = 10000 where α is the coherent field excitation or lsquodis-placementrsquo The broader curve shows the lsquoPoissonianrsquo distribution which describes thecounting statistic of mutually independent particles ie those of the coherent state Dueto the large value of α the distribution is almost Gaussian with a standard deviation ofplusmnradicn The narrow curve corresponds to the equally displaced 10 dB squeezed state which

obviously has a lsquosub-Poissonianrsquo photon statistic Note that squeezed states with smallor even without any coherent excitation (squeezed vacuum states) exhibit quite differentphoton statistics ndash see Fig 13 for example

4

If the coherent state is highly excited and thus the average number ofphotons n per detection interval is large the Poissonian distribution canbe approximated by a Gaussian distribution with a standard deviation ofplusmnradicn During the past decades squeezed states of light have attracted a lot

of attention because they can exhibit less quantum noise than a coherent stateof the same coherent excitation ie they can show sub-Poissonian countingstatistic see Fig 1

θ

Shot noise Squeezed noise

(a) (b)

Brightlaser input

Squeezedvacuum input

Faradayrotator

Photo diode

Michelsoninterferometer

Signal5050

(i)Ph

oto

curr

ent [

rel

units

]

Time [ms]

(ii)

0 5 10

Figure 2 Squeezed-light enhanced Michelson interferometer ndash (a) In addition tothe conventional operation of a Michelson laser interferometer with bright coherent lighta broadband squeezed-vacuum field is injected into the signal output port and overlappedwith the bright interferometer mode The interferometer is operated close to a dark fringesuch that most of the bright coherent light as well as most of the squeezed vacuum areback-reflected from the Michelson interferometer respectively Due to interference withthe broadband squeezed vacuum the interferometerrsquos output light on the photo diodeshows reduced variance in the photon number statistic as shown in Fig 1 Overlappingthe two light fields is possible with theoretically zero loss by the combination of a Faradayrotator and a polarizing beam splitter (PBS) A signal is produced by modulating therelative arm length (b) Simulated data for photo diode measurements Without squeezing(i) the signal of the laser interferometer is not visible With squeezing (ii) the shot noiseis reduced and here a sinusoidal signal visible

Squeezed states belong to the class of lsquonon-classicalrsquo states which areconsidered to be at the heart of quantum mechanics These states are de-fined as those that cannot be described as a mixture of coherent states Inthis case their Glauber-Sudarshan P -functions [Sudarshan (1963) Glauber(1963)] do not correspond to (classical) probability density functions ie theyare not positive-valued functions As a lsquoclassicalrsquo example the P -function ofa coherent state corresponds to a δ-function

5

But the question remains what property of coherent states justifies thename lsquoclassicalrsquo even though coherent states are quantum states and showquantum uncertainties My answer to this question is the following Allexperiments which only involve coherent states and mixtures of them allowfor a description that uses a combination of classical pictures As we will seebelow this description swaps between two different classical pictures and isthus not truly classical but semi-classical (A more precise description of thenature of coherent states uses the term lsquosemi-classicalrsquo)

Let us consider a laser interferometer that uses light in a coherent stateFirstly the light beam is split in two halves by a beam splitter The twobeams travel along different paths and are subsequently overlapped on abeam splitter where they interfere exactly as classical waves would do Theelectric fields superimpose thereby producing the phenomenon of interfer-ence Up to this point there is no reason to argue light might be composedof particlesSecondly the new (still coherent) beams that result from the interference areabsorbed for instance by a photo-electric detector In the case of coherentstates the detection process can be perfectly described in the classical parti-cle picture in which the particles appear independently from each other in atruly random fashion yielding the aforementioned Poisson statistic Duringthe detection process no wave feature of the light is present Let us havea closer look A truly random (lsquospontaneousrsquo) event is an event that hasnot been triggered by anything in the past This allows us to make a clearcut between the first part of the experiment described by the classical wavepicture and the second part of the experiment described by the classicalparticle picture Both lsquoworldsrsquo are disconnected The subsequent applicationof two classical pictures is not truly classical but lsquosemi-classicalrsquo It is in-deed the observation that the photons occur individually with truly randomstatistics that allows this semi-classical description In the case of a mixtureof coherent states the photon statistics are super-Poissonian which can beunderstood as a mixture of different Poissonian distributions In the caseof a slowly changing coherent state the mean value n depends on time Inall these cases the semi-classical description is appropriate Let me pointout that in this very reasonable description photons do not exist before theyare detected eg absorbed Further note that the famous double-slit exper-iment with coherent states also allows for the same semi-classical description

For squeezed states [Yuen (1976) Walls (1983)] the situation is different

6

As before the interference can be fully described by the classical wave pic-ture The result of the detection process however is different from that ofmutually independent random events It is also different from any super-Poissonian statistics that could be produced by mixing an arbitrary num-ber of different andor time-dependent Poissonian distributions Insteadthe squeezed probability distribution in Fig 1 suggests that the probabil-ity of detecting a photon decreases with the more photons that are alreadydetected in the same time interval over which a single measurement is in-tegrated From this observation one must conclude that the photons donot individually appear in a random fashion upon detection There mustbe lsquoquantumrsquo correlations between the photons These correlations mustexisted before detection since there is no interaction between the photonsduring their detection Pre-existing correlations between detected photonsseem to imply that the photons themselves existed before detection ie attimes when interference occurred In a semi-classical description howeverphotons are classical particles and cannot interfere for instance on a beamsplitter At this point the semi-classical picture breaks down Squeezedstates are therefor lsquononclassicalrsquoThe failure of the semi-classical model described above generally certifiesnonclassicality

Squeezed states are usually not characterized by counting their pho-tons but by measuring canonical continuous-variable phase-space observ-ables Measurements are performed as usual on an ensemble of identicalstates and quasi-probability density functions are calculated from the dataThe Glauber-Sudarshan P -function is the quasi-probability density distribu-tion over coherent states If the P -function of a state is entirely positivethe state is a coherent state or a (classical) mixture of coherent states Thestate is considered as semi-classical If the P -function is not a positive-valuedfunction the state cannot be expressed as a (classical) mixture of coherentstates and is thus nonclassical [Gerry and Knight (2005) Vogel and Welsch(2006)] A non-positive-valued P -function is the sufficient and necessary con-dition for the failure of the semi-classical model The Wigner function is thequasi-probability phase-space representation over the canonical continuous-variable phase-space observables themselves [Gerry and Knight (2005)] TheWigner functions of squeezed states are entirely positive Although subject todiscussion this fact does not mean that squeezed states are less nonclassicalthan Fock states or cat states which not only have a nonclassical P -function

7

but also a partially negative Wigner function (A cat state is a quantum su-perposition of two macroscopically distinct states [Monroe (2002)] referringto Schrodingerrsquos-cat gedanken experiment [Schrodinger (1935)]) In practicesqueezed states can even be regarded as superior nonclassical states becausethey represent the only nonclassical state that has been produced in a steadystate fashionIn almost all experiments so far the generation of Fock states and cat statesinvolves a probabilistic event such as the detection of a photon in anotherbeam path to herald these states In fact squeezed states provide the non-classical resource for the probabilistic preparation of Fock states as well as catstates But only the squeezed states themselves show a nonclassical effect in astationary way Limited only by the time duration and the frequency span ofthe mode that is in a squeezed state the squeezing effect can be continuouslyobserved independently of the time when the measurement is performed andalso independently of the measurement integration time This fact is of greatimportance for applications of squeezed states in measurement devices sincea squeezed-light-enhanced measurement remains unconditional and the ef-fective measurement time is not reduced

In past decades squeezed states of light were used in many proof-of-principle experiments to research their potential for improving the sensitivityof laser interferometers [Grangier et al (1987) Xiao et al (1987) McKen-zie et al (2002) Vahlbruch et al (2005) Goda et al (2008) Taylor et al(2013)] or the performance of imaging beyond the shot-noise limit [Lugiatoet al (2002) Treps et al (2003)] both accompanied by a huge number oftheoretical works Potential applications in secure optical communication(quantum key distribution) were also proposed and proof-of-principle ex-periments demonstrated [Ralph (1999) Furrer et al (2012) Gehring et al(2015)] This review restricts itself to the improvement of laser interfero-meters since only here has the application of squeezed light gone beyondproof-of-principle The gravitational-wave detector (GWD) GEO 600 hasoperated with squeezed light now for more than seven years starting in 2010[Abadie (2011) Grote et al (2013)] GEO 600 is a 600 m long Michelsonlaser interferometer built for the detection of gravitational waves Thesewaves are audio-band and sub-audio-band changes of space-time curvatureoriginating from cosmic events such as the merger of neutron stars or blackholes as detected recently [Abbott (2016)] In GWDs such as GEO 600 [Doo-ley et al (2016)] Advanced LIGO [Aasi (2015)] Advanced Virgo [Acernese

8

(2015)] and KAGRA [Aso et al (2013)] conventional laser technology hasbeen pushed to extremes over the past decades Noise spectral densities nor-malized to space-time strain of less than 10minus23 Hzminus12 have been measured[Abbott (2016)] Progress will continue and based on the successful appli-cation in GEO 600 squeezed light is now widely accepted to provide a newadditional technology to contribute to the new field of gravitational-waveastronomy It was also successfully tested in one of the LIGO detectors in2013 [LSC (2013)] and is an integral part of the European design study forthe 10 km Einstein-Telescope [Punturo et al (2010)]

GEO 600 has already taken several years of lsquosqueezedrsquo observational datawhich has increased its sensitivity at signal frequencies above 500 Hz Withthe implementation of a squeezed light source in GEO 600 the applicationof nonclassical states in metrology has been pushed beyond merely proof-of-principle

lsquoTwo-mode squeezed statesrsquo show a squeezed uncertainty in at least onejoint continuous variable of two subsystems lsquoArsquo and lsquoBrsquo Examples of jointvariables are differences and sums of phase-space observables of A and BTwo-mode squeezed states not only belong to the class of nonclassical statesbut due to their bi-partite character also to the class of lsquoinseparablersquo orlsquoentangledrsquo states They are the ideal states to demonstrate the Einstein-Podolsky-Rosen paradox [Einstein et al (1935)] as first achieved in [Ouet al (1992)] Apart from fundamental research on quantum mechanicsrecent proof-of-principle experiments demonstrated their usefulness in inter-ferometric measurements that go beyond the application of simple squeezedstates [Steinlechner et al (2013) Ast et al (2016)] This experiment is thefinal topic of this review

2 Observations on light fields in squeezed states

Generally there are two different kinds of observables that can be subjectof a measurement performed on a quantum system The first kind is associ-ated with the systemrsquos wave property In optics it corresponds to the electricfield strength at a given phase angle ϑ The according (dimensionless) opera-tors are called the quadrature amplitudes Xϑ and have a continuous spectrumof eigenvalues Quadrature amplitudes are measured in very good approx-imation with a balanced homodyne detector using the interference with a

9

bright local oscillator beam see Fig 3 (a) In practice any measurement ofXϑ integrates over some sideband (Fourier) spectrum within the angular fre-quencies Ω plusmn ∆Ω2 The sideband information always needs to be quotedA straight forward but rather untypical way is by adding subscripts whichleads to Xϑ

Ω∆Ω The classical analogue of the quadrature amplitude operator

XϑΩ∆Ω is the modulation depth of the optical field at modulation phase angle

ϑ and at angular modulation frequency Ω measured over the band ∆Ω lt 2ΩThe uncertainties of the statersquos quadrature amplitudes at different phases ϑare limited by a Heisenberg uncertainty relation see section 3 The secondkind of measurement is associated with the systemrsquos particle property andis given by the photon number operator n associated with a measuring timeinterval ∆t Its precise measurement requires a photon counter ideally withsingle photon resolution The measurement result obviously has a discretespectrum Continuous as well as discrete observables are usually subject toquantum uncertainties and thus quantum noiseUsually the measurementrsquos integration time and frequency band actuallydefine the physical system that is characterized In quantum optics experi-ments the interrogated physical system is called a lsquomodersquo

21 Definition of a lsquosingle modersquo

Let us define a light field or generally any quantum system to be asingle mode if it corresponds to the lsquosmallest entity of a waversquo In this caseits spectral and temporal distributions as well as waist size and divergenceare at their Fourier limits and all other properties such as optical axis waistposition and polarization are well defined For instance a linearly polarizedlongitudinal resonance of an optical standing-wave cavity defines such a singlemode if the cavity finesse is high and transversal modes are non-degenerateThe complete photo-electrical detection of a cavity mode however is notstraight forward Most quantum optical experiments are instead performedon propagating light In this case single modes are defined by spatial filtersand by temporal-spectral measurement windows both being at the Fourierlimit Examples for single modes are a laser pulse and a spectraltemporalcutout from a continuous observation of a quasi-monochromatic continuous-wave light beam in the spatial TEM00 mode both at the Fourier limits

In classical physics the only remaining free parameter of a given singlemode is its excitation energy In quantum physics the situation is differentFor a given energy a single mode can be in many different quantum states

10

which differ in their quantum statistics Examples are coherent states num-ber (Fock) states and squeezed states

22 Observations on squeezed states using a single PIN photo-diode

An ideal PIN photo-diode absorbs the full energy of a light mode andproduces one photo electron for every absorbed photon energy It uses theinternal photo-electric effect inside a semiconductor such as silicon or In-GaAs In contrast to avalanche photo-diodes PIN photo-diodes operatewith unity gain lsquoPINrsquo stands for lsquopositiversquo lsquointrinsicrsquo and lsquonegativersquo and isdescribing the doping of the semiconductor layers A PIN photo-diode is op-timally suited for the continuous monitoring of a rather bright light field ofup to several tens of milliwatts An example is the photo-diode in the outputport of a gravitational-wave detector as shown in Fig 2 (a) The prominentwavelength of 1064 nm which is emitted by NdYAG lasers has an opticalfrequency of ν = ω(2π) = 282 middot 1014 Hz The period of the field oscillationis a few femtoseconds and cannot be directly resolved with photo-electric de-tectors However variations of the electric field around the averaged opticalfield oscillation on longer time-scales can be resolved Applying an electronicbandpass filter at the sideband angular frequency Ω plusmn ∆Ω2 to the photovoltage provides information about the lsquodepth of the lightrsquos amplitude mod-ulationrsquo which is also called the lsquoamplitude of the amplitude quadraturersquo Itcan also slowly vary in time and reads

Xϑ=0

Ω∆Ω(t) equiv XΩ∆Ω(t) equiv X (1)

The subscript is usually skipped as it is done with the time dependenceas indicated on the right Applying the electronic bandpass filter in factdefines the mode of the light being detected The structure of the defini-tion in Eq (1) forms the basis of interferometric signals and quantum noisealso in the semi-classical case of coherent states Lets take an example Inthe recent observation of gravitational waves [Fig 1 bottom row in Abbott(2016)] the time-frequency representation of the gravitational-wave signalcorresponded to the amplitude quadrature amplitude XΩ∆Ω(t) of the inter-ferometer output light Note that a larger value of ∆Ω allows for changes ofthe quadrature amplitude on shorter time scales

If the light fieldrsquos lsquomodulation modersquo does not contain any quanta simplybecause there are no photons that have a frequency difference of plusmnΩ with

11

respect to the carrier it is in its ground state In this case lsquovacuum noisersquois observed which originates from the ground state uncertainty Since thevacuum noise only becomes measurable as a beat with a bright light field itcan also be seen as the carrierrsquos band-path filtered shot noise A modulationmode in a displaced vacuum state (a coherent state) corresponds to nonzerocoherent modulationThe measured level of the vacuum noise generally depends on the power ofthe bright carrier light and on the electronic amplification In any case itprovides the reference for certifying lsquosqueezingrsquo Observations using a singlePIN photo-diode require an independent measurement to quantify vacuumnoise A necessary condition is that attenuating the total fieldrsquos light powerresults in the same attenuation of the measured XΩ∆Ω values If they showa stronger attenuation a coherent modulation or thermal noise might bepresent If they show a weaker attenuation the photo-diode and its electron-ics might be saturated

Fig 2 (b) illustrates how a broadband squeezed field improves the mea-surement of an amplitude modulation in time domain based on a PIN photo-diode Shown is a simulated time sequence of XΩ∆Ω-data sampled from thephotoelectric voltage In this simulation all sideband frequencies from zero(DC) to the cutoff frequency of the detector electronics (Ωcut) are included(Ω = ∆Ω2 = Ωcut2) No additional band pass filter is applied making it amaximally broadband detection Although the data in Fig 2 (bi) contains aclassical amplitude modulation of the detected light this signal is not visibledue to random noise here representing shot noise Fig 2 (bii) shows thesame situation but with shot noise that is squeezed over the full detectionband The quantum uncertainty of the modulation depth is squeezed andthe classical signal becomes visible

It needs to be noted that a single PIN photo-diode can only measure theamplitude of the amplitude quadrature XΩ∆Ω(t) but not the non-commutingobservable the lsquoamplitude of the phase quadraturersquo

Xϑ=90

Ω∆Ω (t) equiv YΩ∆Ω(t) equiv Y (2)

For values that are small compared to the field strength of the bright fieldthe quantity Y approximately describes the bright fieldrsquos lsquophase modulationdepthrsquo

12

23 Observations on squeezed states using a balanced homodyne detector

-72

-69

-66

-63(i)

(ii)

(iii)

(a)

LO

(Squeezed)signal input

Phaseshifter

PD1

PD2

5050

01 0200 03

-75

-60

-57Balanced homodyne detector

(b)

Time [s]

Noi

se p

ower

[dB

m]

Figure 3 Balanced homodyne detection (BHD) ndash (a) Setup The quadrature atchoosable angle ϑ of the signal field is measured by overlapping the latter with a localoscillator (LO) field of the same mode parameters on a balanced beam splitter and record-ing the difference voltage from two PIN photo-diodes as shown In order to meet theBHD approximation the LO needs to be much more intense than the signal field A closeto perfect mode overlap between LO and signal input field is crucial For a non-perfectoverlap the detector measures the input state with unwanted contributions of the vacuumstate (b) Noise power measurements (i) on an electronically amplified and band-passfiltered quadrature amplitude of the vacuum field (Xvac

Ω∆Ω) (signal input blocked) (ii) on

a squeezed quadrature (XsqzΩ∆Ω) of a squeezed vacuum state (ϑ = 0) and (iii) on re-

spective quadratures of the same squeezed state where the phase angle ϑ was continuouslyshifted by changing the optical path length of the LO The measurement data shows about5 dB of squeezing and was first published in [Chelkowski et al (2007)] Ω2π = 5 MHz∆Ω2π = 100 kHz

In contrast to a single PIN photo diode a balanced homodyne detector(BHD) is suitable to measure the quantum statistic of all types of modu-lations ie for all angles ϑ Such a detector consists of two identical PINphoto-diodes a balanced beam splitter and an external homodyne local os-cillator field that is much brighter than the signal beam and that has anadjustable phase The signal beam corresponds to the squeezed field whichin many experiments is in a squeezed vacuum field having an optical powerthat usually corresponds to just a few photons per mode The two beams areoverlapped on the balanced beam splitter with close to perfect mode match-ing and the two interference outputs are focussed onto the photo diodes seeFig 3 (left) The electric output signal of the BHD is the difference of the

13

photo diode voltages The LO takes over the role of the carrier light fieldbut with the possibility to choose the phase shift ϑ This way eigenvaluesof X Y or Xϑ can be measured where the latter is given by the followinglinear combination of the first two

Xϑ(t) = cos(ϑ) X(t) + sin(ϑ) Y(t) (3)

If the modulation depths of signal and local oscillator beams are weak com-pared to their coherent amplitudes |α| and |αLO| the output voltage of aBHD corresponds to eigenvalues of the following operator

V (t) prop 2cos(ϑ) |αLO||α|+ |αLO| Xϑ(t) + |α| XϑLO(t) (4)

The lsquohomodyne approximationrsquo further involves |αLO| |α| such that theterm on the right can be neglected even if the local oscillator shows someclassical quadrature excitation The output voltage of a BHD is usually spec-trally analysed or at least spectrally filtered which removes the DC part infull analogy to a single photo diode (see previous subsection) Sampling thefiltered voltage provides eigenvalues proportional to the generalized quadra-ture amplitude in Eq (3)

V BHDΩ∆Ω(t) prop |αLO| Xϑ

Ω∆Ω(t) (5)

Fig 3 (a) shows the setup of a balanced homodyne detector for the char-acterization of squeezed states Setting ϑ = 0 eigenvalues of the ampli-tude modulation depths can be sampled from the photo voltage accordingto Eq (5) Setting ϑ = 90 eigenvalues of the phase modulation depths aremeasured The datarsquos expectation values 〈Xϑ〉 provide the coherent displace-ment of the squeezed state The datarsquos variances

∆2Xϑ equiv 〈(Xϑ)2〉 minus 〈Xϑ〉2 (6)

provide the statersquos (quantum) noise A pure squeezed state as well as asqueezed state that experienced photon loss have Gaussian quantum statisticsand are thus fully described by the expectation values and variances (first andsecond moments) of two orthogonal quadratures but only if one quadraturereflects the lowest quadrature variance

14

In most experiments with squeezed light the photo electric voltage ac-cording to Eq (5) is not sampled with a data aquisition system but the signalis directly fed into a spectrum analyser measuring the noise power of the volt-age If the expectation value 〈Xϑ〉 is zero the noise power is proportionalto the variance ∆2Xϑ in Eq (6) The reference for quantifying the squeezefactor is measured by blocking the (squeezed) signal field in Fig 3 (a) Themeasured vacuum noise level corresponds to the LOrsquos (electronically ampli-fied) shot noise level

Traces (ii) and (iii) in Fig 3 (b) show measured noise powers of the mod-ulation mode (Ω2π = 5 MHz ∆Ω2π = 100 kHz) being in a squeezed vac-uum state (i) is proportional to the variance of the ground state uncertainty∆2Xvac

Ω∆Ω (ii) is proportional to the quantum noise variance of the squeezed

quadrature amplitude ∆2XsqzΩ∆Ω (iii) is proportional to the quantum noise

variance of the quadrature amplitude with scanned phase ∆2XΩ∆Ω(ϑ(t))

To fully characterize a quantum state ie to do quantum state tomogra-phy [Vogel and Risken (1989)] a BHD is a prerequisite But also interfero-metric measurements with balanced homodyne detectors instead of singlePIN photo-diodes have several advantages A correctly implemented BHDreadily provides the vacuum noise level when the signal beam is blockedWith a BHD the optimum operating point of the interferometer is preciselyat a dark fringe If a perfect dark fringe can practically be achieved ampli-tude noise of the laser does not couple into the signal port If the interfero-meter has balanced arm length also frequency noise of the laser then does notcouple into the signal port Some quantum non-demolition schemes with theprospect of evading quantum radiation pressure noise require the detection ofa non-canonical quadrature angle [Jaekel and Reynaud (1990) Kimble et al(2001)] Here the adjustable phase of a BHD provides a straight forwardapproach The experimental exploration of BHDs for gravitational-wave de-tectors only has started recently [Steinlechner et al (2015)]

A light field can be analysed with respect to many different modulationfrequencies Ω The result constitutes a spectrum [Breitenbach et al (1998)]where in principle every modulation mode can be in a different quantumstate Fig 4 shows spectra of squeezed states from 5 MHz to 100 MHz with∆Ω2π = 1 MHz The lower curve shows the spectrum of the most strongly

15

-14-12-10-8-6- 4-20 2 4 6 8

10 12 14 16 18

6 8 10 20 40 60 80 1005

Squeezed noise

Anti-squeezed noise

Vacuum noise

Frequency [MHz]

Nor

mal

ized

noi

se v

aria

nce

[dB

]

Figure 4 Spectrum of quadrature amplitude variances ndash Shown are the quantumnoise properties of a large number of modulation modes having a resolution bandwidth of∆Ω(2π) = 1 MHz For all traces the balanced homodyne detector output was analysedwith a spectrum analyser Squeezing of XΩ∆Ω (bottom trace) and anti-squeezing of

YΩ∆Ω (top trace) versus f = Ω(2π) are shown relative to the vacuum noise variance Thespectrum below 5 MHz is not shown since it contained less squeezing due to laser relaxationoscillation in the carrier field as well as disturbances from back-scattered light [Vahlbruchet al (2007)] Disturbances at frequencies above 70 MHz originated from relatively largedetector dark noise which was subtracted from all traces shown here The thin linerepresents a theoretical model that takes into account for the linewidth of the squeezingcavity The data was first presented in in Ref [Mehmet et al (2010)]

squeezed variances in this case the variances of XΩ∆Ω The upper spectrum

shows the variance in the orthogonal quadrature amplitude (YΩ∆Ω) Allvariances are normalized to those of the corresponding vacuum state Thesqueeze factor reduces towards higher frequencies due to the linewidth ofthe squeezing cavity The anti-squeezing is always higher than the absolutevalue of the squeezing due to Heisenbergrsquos uncertainty relation and due to thepresence of optical loss The curves do not represent pure squeezed states butmixed squeezed states with a significant contribution from vacuum statesdue to optical loss Pure squeezed states can only be produced by making

16

the influence of all decoherence processes negligibleThe choice of the resolution bandwidth (RBW ∆Ω) during data taking

and processing defines the spectral-temporal modulation modes includingtheir number within the detected spectrum For any setting of the RBWthe quantum mechanical properties of the quadrature amplitudes XΩ∆Ω and

YΩ∆Ω [Caves (1985)] fully correspond to those introduced for quadratures instandard text books and which are reviewed in Sec 3

24 Observations on two-mode squeezed states using balanced homodyne de-tectors

Two-mode squeezed states are composed of two subsystems lsquoArsquo and lsquoBrsquoand are bi-partite entangled states with a Gaussian quantum statistic Toavoid conflicts with different usage of the term lsquomodersquo they can synony-mously be named lsquobipartite Gaussian entangled statesrsquo or lsquobipartite squeezedstatesrsquo which will be mainly used in this Review In the same way multi-partite Gaussian entangled states correspond to multi-partite squeezed states

The measurement observables that prove or disprove the bi-partite squeez-ing property are XA

Ω∆Ω minus XBΩ∆Ω and Y A

Ω∆Ω + Y BΩ∆Ω where the minus and

plus signs may be swapped Bi-partite squeezed states are precisely thosestates that were discussed by Einstein Podolsky and Rosen (EPR) in theirseminal paper [Einstein et al (1935)] Fig 5 shows a measurement resulton bi-partite squeezed light [Eberle et al (2013)] The variances of bothjoined observables are squeezed as shown in the two lower traces They wererecorded consecutively by adding or subtracting the outputs of two balancedhomodyne detectors But by interfering the subsystems on a beam splitterone could even measure both joined observables simultaneously This pos-sibility is correctly described in quantum theory since their commutator iszero

The so-called EPR paradox arises as follows If we either measure XAΩ∆Ω

and XBΩ∆Ω or Y A

Ω∆Ω and Y BΩ∆Ω it is obvious from the data in Fig 5 that

we can always predict the measurement result at subsystem lsquoBrsquo when know-ing the result at subsystem lsquoArsquo This seems to suggest that both quantitiesat lsquoBrsquo are precisely defined simultaneously before the measurement on lsquoArsquowhich contradicts the rigorous (and correct) interpretation of their non-zerocommutator that they are not precisely defined simultaneously

To solve this paradox EPR conjectured that the wavefunction as definedby quantum theory does not provide the full information This led to a

17

discussion of whether hidden variables existed that needed to be included ina complete theory of quantum mechanics (see also Bell [Bell (1966)]) Theexperimentally observed violation of Bellrsquos inequality [Bell (1964) Aspectet al (1981) Giustina et al (2013) Hensen et al (2015)] however ruled outthe existence of (local) hidden variables

Based on that the EPR paradox needs to be solved in a different wayContrary to what EPR assumed it is in fact possible to predict the valueof an arbitrary observable of a physical system A with certainty via a mea-surement on system B although this observable was not defined before themeasurement Without any interaction a measurement on subsystem lsquoArsquo notonly creates lsquorealityrsquo of eg XA

Ω∆Ω simultaneously lsquorealityrsquo is also created

regarding the observable XBΩ∆Ω describing subsystem lsquoBrsquo Here the term

lsquorealityrsquo has the meaning as defined by EPR [Einstein et al (1935)] Simi-larly the detection of one photon of a two photon entangled number statenot only produces the reality of this photon but also that of a second oneA discussion of Einstein-Podolsky-Rosen entanglement can also be found in[Schnabel (2015)] Note that the EPR paradox can also be described aslsquoquantum steeringrsquo [Schrodinger (1935) Cavalcanti et al (2009) Handchenet al (2012)] It should also be mentioned that two-mode squeezing beingdetected with BHDs and not with photon counters cannot be used to violatea Bell inequality The latter topic is outside the scope of this Review

Bi-partite squeezed states were first characterized with balanced homo-dyne detectors by the group of J Kimble in 1992 [Ou et al (1992)] Gener-ally the EPR paradox becomes more pronounced the stronger the bi-partitesqueezing is A measure of the strength of EPR entanglement was introducedby M Reid [Reid and Walls (1985)] According to this measure the resultin Fig 5 can be quantified to ε2 = 00309 where the critical value is one Itcorresponds to the strongest Gaussian EPR entangled state generated so far

For a long time it looked like that two-mode squeezed states are notuseful for laser interferometers The reason for that belief was that a laserinterferometer as any other measurement device too is built to measureone observable It seems to be ideal already if the quantum noise in thissingle observable is squeezed The increased quantum noise in the orthog-onal observable is not harmful in this case and squeezing in two differentobservables useless Only recently realistic scenarios were discussed in whichtwo-mode squeezing in fact does improve the performance of a laser inter-

18

-12

-10

-8

-6

-4

-2

0

0 1 2 3 4 5 6 7 8 9 10

100

dB

109

dB

Δ (XAvac + X B

vac ) = Δ (Y

Δ (XA + X B)

Avacminus Y B

vac )2 2

2

Δ (YAminus Y B)2

Nor

mal

ized

noi

se v

aria

nce

[dB

]

Times [s]

Figure 5 Two-mode squeezing measurement ndash For this measurement the outputs oftwo balanced homodyne detectors are added or subtracted and the variances (noise pow-ers) of the results recorded The upper trace was measured with modes lsquoArsquo and lsquoBrsquo beingin their ground states This measurement served as a reference level Strong two-modesqueezing was observed as shown by the lower two traces The sideband frequency wasΩ(2π) = 8 MHz and the resolution bandwidth was ∆Ω(2π) = 200 kHz The measure-ment results were first published in Ref [Eberle et al (2013)]

ferometer [Steinlechner et al (2013)] The proof-of-principle experiment isreviewed in Sec 7

25 Observations using photon counters

Alternatively to field quadratures an optical mode in a squeezed statecan also be characterized at least partly by detecting its photon numberdistribution For a pure squeezed vacuum state such a measurement wouldreveal the existence of solely even photon numbers including a large prob-ability for zero photons The average photon numbers of squeezed vacuumstates with feasible squeeze factors are very small of the order of one persecond and bandwidth in hertz see Fig 13 (a) ndash (c) A distribution with closeto zero probability of odd photon numbers however has not been measuredso far The reason is the lack of ideal photon counters First of all theefficiency of these detectors ie their probability of converting one photoninto one click and no photon into no click must be almost perfect lsquoLostrsquophotons as well as dark counts wash out the oddeven oscillations Further-more most detectors available can only distinguish between zero and one

19

photon This problem can be solved by distributing the squeezed mode ontoa large number of single photon detectors using an array of beam splitterssuch that all paths have a low probability of carrying more than one pho-ton Photon number measurements on squeezed vacuum states neverthelessplay an extremely important role in quantum optics When the squeezingstrength is very low the probability of detecting more than 2 photons can beneglected and the detection of a photon heralds the existence of a second one

0

300

600

900

1200

-40 -20 0 20 400

5

10

15

20

25

Two-

fold

coi

ncid

ence

s [1

(4s

)](a) (b)

Non-degeneratetwo-mode squeezing

5050

APDA1

APDA2

APDHerald

Photon coincidencedetection

Delay [ns]

Thr

ee-f

old

coin

cide

nces

[1

(4s)

]1500

Figure 6 Coincidence clicks from non-degenerate photon pairs ndash The first suchexperiment was reported in Ref [Hong and Mandel (1986)] (a) shows a setup with threeavalanche photo-diodes (APDs) for proving the successful heralding of a single photonnumber state (b) Histograms of the two-fold coincidence detections at APDHerald andAPDA1 (red) and at APDHerald and APDA2 (yellow) with theoretical models (solid lines)If the two-mode squeezing just carried one photon in each spatial subsystem the three-foldcoincidence detection should be zero Indeed the according histogram (grey points righty axis) shows only a few events These are produced by false (dark) counts of the APDsThe delay for the three-fold coincidences is defined as the time between counts at lsquoA1rsquo andlsquoA2rsquo given that the trigger APDHerald detected a photon (within a 100 ns time window)The data was taken on photons that were up-converted from 1550 nm to 532 nm and itwas first published in Ref [Baune et al (2014)]

If a mode of light is always excited by either zero or two photons lsquocondi-tionalrsquo or lsquoheraldedrsquo one-photon Fock states can be realized (Measurementson an ensemble of the n-photon Fock state would always produce the mea-surement result n ie Fock states have a zero photon number uncertaintyThey are also called lsquonumber statesrsquo) The above concept of producing aone-photon Fock state obviously requires the deterministic and balanced dis-tribution of the down-converted signal and idler fields into two different paths

20

In order to achieve this the signal and idler fields need to be non-degenerateUsually a mode in a squeezed state is composed of degenerate signal andidler fields and this degeneracy thus needs to be removed Possible waysare producing the down-converted fields at well separated wavelengths [Vil-lar et al (2005) Su et al (2006) Li et al (2010) Samblowski et al (2011)]separating the upper and lower sidebands belonging to an ordinary squeezedmode by frequency filters [Schori et al (2002) Hage et al (2010)] and us-ing spatial filters [Hong et al (1987)] A frequently used approach is usingtype II parametric down-conversion where the photons within a pair are al-ways orthogonally polarized [Ou et al (1992) Kiess et al (1993) Kwiat et al(1995)]The list of experiments with conditional or heralded photon number states islong They showed for instance nonclassical g(2)-functions [Hong et al (1987)]and violations of Bell inequalities [Weihs et al (1998)] Fig (6) shows a re-sult from a more recent experiment in which a bipartite-squeezed state withsubsystems at 1550 nm and 810 nm was produced the subsystem at 1550 nmsubsequently up-converted to 532 nm and the lsquoquantum non-Gaussianityrsquo ofheralded up-converted single photons demonstrated [Baune et al (2014)]Squeezed states are also the resource for the conditional generation of super-positions of coherent states [Ourjoumtsev et al (2006) Neergaard-Nielsenet al (2006)] and so-called N00N-states [Afek et al (2010)]

The generation of nonclassical states mentioned in the paragraph aboveis not stationary but relies on a probabilistic trigger event The produc-tion of squeezed states themselves usually happens in a stationary fashionThis distinction has an important consequence for applications of nonclassi-cal states in measurement devices Only (stationary) squeezed states allowfor a continuous improvement of a measurement Avoiding any loss of mea-suring time is generally of high relevance for the detection of short-livedsignals with unknown arrival time as well as for the detection of long-livedquasi-monochromatic signals since the signal-to-noise-ratio (SN) improveswith measuring time

26 Conclusions

The detection of squeezed light produces measurement results that canbe considered as remarkable Let us focus on experiments where a mode in abright coherent state is overlapped with a mode in a squeezed vacuum state

21

as shown in Figs (1) and (3) In both setups the squeezed vacuum field caneasily be blocked which allows us to compare the measurement results ona bright coherent state with and without the interference with the squeezedvacuum state Without squeezing the photo-electric detectors measure alarge number of photon events with a large quantization noise (shot noise)The large noise reflects the fact that all photon events were independent fromeach other as shown in Fig 2 (bi) With squeezing the photo-electric de-tectors again measure a large number of photon events with an expectationvalue that is even slightly higher but nevertheless the quantization noise ofall detected photons is significantly reduced Fig 2 (bii)Based on the discussion of EPR entanglement in Subsec 24 the photo-electric detection of the output light of a squeezing-enhanced laser inter-ferometer (with αlowastα1) produces the reality of photons This way we cankeep the lsquowave picturersquo in which no photons exist when light travels alongthe interferometer arms and when it interferes at the beam splitter Whenthe energy of the beam is elevating electrons to the conductance band ofthe photo-diodersquos semi-conductor n photon events simultaneously appearwithin the measuring interval with probability P (n) What conclusion hasto be drawn if the probabilities resemble a sub-poissonian statistic ndash Theoccurrence of photon events is still truly random but in this case not forindividual photons The occurrence of photons is correlated in such a waythat the probability of detecting an additional photon in the same time in-terval reduces the larger the number of already detected photons is Whatfollows from the discussion of EPR entanglement for a photon counting ex-periment with pure squeezed vacuum and ideal photon counters Here theprobabilistic detection of one photon entails the detection of a second onewith certainty With some smaller probability a third photon is detectedwhich entails the detection of a fourth photon with certainty and so on

If a photon of a mode that was not interrogated by the environment beforeis absorbed its reality is created in this very moment If the photon belongsto a squeezed state this process instantaneously influences the probability ofother photons becoming reality

Of course a more general statement can be made based on the insightthat interaction with the environment creates the reality of any kind ofquanta including electrons atoms and molecules

22

3 Theoretical description of squeezed states

31 The quadrature amplitude operators

Consider a single mode of light at optical frequency ω Its Hamiltonoperator reads

Hω = ~ω(n+

1

2

)= ~ω

(adaggerωaω +

1

2

)= ~ω

(X2ω + Y 2

ω

) (7)

where n is the photon number operator and aω and adaggerω are the annihilationand creation operators which obey the commutation rule

[aω a

daggerω

]= 1 The

operator aω has a complex-valued dimensionless eigenvalue spectrum andcorresponds to the complex amplitude αω in classical optics Xω and Yω arethe hermitian amplitude and phase quadrature operators The eigenvaluesof the quadrature operators are also dimensionless and proportional to theelectric fields at the oscillationrsquos antinode and at the oscillationrsquos node Inthe above equation they are defined such that their variances are ∆2Xω =∆2Yω = 14 if the oscillator is in its ground state ie if 〈n〉 = 0

Although Eq (7) simply describes the energy of an harmonic oscillator itis the essence of quantum theory since it mathematically describes the wave-particle dualism Whereas the eigenvalues of n have a discrete spectrum theeigenvalues of Xω and Yω have a continuous spectrum In classical opticsthe phase quadrature is zero In quantum optics its expectation value is alsozero but its uncertainty contributes to the overall energy

Eq (7) describes a cavity mode as well as a section that is cut from apropagating quasi-monochromatic light beam The latter example is of highrelevance in actual experiments By setting the sectionrsquos time window ie themeasuring time interval the time-frequency (lsquomodulationrsquo) mode is defined

The quadrature operators introduced in Eq (7) and displayed in Fig 7 donot correspond to lsquoXrsquo and lsquoY rsquo that are of relevance in laser interferometry andin optical communication and which were already discussed in Subsec 22and 23 The optical frequency of visible and near-infrared light is far too highto be transferred to an oscillation of photoelectric voltage Quite general alaser interferometer targets signals at audio or radio band frequencies Ωi ωSuch a measurement is achieved as stated before by decomposing the photo-electric voltage from the photo diode at the interferometer output into asingle-sided spectrum (positive frequencies only) of intervals of Ωplusmn∆Ω2

23

(a)

(b)

(c)

(d)

2π 4π

2π 4π

2π 4π

2π 4π

Figure 7 Phase spaces and electric field oscillations of monochromatic lightndash Top Left Monochromatic light in a coherent state is represented by a phasor (whitearrow) including its quantum uncertainty (white dashed circle and fuzzy area) located inthe phase-space spanned by the quadratures Xω and Yω When the phase space rotateswith optical frequency ω2π the projection of the quantum phasor onto a fixed (vertical)axis corresponds to the electric field E(t) as shown on the right side (a) Weakly displacedcoherent state (b) Corresponding amplitude squeezed state The electric field uncertaintyaround the zero average field region is anti-squeezed (c) Vacuum state at the same opticalfrequency (d) Corresponding squeezed vacuum state The meaning of the uncertaintycould be carved out by supplementing them with monochromatic waves all having theoptical frequency ω2π Changing amplitudes then display amplitude quadrature noiseChanging shifts along the time axis model the electric field uncertainty at the expectedzero crossing They are not implemented in the graphics here however since any of thosewaves does not exist due to Heisenbergrsquos uncertainty relation

24

The signals as well as the quantum uncertainties carried by a beam oflight are thus described by a spectrum of pairs of non-commuting quadratureoperators Mathematically every such operator is defined by an integral overthe Fourier components within the bandwidth The spectral weighting of theFourier components is called the lsquowindow functionrsquo By going to sidebandintervals a spectrum of a new type of optical mode is defined which describesthe modulation of the electric field in the respective frequency interval Ω plusmn∆Ω2 In this Review we call it a lsquomodulation modersquo

The quadrature operators that are defined around a modulation frequencyΩ with a bandwidth of ∆Ω are the quadrature amplitude operators thatare relevant in laser interferometry Whenever they are not related to aspecific band we use the short form XΩ∆Ω(t) equiv X and YΩ∆Ω(t) equiv Y cfEqs (1) and (2) These operators can slowly vary with time where thetime dependence is limited by ∆Ω (The time dependence is not due toquantum uncertainty which usually is time independent but for instancedue to the time dependence of the signal eg a passing gravitational wave)Let us consider now a pair of quadrature operators for a particular sidebandΩ plusmn ∆Ω2 The Hamilton operator of the corresponding modulation modeis found by switching to the frame rotating at optical frequency ω Thetransition is done by applying the unitary transformation U = exp(iωadaggerat)generating a new Hamiltonian H = U daggerHωU minus i~UpartU daggerpartt The Hamiltonianof the modulation mode reads

H = ~Ω

(nΩ +

1

2

)= ~Ω

(adaggera+

1

2

)= ~Ω

(X2 + Y 2

) (8)

where nΩ is the (occupation) number operator for the modulation modeand a and adagger its annihilation and creation operators The commutation rule[a adagger

]= 1 is unchanged X and Y are the amplitude and phase quadra-

ture amplitude operators respectively They correspond to the depth of theamplitude modulation and for weak excitations to the depth of the phasemodulation respectively They are the conventional hermitian field operatorsin experimental quantum optics Note that modulation modes at angularfrequency Ω can be described by a superposition of three optical frequenciesa carrier at ω an upper sideband at ω + Ω and a lower sideband at ω minus ΩThe quantum mechanical description of modulation states in connection tooptical carrier and upper and lower sidebands is known as the lsquoTwo-PhotonFormalismrsquo [Caves and Schumaker (1985) Schumaker and Caves (1985)]

The quadrature amplitude operators in Eq (8) are again defined such

25

that the variances of the uncertainty of a modulation field in its ground stateor in a coherent state are

∆2Xvac = ∆2Yvac = 14 (9)

Generally quadrature operators X and Y as defined in Eqs (7) and (8) arethe real and imaginary parts of the annihilation operator

a = X + iY hArr adagger = X minus iY (10)

hArr X =1

2

(a+ adagger

) Y =

1

2i

(aminus adagger

) (11)

They satisfy the commutation relation[X Y

]=i

2 (12)

and their variances are limited by a Heisenberg uncertainty relation of thefollowing form

∆2X∆2Y ge 1

16 (13)

A quantum state is called a ldquosqueezed staterdquo [Bachor and Ralph (2004)]if ∆2Xϑ lt 14 for an arbitrary field quadrature Xϑ = X cosϑ + Y sinϑ seeEq (3) The angle of the lowest variance below 14 is called the squeeze angleθ The largest factor by which the variance is below 14 is called the squeezefactor often given on a decibel (dB) scale using the following transformation

minus 10 middot log10

(∆2Xθ

∆2Xvac

) (14)

The squeeze factor can also be described by the squeeze parameter r

eminus2r =∆2Xθ

∆2Xvac

(15)

When a squeezed state experiences optical loss it remains squeezed butthe squeeze factor is reduced Also the statersquos purity is reduced ie theproduct of the quadrature uncertainties increases above the minimum valueOptical loss corresponds to mixing the state with the vacuum state Let∆2Xϑ be the variance of a quadrature amplitude ∆2Xvac the variance of the

26

(quadrature angle independent) ground state uncertainty and (1 minus η2) therelative energy loss Then the resulting quadrature variance reads

∆2Xprime

ϑ = η2∆2Xϑ + (1minus η2)∆2Xvac (16)

To maximize the benefit from squeezed states in applications stronglysqueezed states need to be generated and optical loss minimized Opticalloss occurs due to absorption and scattering in the optical components inthe path of the squeezed beam including the squeezing resonator itself anddue to non-perfect matching to the interferometer mode non-perfect inter-ference contrast of the interferometer and non-perfect quantum efficiency ofthe photo diodes The sum of all losses including those outside the inter-ferometer need to be less then 10 to allow a nonclassical quantum noisesuppression of a factor of 10 in power ie 10 dB

32 Phase space representations of squeezed states

The Wigner function ndash The properties of squeezed states are nicely dis-played by the Wigner function W (X Y ) [Wigner (1932)] An example interms of a squeezed vacuum state is shown in Fig 8 It is a quasi-probabilitydistribution which contains the statersquos full information including its quan-tum statistic There are two ways how a Wigner function provides a sufficientcriterion for nonclassicality First by containing negative values second byfeatures that have a smaller (squeezed) width compared with the Wignerfunction of the ground state Integrating the Wigner function over Y pro-vides the probability density of measurement results ie of the eigenvaluesof the observable X and vice versa

infinintminusinfin

W (X Y )dY = p(X)

infinintminusinfin

W (X Y )dX = p(Y ) (17)

where p(X) and p(Y ) are the observed probability distributions also exem-plarily shown in Fig 8

The ground state coherent states as well as (quadrature) squeezed stateshave quadrature eigenvalue probability densities that are Gaussian TheirWigner functions are also Gaussian and thus entirely positive Wigner func-tions of other nonclassical states for instance Fock states exhibit negativevalues For this reason the Wigner function is called a quasi -probabilityfunction

27

X Y XY

Figure 8 Wigner function and its projections ndash Displayed is the full information ofa squeezed vacuum state The quasi probability density of the Wigner function (centre)is plotted along a third dimension and is color-coded Also shown are the actual mea-surement results from which the Wigner function is reconstructed They are representedby the squeezed and anti-squeezed Gaussian projections onto the X and Y axes Theirprobability densities p(X) and p(Y ) are given on the vertical axes The Gaussian measure-ment statistic of the first clearly shows squeezing compared to the ground state statistic(dashed) The squeeze factor is 116 dB and the squeeze parameter r = 1335 [Mehmetet al (2010)]

Fig 9 shows the Wigner functions for (a) a coherent state (b) the ground(vacuum) state (c) a displaced squeezed state and (d) a squeezed vacuumstate All Wigner functions describe a modulation of the carrier light atsideband frequency Ω integrated over the frequency interval ∆Ω The car-rier light is not part of these Wigner functions The displacement in (a)represents a classical amplitude modulation (b) corresponds to the absenceof any photons with a frequency offset of plusmnΩ from the local oscillator field(c) and (d) represent states whose amplitude modulation depth is more pre-cisely defined than that of the ground state Fig 10 shows Wigner functionspectrum for a broadband squeezed vacuum field Every Wigner functiondescribes the modulation field at some modulation frequency Ωi integrated

28

X

Y

X

(a) (b)

X X

(c) (d)

Y

Y Y

Figure 9 Simplified representation of Wigner functions ndash The darker the shadedareas the larger is the phase-space quasi-probability Shown are four different (time-independent) states of a modulation mode at frequency Ω for a specific resolution bandwidth ∆Ω Panel (a) represents a coherent state the displacement (α) corresponds to aclassical amplitude modulation Panel (b) represents the ground (vacuum) state (c) adisplaced squeezed state and (d) a squeezed vacuum state both with squeeze angle θ = 0The latter is in analogy to Fig 8 Again the light field that carries the modulation is notpart of the pictures

over the resolution bandwidth (RBW) of ∆Ω

The Glauber-Sudarshan P -function ndash The P -function [Glauber (1963)Sudarshan (1963)] is calculated by de-convoluting the Wigner function fromthe ground state uncertainty [Gerry and Knight (2005)] For displaced vac-uum states (coherent states) the P -function corresponds to a displaced δ-function The mathematical expression of the P -function of a squeezed statecontains infinitely high orders of derivatives of the δ-function [Vogel andWelsch (2006)] Such a function contains negativities but cannot be dis-played It is possible however to define a phase-space quasi probabilityfunction for squeezed states that can be displayed and that does show neg-

29

0

Ω1

Ω2

Ω

Single-sided spectrum

Y

X

Figure 10 Hint of a Wigner function spectrum ndash A single-sided spectrum (positivefrequencies only) with respect to the carrier field can be used to visualize a broadbandsqueezed field Shown are two examples displaying a squeezed vacuum state at Ω1 anda displaced amplitude squeezed state at Ω2 The individual Wigner functions cover theresolution bandwidth ∆Ω gt 0 (not shown) In general the squeezing strength as well asthe squeeze angle and the displacement are a function of sideband frequency

ativities as a sufficient and necessary condition for certifying the squeezingeffect This lsquononclassicality functionrsquo is calculated by de-convoluting theWigner function from an uncertainty distribution that is steeper than theGaussian distribution A pronounced negativity of a squeezed vacuum stateof up to 69 standard deviations was found [Kiesel et al (2011)]

The double-sided phasor picture ndash This phasor picture links quantumstates of modulations with the quantum states of the contributing opticalfields [Bachor and Ralph (2004)] and is mathematically described by thetwo-photon-formalism [Caves and Schumaker (1985) Schumaker and Caves(1985)] Generally a weak amplitude or phase modulation at frequency Ω ofa carrier field at optical frequency ω can be understood as the carrierrsquos beatwith two optical frequencies at ωplusmnΩ The double-sided phasor picture is ableto display a spectrum of different and independent modulation frequenciesin the rotating frame of the carrier field The carrier light field is time-independent but the upper and lower sidebands are not They rotate withplusmnΩi(2π) respectively around the frequency axis

30

ω0

Yω

Xω

Upper sideband

Lower sideband

ω0+Ω1

ω0+Ω2

ω0-Ω2

ω0-Ω1

ω

Upper sideband

Lower sideband

Yω

Xω

ω0+Ω1

ω0+Ω2

ω0-Ω2

ω0-Ω1

ω

ω0

Figure 11 Double-sided phasor pictures ndash Phase spaces at optical frequency ω0 plusmnΩirotate around the frequency axis with sideband frequency plusmnΩi Its sign determines thedirection of rotation with respect to the local oscillator in the rotating frame at ω0 Apair of phase spaces need to be superposed to provide a description of a modulation fieldat |Ωi| Top Amplitude quadrature squeezed field with a coherent displacement at |Ω2|The displacement corresponds to a classical amplitude modulation The uncertaintiesof all optical frequencies are circular and larger than that of the ground state (dashed)Quantum correlations are indicated by additional symbols More details are given in themain text Bottom Corresponding spectrum of (displaced) vacuum states which do nothave any quantum correlations

31

Fig 11 shows such a double-sided phase space picture where the carrierrsquosmodulation at Ω1 is in a squeezed vacuum state and where the modulationat Ω2 is in a displaced squeezed state The picture shows how a classicalamplitude modulation as well as the quantum statistic of a modulation fieldis decomposed into contributions from upper and lower sidebands For asqueezed modulation field the upper and lower sidebands show no squeezedbut circular thermally excited quantum uncertainties The uncertainties of apair of sidebands however show correlations as well as anti-correlations InFig 11 these (anti-) correlations are marked with times and + for the modulationfrequency Ω1 and with N and bull for the modulation frequency Ω2

33 Covariance matrix representation of (single-party) squeezed states

Since squeezed states have a Gaussian quantum statistic four numbersare sufficient for their full description These numbers are the second mo-ment of the quadrature amplitude showing the strongest squeezing and thesecond moment of its orthogonal quadrature amplitude as well as their firstmoments describing the displacement These four numbers are sufficientto calculate the Wigner function shown in Fig 8 In general the quadra-ture of strongest squeezing is not perfectly aligned with one of the axesof the measurementrsquos coordinate system The so-called covariance matrix(VXXVXY VY XVY Y ) [Simon et al (1994)] accounts for phase space rotationsand enables the calculation of how these states evolve within an interfero-metric arrangement Their components are normalized to the vacuum noisevariance ∆2Xvac = ∆2Yvac and read

VXY =

langXY + Y X

rangminus 2

langXranglang

Yrang

2∆2Xvac

(18)

The following examples represent the ground state a pure 10 dB am-plitude quadrature squeezed state and a pure 10 dB squeezed state with asqueeze angle of 45

Vvac =

(1 00 1

) V0

01 =

(01 00 10

) V45

01 =

(505 495495 505

) (19)

with V4501 =RT

45V001R45 where Rα = (cosα minussinα sinα cosα) is the rota-

tion matrix

32

34 Phase space representation of two-mode (bi-partite) squeezed states

XA

YA

XB

YB

Figure 12 Bi-partite squeezed vacuum state ndash Shown is a Wigner-function-basedphase space representation in close analogy to the single party version in Fig 9 (d) Thepicture describes a single modulation at frequency Ω with bandwidth ∆Ω Measurementsat party A and B reveal local Wigner functions that correspond to thermal states sincethe uncertainties (indicated by the color and by the large circles) are symmetric and largerthan that of the ground state (indicated by the small dashed circles) The uncertaintieshowever show correlations and anti-correlations here indicated by + and times respectivelyThe strength of these (anti-)correlations are indicated by ellipses Bi-partite squeezingie entanglement is present if the short axes of the ellipses are shorter than the diameterof the ground state uncertainty The picture in fact represents Einstein-Podolsky-Rosenentanglement [Einstein et al (1935)] From a measurement of XA or YA the correspondingmeasurement result at party B can always be inferred with an uncertainty that is smallerthan the ground state uncertainty

A bi-partite state enables a measurement on subsystem A and simulta-neous a measurement on subsystem B For a large number of simultaneousensemble measurements of the same quadrature amplitude Xϑ the followingtwo joint quadrature variance can be calculated

∆2(XAϑ plusmn XB

ϑ ) (20)

A state that is symmetrically shared between two parties (A and B) is calleda two-mode squeezed state if the variances of joint quadrature measurementsfulfill the following inequality [Duan et al (2000)] ie

∆2(XA minus XB)

∆2(XAvac minus XB

vac)+

∆2(Y A + Y B)

∆2(Y Avac + Y B

vac)lt 2 (21)

33

with ∆2(XAvac plusmn XB

vac) = ∆2(Y Avac plusmn Y B

vac) = 2∆2Xvac A lsquotwo-mode squeezedstatersquo reveals entanglement in the second moments of the measurement statis-tics It is thus a lsquobi-partite Gaussian entangled statersquo

Fig 12 displays a (pure) bi-partite squeezed vacuum state distributedbetween A and B The state shows full symmetry regarding its subsystemsat the two sites The large circles and the colored area represent Wignerfunctions of the subsystems Measurements of the quadrature amplitudesXA YA XB and YB show identical variances and the correlations and anti-correlations have identical strength since ∆2(XA minus XB) = ∆2(YA + YB) lt12 for our normalization of quadrature amplitudes having a ground statevariance of 14

Generally a symmetric bi-partite squeezed state fulfills another quantita-tive (Gaussian) entanglement criterion if less than 50 of the vacuum state issymmetrically mixed into the initially pure state Bi-partite squeezed statesare always entangled but in this case they are even Einstein-Podolsky-Rosen(EPR) entangled [Reid (1989)] allowing the demonstration of the quantumsteering effect [Einstein et al (1935) Schrodinger (1935) Reid (1989) Cav-alcanti et al (2009)] The first such experiment was performed by Ou etal [Ou et al (1992)] using type II parametric down-conversion (PDC) Laterexperiments produced bi-partite squeezed vacuum states by overlapping twosqueezed vacuum states each produced with type I PDC on a balanced beamsplitter and used the entangled output for the demonstration of quantumteleportation [Furusawa et al (1998) Bowen et al (2003ca)] The criterionin Eq (21) and the EPR criterion from [Reid (1989)] was experimentallycompared in Ref [Bowen et al (2003b)] The steering effect in asymmet-ric bi-partite squeezed states were recently experimentally characterized inRef [Handchen et al (2012)]

Fig 12 shows features similar to those in the top part of Fig 11 Thisis not a coincidence and shows that a bi-partite squeezed state can also begenerated by spatially splitting the upper and lower sideband of a (single-party) squeezed state This was first experimentally demonstrated by thegroup of E Polzik [Schori et al (2002)] and later used for EPR multiplex-ing of a single longitudinal mode of a squeezing resonator [Hage et al (2010)]

35 Covariance matrix representation of bi-partite squeezed states

Also the full information of bi-partite states including the entanglementcan be cast by the covariance matrix [Simon et al (1994)] which can be used

34

to calculate the propagation of these states in laser interferometers Againall variances are normalized to the vacuum noise variance in full analogy toEq (18) The generic bi-partite covariance matrix has dimension 4times4 andreads

Vbp =

VXAXA

VXAYA VXAXBVXAYB

VYAXAVYAYA VYAXB

VYAYBVXBXA

VXBYA VXBXBVXBYB

VYBXAVYBYA VYBXB

VYBYB

with

VXAYB =

langXAYB + YBXA

rangminus 2

langXA

ranglangYB

rang2∆2Xvac

ϑ

(22)

Due to the symmetry in Eq (22) the 4times4 covariance matrix is fully spec-ified by just ten independent coefficients If the phase spaces at A and Bare aligned along the strongest correlations and anti-correlations the ma-trix components referring to different quadrature amplitudes eg VXAYA arezero Such entangled states can be produced by overlapping two squeezedfields with a squeeze angle difference of 90 on a balanced beam splitter

A symmetric bi-partite squeezed vacuum state which is also called anlsquoS-classrsquo [DiGuglielmo et al (2007)] bi-partite squeezed vacuum state shows(anti-)correlations in two joint quadratures as defined in Eq (21) For a puresuch state of 10 dB squeezing the covariance matrix reads

Vbp10|10 =

505 0 495 0

0 505 0 minus495495 0 505 0

0 minus495 0 505

The following covariance matrix describes a so-called lsquoV-classrsquo 10 dB bi-partite squeezed vacuum state Here only one joint quadrature shows 10 dBsqueezing whereas the orthogonal joint quadrature shows vacuum noise Thestate is obtained by overlapping one 10 dB squeezed state with a vacuumstate on a balanced beam splitter

35

Vbp0|10 =

055 0 045 0

0 55 0 minus45045 0 055 0

0 minus45 0 55

The first measurement of all elements of such a covariance matrix was achievedin [DiGuglielmo et al (2007)]

36 Photon numbers of squeezed states

In contrast to the ground state squeezed vacuum states do have photonexcitations As said earlier quantum theory links the wave and the particlepictures Indeed the squeeze factor of a modulation mode is directly con-nected to a certain photon number excitation Squeezed states of light areproduced via spontaneous photon pair generation eg by parametric down-conversion The following operator S is called the lsquosqueeze operatorrsquo [Gerryand Knight (2005)] It creates and annihilates photon pairs

|r θ〉 = S(r θ) |0〉 (23)

where |r θ〉 is a squeezed vacuum state with squeeze parameter r and squeezeangle θ and |0〉 is the vacuum state The definition of the squeeze operatoris

S(r θ) = exp

[1

2

(reminusiθa2 minus reiθadagger 2

)] (24)

The following shows that this definition indeed results in a state with squeezedquadrature amplitude variances Lets set θ = 0

〈0| Sdagger(r 0) X S(r 0) |0〉 =1

2〈0| Sdagger(r 0)

(a+ adagger

)S(r 0) |0〉 (25)

〈0| Sdagger(r 0) Y S(r 0) |0〉 =i

2〈0| Sdagger(r 0)

(aminus adagger

)S(r 0) |0〉 (26)

Using the Baker-Hausdorff formula we get

Sdagger(r θ) a S(r θ) = a cosh r minus adaggereiθsinh r (27)

Sdagger(r θ) adagger S(r θ) = adaggercosh r minus aeminusiθsinh r (28)

36

Since 〈0| X |0〉 = 〈0| Y |0〉 = 0 also Eqs (25) and (26) are zero To finallycalculate the variances we need

〈0| Sdagger(r 0) X2 S(r 0) |0〉 =1

4〈0| Sdagger(r 0)

(a2 + adaggera+ aadagger + adagger2

)S(r 0) |0〉

〈0| Sdagger(r 0) Y 2 S(r 0) |0〉 = minus1

4〈0| Sdagger(r 0)

(a2 minus adaggeraminus aadagger + adagger2

)S(r 0) |0〉

Given that SSdagger is the identity and using again Eqs (27) and (28) we get theexpected variances

∆2X =1

4

[cosh2 r minus 2cosh r sinh r + sinh2 r

]=

1

4eminus2r

∆2Y =1

4

[cosh2 r + 2cosh r sinh r + sinh2 r

]=

1

4e2r

Since the squeeze operator can only create and annihilate photon pairsa squeezed vacuum state without photon loss must correspond to an evennumber of photons But not only photon loss also a coherent displacementleads to flattening out the odd-even oscillations The probability of detectingN photons in a pure displaced squeezed state are derived for instance in[Gerry and Knight (2005)] and read

P (N) = |〈N |α r θ〉|2 =(05 tanh r)N

N cosh rexp

[minus|α|2 minus 1

2(αlowast2eiθ + α2eminusiθ)tanh r

]times∣∣∣HN

[(αcosh r + αlowasteiθsinh r)

radiceiθsinh(2r)

]∣∣∣2 (29)

where HN is the N th Hermite polynomialFig 13 shows the photon number distributions for 5 different pure squeezed

states according to Eq (29) Panels (a) to (c) show squeezed vacuum stateswith 43 dB 86 dB and 172 dB of squeezing Panel (d) shows the moregeneral case of a squeezed state with a coherent displacement α 6= 0 Dueto θ = 0 the state is amplitude quadrature squeezed Panel (e) refers to thecorresponding phase quadrature squeezed state For comparison panel (f)shows the photon number distribution of the coherent state with the samedisplacement

The panels in Fig 13 represent the diagonal elements of the statersquos den-sity matrix in number basis Only the latter also contains the coherences

37

002

004

006

000 000

002

004

006

008

403020100 Photon number n

403020100Photon number n

Pro

babi

lity

403020100000

005

010

015

100 2 4 6 8100 2 4 6 800

04

06

02

100 2 4 6 800

03

02

01Pro

babi

lity

Pro

babi

lity

08

00

04

06

02

08

(a) (b)

(c) (d)

(e) (f)

Figure 13 Photon number distributions ndash All panels represent pure states (a) 43 dBsqueezed vacuum state (r = 05 α = 0) (b) 86 dB squeezed vacuum state (r = 1 α = 0)(c) 172 dB squeezed vacuum state (r = 2 α = 0) (d) displaced 86 dB squeezed state(r = 1 θ = 0 α = 4) (e) displaced 86 dB squeezed state (r = 1 θ = π2 α = 4) (f)Coherent state (r = 0 α = 4) The average photon numbers are about 027 14 131174 174 and exactly 16 see Eq (30)

between photon numbers [Gerry and Knight (2005)] Figures as shown heregenerally do not give full descriptions of the states

38

A squeezed vacuum state (r 6= 0) always has a non-zero photon numberand can not be the ground state The average photon number of a puresqueezed vacuum state can be calculated using Eq (8) With the maximallysqueezed quadrature variance ∆2Xsqz the average photon number is given by

n = 〈n〉|α=0θr〉 = ∆2Xsqz +(∆2Xsqz)

minus1

16minus 1

2=eminus2r

4+e2r

4minus 1

2 (30)

with the vacuum noise variance normalized to one quarter A coherent dis-placement further adds |α|2 photons on average

4 Squeezed-light generation

41 Overview

Squeezed light was first produced in 1985 by Slusher et al using four-wave-mixing in sodium atoms in an optical cavity [Slusher et al (1985)]Shortly after squeezed light also was generated by four-wave-mixing in an op-tical fibre [Shelby et al (1986)] and by degenerate parametric down-conversion(PDC) in a 2nd-order nonlinear crystal placed in an optical cavity [Wu et al(1986)] The pumped cavity was operated below its oscillation thresholdie the parametric gain did not fully compensate the round trip losses whichis also called lsquocavity-enhanced optical-parametric amplification (OPA)rsquo

The early day experiments achieved squeeze factors of a few percent up toabout 3 dB Today squeeze factors of more than 10 dB are directly observedin several experiments [Vahlbruch et al (2008) Eberle et al (2010) Stefszkyet al (2012) Vahlbruch et al (2016)] All of them are based on cavity-enhanced OPA (below threshold) The parametrically amplified mode isdegenerate ie signal and idler modes are identical In particular the down-conversion process is of lsquotype Irsquo which means that the amplified mode has awell-defined polarization Squeezed states can also be generated above oscil-lation threshold In Refs [Villar et al (2006) Jing et al (2006)] bi-partitesqueezing was generated with above-threshold PDC Both experiments usedtype II PDC which provides orthogonally polarized signal and idler fieldsType II PDC below threshold was also used to generate squeezed and bi-partite squeezed fields [Grangier et al (1987) Ou et al (1992)] All these

39

experiments were performed in the continuous-wave regime which is also thefocus of this Review Squeezed states of modulations of trains of laser pulseshowever have been also generated since the 1980s using either PDC or theoptical Kerr effect [Slusher et al (1987) Bergman and Haus (1991) Our-joumtsev et al (2006) Dong et al (2008)] For an overview of the develop-ments in squeezed-light generation in the continuous-wave as well as pulsedregime see Ref [Bachor and Ralph (2004)] Squeezed-light generation inopto-mechanical setups [Aspelmeyer et al (2014)] which use the intensitydependent phase shift from radiation pressure was discussed in Refs [Paceet al (1993) Rehbein et al (2005) Corbitt et al (2006)] and recently ex-perimentally achieved by several groups [Brooks et al (2012) Safavi-Naeiniet al (2013) Purdy et al (2013)]

42 Degenerate type I optical-parametric amplification (OPA)

This section provides a graphical description of how degenerate type IOPAPDC turns a vacuum state into a squeezed vacuum state and a coher-ent state into a displaced squeezed state The process requires a bright pumpfield and a 2nd-order nonlinear crystal For simplicity we set all nonlinearitiesabove 2nd-order to zero

Let us consider a short segment of the second-order nonlinear crystalpumped with light of optical frequency 2ν All other modes that enter thecrystal shall not contain any photons ie are in their vacuum states Ofthese the only mode of interest is that at optical frequency ν which spatiallyoverlaps with the pump mode Fig 14 shows the total electric field of theoptical input E in and the 2nd-order nonlinear dielectric polarisation of thecrystal P The latter is proportional to the total electric field of the outputEout The pump field at 2ν periodically drives the vacuum field at ν betweenregions of low and high polarisation This process transforms the vacuumstate into a squeezed vacuum state in the output [Bauchrowitz et al (2013)]The output further contains the hardly depleted pump field and frequencydoubled parts of the pump field at 4ν It is again emphasized that Fig 14displays OPA in a small segment of the crystal In reality the nonlinear effectaccumulates over the crystal length or even over several passages since thecrystal is usually put into an optical resonator A noticeable effect is achievedif all infinitesimal contributions constructively interfere This is achieved incase of phase matching ie if the wave fronts of the modes at 2ν and νpropagate with the same speed and thus do not run out of phase Note that

40

t

t

Squeezing

Figure 14 Optical parametric amplification of a vacuum state ndash The upper leftcorner shows the crystal polarization P(E) = ε0

(χ(1)E + χ(2)E2

) ie the separation of

charge carriers by the electric component of an optical field E The graph illustrates howan input quantum field (from below) is projected into an output quantum field (towardsthe right) In the example shown here the input field is composed of a classical pumpfield E in

2ν at frequency 2ν and zero-point fluctuations E inν of a field at frequency ν cf

Fig 7(c) The superposition E in of these two fields is transferred into a time-dependentdielectric polarization that is the source of (and thus directly proportional to) the elec-tric component of the output field Eout The quantum uncertainty of the output fieldshows a phase-dependent (parametric) amplification at frequency 2ν Spectral decompo-sition of the output field Eout reveals coherent amplitudes at frequencies 2ν and 4ν and asqueezed vacuum state Eout

sqzν cf to Fig 7(d) The concept of this figure was published inRef [Bauchrowitz et al (2013)]

41

t

t Amplitudequadraturesqueezing

Figure 15 Optical parametric amplification of a coherent state ndash The pictureshows how a displaced vacuum state is transformed into a displaced squeezed state Thepumprsquos electric field is maximal when the input field at fundamental frequency ν shows azero crossing The phase relation described results in an output state that is amplitudequadrature squeezed If the pump field at the input in phase was shifted by half of itswavelength the squeezed output field were a phase quadrature squeezed The squeezinggeneration displayed here corresponds to the transition from Fig 7(a) to Fig 7(b) but withan additional reduction of the displacement The concept of this figure was published inRef [Bauchrowitz et al (2013)]

in actual squeezing experiments the 4ν component is usually suppressed byphase miss-matching

Fig 15 shows the same process but now for an input field at frequencyν in a coherent state In this case the relative phase between the two inputstates ∆ϕ = ϕ2ν minus 2ϕν is relevant In Fig 15 the relative phase is set suchthat the expectation value of the field at frequency ν is zero when the pump

42

(a) (b)

(c) (d)

X

Y

X

YΔϕ = ndash50deg

X

Y

Δϕ = 0deg

X

Y

Δϕ = 90deg

Figure 16 Phase-space illustration of degenerate OPA ndash The (displaced) dashedcircle in each diagram represents the uncertainty of the initial state at optical frequencyν The (displaced) shaded area represents the state after degenerate optical parametricamplification All quantum uncertainties shown correspond to pure states The boldgreen arrow describes the bright second-harmonic pump field whose uncertainty can beneglected The phase between the 2nd harmonic pump and the initial state (∆ϕ = ϕ2ν minus2ϕν) determines the result of the parametric amplification

field reaches its maximum (∆ϕ = 90) The output at the fundamentalfrequency is then an amplitude squeezed state with a deamplified coherentamplitude

Fig 16 summarizes the squeezing operation on the vacuum state as wellas on displaced vacuum states for different phase relations ∆ϕ between thetwo input fields

43 Cavity-enhanced OPA

Placing the nonlinear crystal inside a cavity can greatly enhance thedown-conversion efficiency but not only that A cavity introduces a thresholdfor the pump power above which the parametric gain is infinite just limited

43

by the finite pump power In this case the vacuum uncertainty of the inputfield at frequency ν is amplified to a bright laser field at frequency ν Thedevice is then called an optical-parametric oscillator (OPO) For the gener-ation of squeezed states however the pump power is usually kept (slightly)below threshold Due to nonzero optical loss there exists a pump powersmaller than the threshold above which the tiny improvement of squeezingis not noticeable anymore Getting the pump power closer to the thresholdcould even reduce the observed squeeze factor if a fluctuating squeeze an-gle projects anti-squeezing into the observed quadrature amplitude [Franzenet al (2006) Suzuki et al (2006) Dwyer et al (2013)] The cavity has an-other important purpose It confines the transverse spatial mode usuallyto TEM00 This mode confinement is crucial for any efficient application ofthe squeezed state in laser interferometry since it allows the suppression ofanti-squeezing from other transversal modes The squeezing process requiresa nonlinear material that should show negligible absorption at both opticalfrequencies involved in particular at the wavelength of the squeezed modeIn Refs [Vahlbruch et al (2008) Mehmet et al (2009)] 10 dB and 116 dBof squeezing were achieved using MgOLiNbO3 The highest squeeze factorstoday are produced in (quasi phase matched) periodically poled KTP [Eberleet al (2010) Mehmet et al (2011) Stefszky et al (2012) Vahlbruch et al(2016)]

The optical cavity that is built around the nonlinear crystal is vital forsqueezed-light generation and it deserves a detailed consideration Gener-ally the mode propagating away from a cavity is the result of interferenceat the cavity coupling mirror One contribution is given by the intra-cavityfield attenuated by the amplitude transmission coefficient t of the outcouplingmirror The second contribution is given by the outside field that is reflectedby the same mirror with amplitude reflectivity r =

radic1minus t2 and spatially

overlapped with the first Also the mode from a squeezing resonator is suchan interference product

The impedance matched resonator

Let us consider first an empty optically stable and loss-less Fabry-Perotresonator built from two identical mirrors each with amplitude reflectivityr = r1 = r2 lt 1 A propagating field be perfectly mode-matched to one ofthe cavity resonances In this setup the resonator shows zero reflection andthe resonator is said to be impedance matched (for all such input fields)

44

|α〉

r1α

r12

= +

|α〉 r22 = r1

2

minusr1α equiv t1 αcav = 1minus r12 minusα sdot r1

1minus r12

0

XY

Figure 17 Empty impedance-matched resonator ndash Mode-matched and resonant lightthat enters the cavity from the left here displayed by a quantum phasor for a coherentstate is fully transmitted including its quantum uncertainty The back-reflected statedestructively interferes with itself for all frequencies well within the cavity linewidth The180 phase shift of the transmitted cavity field amplitude minusr1α is a necessary conditionin order to fulfill energy conservation on cavity resonance Impedance matching is forinstance achieved for a resonant cavity with matched mirror reflectivities (r2

2 = r21) and

zero optical loss The complex amplitude of the field traveling towards left inside the cavityis readily deduced from the figure and reads αcav = minusαr1(1minus r2

1)minus12 (highlighted by thedashed circle) In the displayed setup zero field uncertainties are reflected however alsothe vacuum state that enters the cavity from the right is fully transmitted (not shown)

r1α

XY

r12 r2

2 =1

= + |α〉

minus 1+ r1( )α equiv t1 αcav = 1minus r12 minus 1+ r1( )α

1minus r12

|α〉

Figure 18 Empty maximally overcoupled resonator ndash Maximal overcoupling isachieved for a resonant cavity with a perfect end-mirror reflectivity (r2

2 = 1) and zerooptical loss For a given input-mirror reflectivity r2

1 the intracavity light power is maximalMode-matched and resonating light entering the resonator from the left is fully reflectedThe complex amplitude of the field traveling towards left inside the cavity is readilydeduced from energy conservation to αcav = minusα(1 + r1)(1 minus r2

1)minus12 (dashed circle) Inthis setup no uncertainty from the right couples to the left side of the cavity

45

Obviously the interference described in the previous paragraph is fullydestructive The same resonator also shows zero reflection of the input fieldrsquosquantum uncertainty since the interference happens between parts of thesame quantum state The mode propagating away from such a resonatorhowever is not in a nonclassical but in a vacuum state because the vacuumstate that enters the cavity through the opposite site is also fully transmittedThe interference at the coupling mirror of an impedance matched resonatoris displayed in Fig 17

r1α

r12

+

=

OPA PDC

Xcavg r1(1+r1) Xcav

Δ2Xg 0 Δ2Yg

XY

r22 =1

|α〉

Figure 19 Squeezing resonator ndash Shown is the interference at the zero-loss squeezingresonator operated at threshold The lower line represents the perfectly squeezed modepropagating away from the cavity towards the left The parametric gain medium inside thecavity deamplifies the X quadrature of the cavity mode (Xcav) by the factor r1(1 + r1)which is the ratio of the intra-cavity field amplitudes of the two previous figures The Xquadrature of the field that is back-reflected towards the left destructively interferes withitself similar to the situation of the impedance matched cavity in Fig 17 The parametricpower gain per resonator round-trip (G) needs to mimic the effect of an end mirror withreflectivity r2

2 = r21 For this reason the deamplification of Xcav corresponds to a round-

trip deamplification factor of r1 equivradic

1G The round-trip amplification factor for Ycav

then is 1r1 equivradicG which exactly compensates for the outcoupling and thus determines

the parametric oscillation threshold (threshold for bright lasing) The variances of thequantum uncertainties ∆2Ycav and ∆2Y are thus infinite In this setup no field uncertaintyfrom the right couples to the left of the cavity and a perfectly X-quadrature-squeezed fieldoutside the squeezing resonator is produced

46

The perfectly over-coupled single-ended resonator

We now increase the reflectivity of the far mirror lsquo2rsquo to being perfect (r2 = 1)This way the counter-propagating vacuum state can not enter the cavityAgain a propagating field be perfectly mode-matched through mirror lsquo1rsquo toone of the cavity resonances For frequencies well inside the cavity linewidththe situation is displayed in Fig 18 The setup protects the left side of thecavity against vacuum fluctuations entering through mirror lsquo2rsquo but of coursedoes not squeeze quantum noise The intra-cavity built-up factor is too highfor achieving destructive interference below the vacuum uncertainty on theleft side of the resonator

The impedanced-matched single-ended squeezing resonator

Building on the two previous concepts the straight forward approach now isto start from the perfectly over-coupled single-ended resonator and insert anattenuator into the cavity that does not couple the cavity mode to any bathbut still results in a roundtrip efficiency of precisely r1(lt 1) in amplitudeOptical loss is not appropriate since it increases the coupling of the cavitymode to a thermal bath neither would any phase-insensitive atenuator beappropriate It is easy to show that a phase-insensitive attenuator adds ad-ditional uncertainty since otherwise the commutation relation [a adagger] = 1 isviolated The amplification process that matches our requirement is OPA Toachieve infinite squeezing in X on cavity resonance a second-order nonlinearcrystal needs to be put into the cavity and pumped such that the intra-cavityamplitude quadrature is attenuated by the factor (1 + r1)r1 (on cavity reso-nance) with respect to the empty cavity This factor is readily deduced fromFigs 17 and 18 Due to the symmetry in parametric amplification the intra-cavity phase quadrature is then amplified by (1 + r1)r1 and the round-tripgain has a value of 1r1 in amplitude In this situation not only infinitesqueezing but also the (laser) threshold of the resonator is achieved sincethe round-trip gain of the intra-cavity phase quadrature equals its roundtriploss here fully given by the incoupling mirror

The physical descriptions in Figs 17 to 19 are fully consistent with ob-servations in squeezing experiments The consideration above in particularshows that the intra-cavity field shows a finite squeezing strength while theexternal field shows infinite squeezing The strongest intra-cavity squeezefactor possible is (1 + r1)2r2

1 In the high reflectivity limit this factor corre-

47

sponds to 6 dB Averaged over the full cavity mode the squeeze factor of thecavity mode is in this limit even limited to 3 dB [Walls and Milburn (2008)]Higher intra-cavity squeeze factors are possible for lower mirror reflectivities

44 The generation of squeezed light for laser interferometry

With the insights gained in the previous subsection we now turn to ac-tual experiments The application of squeezed states in laser interferometrycertainly requires large squeeze factors (idealy accompanied with the highestpossible purity) to maximize the impact in terms of sensitivity improvementIn cavity-enhanced OPA the highest parametric gain is achieved on cavityresonance ie at zero sideband frequency But this is not the main reasonwhy this Subsection focusses on the generation of squeezed states at lowsideband frequencies The application of squeezed states in a laser inter-ferometer requires that their sideband frequencies cover the devicersquos signalband Ground-based gravitational wave (GW) detectors have a detectionband from about 10 Hz to 10 kHz frequencies which can be considered aslsquolowrsquo compared to typical frequencies in quantum optics experiments

Squeezing at MHz sideband frequencies is easier to observe than at acous-tic frequencies because the latter are often polluted with excess noise fromlight beams that serve as control beams [Bowen et al (2002) McKenzie et al(2004)] and parasitic interferences from back-scattered light [Vahlbruch et al(2007)] Furthermore the observation of squeezing at low sideband frequen-cies requires a more stable setup since larger measuring times are necessaryThe observation of strong squeezing at MHz frequencies however alreadysets an upper limit to the optical loss of the setup At least the same squeezefactor can be observed at lower frequencies

There are two different main topologies for squeezing resonators TheFabry-Perot-type standing-wave resonator consists of a minimum number ofmirror surfaces and has the advantage of being compact and thus robustagainst mechanical vibrations Usually one or even two mirror coatings aredirectly placed on the spherical and polished surfaces of the nonlinear crys-tal itself [Wu et al (1986) Grangier et al (1987) Breitenbach et al (1998)Vahlbruch et al (2008) Eberle et al (2010) Vahlbruch et al (2016)] TheBowtie traveling-wave resonator has the advantage of providing a separa-tely accessible counter propagating mode for cavity length control [Ou et al(1992) Takeno et al (2007)] It shows no direct back-reflection of incoupledlight which helps reducing parasitic interferences [Stefszky et al (2012)]

48

(a)

(b)

(c)

(d)Squeezingresonator

To inter-ferometer

OPA

SHG

BHD

Mode cleaner

LO

Laser

DBS

DBS

Figure 20 Generation of squeezed light ndash (a) Example of a 2nd-order nonlinearcrystal for the squeezed-light generation at 1064 nm Shown is a bi-convex 65 mm long7MgOLiNbO3 crystal whose polished surfaces also carry the mirror coatings of the res-onator The crystal thus realizes a monolithic squeezing resonator as it was used forthe first demonstration of 10 dB squeezing [Vahlbruch et al (2008)] (b) Optical con-figuration of a half-monolithic (hemilithic) standing-wave squeezing resonator Here thecavity length can be adjusted by displacing the coupling mirror The crystal surface insidethe cavity is anti-reflection coated The photograph shows a 10 mm long PPKTP crystalsqueezing resonator as used for the GEO 600 squeezed-light source [Abadie (2011)] (c)Mechanically stable housing of a standing-wave squeezing resonator The crystalrsquos temper-ature is stabilized at its phase matching condition using Peltier elements (d) Schematicfor the squeezed-light generation After spatial filtering of continuous-wave laser lighttwo hemilithic standing-wave resonators are employed The first generates second har-monic pump light (SHG) The second (OPA) generates a squeezed vacuum field at theinitial wavelength The squeezed states are observed by a balanced homodyne detector(BHD) or alternatively sent and mode-matched to the optical mode of an interferometerbeforehand LO local oscillator DBS dichroic beam splitter

49

Fig 20 (a) and (b) show photographs of typical nonlinear crystals used forsqueezed-light generation at near infra-red wavelengths The crystals shownhere form a monolithic standing-wave squeezing resonator (a) or are partof a half-monolithic standing-wave squeezing cavity (c) shows a tempera-ture stabilized and mechanically stable housing of the squeezing resonator(d) shows a schematic of a full setup for the generation of squeezed vacuumstates of light for an application in a laser interferometer The only brightinput required for the squeezing resonator (OPA) is the second-harmonicpump field The resonator mode at fundamental frequency is thus initiallynot excited by photons ie it is in its ground state characterized by vacuumfluctuations due to the zero point energy see Fig 7 (c) [Gerry and Knight(2005)] The pump field spontaneously decays in the degenerate pair of sig-nal and idler fields The combined down-converted field leaving the resonatorexhibits quantum correlations which give rise to a squeezed photon countingnoise when overlapped with a bright coherent local oscillator beam Thedetection is done either in a balanced homodyne detector (BHD) or with asingle photo diode The squeeze factor increases the closer the pump powerof the squeezing resonator gets to the oscillation threshold and the lower theoptical loss on down-converted photon pairs is

441 High squeeze factors ndash minimizing decoherence

Squeezed states of light have significant impact on the sensitivity of laserinterferometers if large squeeze factors can be produced Squeezing of 3 dBimproves the signal-normalized quantum-noise spectral density by a factorof 2 This factor corresponds to doubling the (coherent state) light powercirculating inside the interferometer Squeezing of 10 dB corresponds to aten-fold power increase The experimentally demonstrated squeeze factorswere considerably improved in recent years [Takeno et al (2007) Vahlbruchet al (2008) Polzik (2008) Eberle et al (2010) Stefszky et al (2012)]culminating in a value of as large as 150 dB [Vahlbruch et al (2016)] Thisvalue corresponds to the same reduction of signal-normalized quantum noisethat is achieved by increasing the light power by a factor of 32 (At this pointit is already noted that squeezing the quantum noise can simultaneouslyreduce quantum measurement noise (shot noise) as well as quantum backaction noise (radiation pressure noise) This is not possible with scaling thelight power of coherent states see Subsec 55)

50

Ideally a parametric squeezed-light source can produce an infinite squeez-ing level see Fig 19 fundamentally just limited by the energy provided bythe pump field In practice the limit is set by decoherence mechanismsThe by far most important one is optical loss Optical loss occurs duringsqueezed-light generation its propagation through the interferometric setupincluding imperfect mode matchings and finally the photo-electric detec-tion Also detector dark noise [Schneider et al (1998)] phase noise [Takenoet al (2007)] and excess noise [Bowen et al (2002)] impair the observablesqueezing strength

Optical loss is usually understood as coupling the squeezed mode to a zerotemperature bath ie overlapping it with a vacuum mode For any amountof loss the resulting state is still squeezed But to be able to directly observesay 10 dB of squeezing the total loss on the state needs to be less than 10in this example cf Eq (16) To minimize optical loss the nonlinear crys-tal as well as lenses and beam splitters in the interferometric path need toshow very low absorption and scattering at the wavelength of the squeezedlight PPKTP shows absorption of about 10minus4cm and below at near-infraredwavelengths Low OH content fused silica is a suitable material for all otheroptics Absorptions of less than 10minus6cm were measured [Hild (2007)] Coat-ings on crystal surfaces and on all other optical components should also showlowest optical loss Total loss of the 10minus6 level are available today Superpol-ished surfaces which show roughnesses with less than 1 A root mean square(integrated over spatial scales from approximately 1 micron to 100 microns)and thus very low scattering are necessary to achieve these low numbersMinimizing the total number of optical components is essential From thisperspective a monolithic squeezing resonator as shown in Fig 20 (a) is theoptimum choice The squeezed mode needs to be matched to the mode ofthe laser interferometer or to the mode of the balanced homodyne detectorVisibilities of up to 998 have been achieved [Eberle et al (2010)] whichcorresponds to a loss of about 04 Of great importance also is the quantumefficiency of the photo-diodes used for detecting the squeezed field (togetherwith the interferometric signal) Recently a quantum efficiency of photo-diodes in a squeezing experiment of (995plusmn 05) was measured [Vahlbruchet al (2016)] To minimize photon loss the photo-diodes had no protectionwindow an anti-reflection coating on the semi-conductor material and theremaining reflection was re-focussed with an external mirror

Also the dark-noise spectral density of the detection electronics reducesthe observable squeezing and needs to be as low as possible Similar to optical

51

noise it also provides a contribution to the observed variance The dark noiseof the detection electronics needs to be much lower than the detected photoncounting noise In [Vahlbruch et al (2016)] it was 28 dB below shot noisebut still reduced the observable squeeze factor from 153 dB to 150 dB

Excess noise emerges if the squeezed mode couples to a nonzero tem-perature bath or to a mode whose excitation is strongly fluctuating (Thecoupling process can always be understood as a beam splitter coupling andis physically described by overlapping electric fields Coupling to a zero tem-perature bath leads to Eq (16)) The captured excess noise variance thenneeds to be added to the initial squeezing variance which deteriorates theobserved squeezing stronger than just mixing in the vacuum mode Excessnoise is less likely to occur at MHz frequencies but can be significant ataudio-band sideband frequencies and below and is thus a serious issue ingravitational-wave detectors [Chua et al (2014)] The reason for that isthat acoustically or thermally excited motions of surfaces produce frequencyshifts of back-scattered light mainly at these low frequencies [Vahlbruch et al(2007)]

Phase noise corresponds to stochastic phase fluctuations between thesqueezed field and the local oscillator within the measuring time It cor-responds to mixing the squeezed mode with itself with a fluctuating squeezeangle [Suzuki et al (2006) Franzen et al (2006)] Phase noise in squeez-ing experiments typically is less of an issue than optical loss [Dwyer et al(2013) Oelker et al (2016) Vahlbruch et al (2016)] The setuprsquos phasenoise can be reduced by making the squeezing resonator more compact andthus mechanically more stable against acoustic and thermal fluctuations ofthe environment and by improving the quality of the servo loops for cavitylength and propagation length controls Operating a squeezed-light resonatorin vacuum might also be beneficial The ability to run a high performancesqueezed-light generator in vacuum was demonstrated in [Wade et al (2015)]

442 Squeezing in the gravitational-wave (GW) detection band

High squeeze factors have been first demonstrated at sideband frequen-cies of a few MHz and above where excess noise is generally negligible whenworking with visible or near-infra-red light Today we know that extendingthe squeezing spectrum towards the audio-band and even below is technicallynot always easy but straight forward once a high squeeze factor is achieved

52

at MHz frequencies In most squeezing experiments the main laser light

Figure 21 Photograph of the GEO 600 squeezed-light source ndash The breadboarddimensions are 135 cm times 113 cm The squeezing resonator is high-lighted by the white ar-row and is set up as a standing-wave hemilithic cavity containing a plano-convex PPKTPcrystal of about 10 mm length (see also Fig 20 b) It is pumped with continuous-wave532 nm light that is produced by frequency doubling of light (at angular frequency ω)from a commercial NdYAG laser Two more laser fields at about 1064 nm having fre-quency offsets of more than 10 MHz with respect to ω(2π) serve as optical control fieldsBoth fields are mode-matched and injected into the squeezing resonator together with thesecond-harmonic pump field

at the squeezing wavelength is accompanied by significant noise up to thelaser relaxation oscillation For this reason laser control fields at the opticalcarrier-frequency in the optical path of the squeezed mode need to be avoided[Bowen et al (2002) Schnabel et al (2004) McKenzie et al (2004)] and thesqueezing resonator length and the optical path stabilized by other means[McKenzie et al (2005) Vahlbruch et al (2006)] Furthermore and mostimportantly excess noise due to back-scattering is an issue Back-scattering(also called lsquoparasitic interferencesrsquo) is produced if DC light scatters out ofthe optical path hits a vibrating surface and re-scatters back into the opticalpath [Vahlbruch et al (2007)] Significant back-scattering can be produced

53

in interferometers for the detection of gravitational waves since light powersof several hundreds of kilowatts are used Even back-scattering from the mil-liwatt local oscillator of balanced homodyne detectors is an issue at acousticsideband frequencies and below The recipe for avoiding parasitic interfer-ences turns out to be threefold (i) avoiding scattering by using ultra-cleansuperpolished optics with close to perfect anti-reflex coatings (ii) avoidingback-scattering by carefully blocking all residual (faint) light fields and (iii)reduce the vibrationally and thermally excited motion of all mechanical andoptical parts that could potentially act as a re-scattering surface with re-spect to the optical path [Vahlbruch et al (2007) McKenzie et al (2007)]The insights described above led to the first demonstration of audio-bandsqueezing at frequencies down to 200 Hz [McKenzie et al (2004)] and laterto the first demonstration of squeezing over the full gravitational-wave de-tection band even from as low as 1 Hz [Vahlbruch et al (2007)] Whilea standing-wave squeezing resonator [Ou et al (1992)] can be built in avery compact way that is rather insensitive against mechanical vibrations[Chelkowski et al (2007)] a traveling-wave bow-tie squeezing resonator [Wuet al (1986)] is more tolerant against back-scattered light [Chua et al (2011)]The strongest squeezing in the audio-band of up to 116 dB was reported inRef [Stefszky et al (2012)]

443 The first squeezed-light source for GW detection

The first squeezed-light source for the continuous operation in GW de-tectors had been designed and completed between 2008 and 2010 [Vahlbruch(2008) Vahlbruch et al (2010)] Since then this source has been producingsqueezed vacuum states in a fully phase controlled way using co-propagatingfrequency-shifted bright control beams [Vahlbruch et al (2006)] as an inte-gral part of the GW detector GEO 600 The source is a turn-key device witha fully automated re-lock system [Vahlbruch et al (2010) Khalaidovski et al(2012)] Re-locking is required if the temperature of the environment changessignificantly which drives the actuators outside their dynamic ranges

Up to 9 dB of squeezing over the entire GW detection band was observedusing a balanced homodyne detector (BHD) located close to the squeezingresonator The squeeze factor has been limited by optical loss due to ab-sorption in the PPKTP crystal transmission of the back-surface and thenon-perfect AR-coating of the crystalsrsquos intra-cavity surface The adjustableair gap has been necessary to allow for an easy way to apply length control

54

-12

-8

-4

0

4

8

12

16

20

10 100 1k 10k

Rel

ativ

e no

ise

pow

er [

dB]

Frequency [Hz]

shot noise (a)

squeezed noise (b)

anti-squeezed noise (c)

~ 9dB

Figure 22 Broadband squeezing spectrum ndash Noise power spectra measured on theoutput of the GEO 600 squeezed-light source shown in Fig 21 with a balanced homodynedetector The traces correspond to the spectra of quadrature amplitude variances normal-ized to vacuum noise The resolution bandwidth used increases towards higher frequenciesto reduce the measurement time (a) Shot noise normalized to unity which serves as thereference level (0 dB) (b) Squeezed noise covering the complete detection band of ground-based GW detectors (c) Anti-squeezed noise Peaks at 50 Hz and 100 Hz are the electricmains frequency and its first harmonic The data was published in Ref [Vahlbruch et al(2010)]

Additional optical loss in the path to the balanced homodyne detector mainlyarose due to a Faraday isolator that eliminated parasitic interferences Fi-nally the mode missmatch to the BHD as well as its non-perfect quantumefficiency provided additional loss Inferring the squeeze factor without theBHD detection loss more than 10 dB of squeezing are provided by the sourceSince 2010 it has been used in basically all observational runs of the GEO 600GW detector see Section 6

444 Generation of two-mode (bi-partite) squeezing

lsquoTwo-mode squeezed lightrsquo or lsquobi-partite squeezed lightrsquo is light that allowsfor joint measurements at two locations A and B These joint quadraturemeasurements reveal correlations and anti-correlations with a remaining un-certainty smaller than the ground-state uncertainty which certifies the pres-

55

ence of entanglement cf Subsec 34 Bi-partite squeezed light has been gen-erated by type I and by type II parametric down-conversion In case of type Ithe squeezed fields from two squeezing resonators as described in Subsec 44are overlapped on a balanced beam splitter with a 90 phase shift The twooutput fields together represent the entangled mode [Furusawa et al (1998)Bowen et al (2003c) Eberle et al (2013)] In case of type II signal and idlerfields are non-degenerate regarding polarisation and a single cavity contain-ing an appropriate crystal and a polarising beam splitter are sufficient for theproduction of bi-partite squeezing Also in this case the measurements ofthe quadrature amplitudes of signal and idler fields show large uncertaintiestogether with bi-partite correlations and anti-correlations that are strongerthan the ground state uncertainty of individual subsystems [Ou et al (1992)Villar et al (2006) Jing et al (2006)]

To date the strongest entanglement of bi-partite squeezed light has beenproduced based on type I parametric down-conversion [Eberle et al (2013)]The requirements of producing strong entanglement are identical to those ofproducing strong squeezing outlined above The strength of bi-partite en-tanglement can be given in decibels in full analogy to the squeeze factorPractically the strength of bi-partite squeezing is always somewhat smallerthan that of single party squeezing since it requires an additional mode-matching that results in additional optical loss

45 Conclusions

The first observation of squeezed light was achieved in 1985 [Slusher et al(1985)] Shortly after cavity-enhanced optical parametric amplification forsqueezed-light generation was demonstrated [Wu et al (1986)] which todayenables the observation of up to 15 dB of squeezing [Vahlbruch et al (2016)]Quite generally the maximum squeezing level that is observed does not de-pend on the strength of the optical nonlinearity Squeezing cavities can easilybe operated at their oscillation threshold where they should provide infinitesqueezing if decoherence is zero The main limiting factor is optical lossincluding that of the photo-electric detection

Dedicated experimental research and development towards a squeezed-light source for applications in gravitational-wave detectors can be tracedback to 2002 [McKenzie et al (2002) Bowen et al (2002)] Since then a sur-prising amount of progress has been made culminating in the first squeezed-light source specifically built for the integration into a gravitational-wave

56

detector For the future squeeze factors above 15 dB will certainly be possi-ble by further reducing optical loss This statement is supported by the highdegree of matching between experimental data and a theoretical loss modelas presented in Fig 3 of [Vahlbruch et al (2016)]

5 Quantum noise in laser interferometers

51 Interferometric measurements

The purpose of a laser interferometer is the precise measurement of smallchanges of an optical path length with respect to a reference path Forthis the interferometer transfers the change of the phase difference betweentwo light fields into an amplitude quadrature change of the interferometerrsquosoutput light The latter can easily be detected by a single photo diode Ofgeneral interest are differential changes of the optical path length that aremuch smaller than the laser wavelength ie differential phase changes thatare much smaller than 2π In this case the differential phase change can bedescribed in very good approximation as a differential change of the phasequadrature instead

In order to transfer the phase quadrature signal with minimum loss ahigh interference contrast at the interferometerrsquos beam splitter is essentialAdditionally instrumental noise in terms of unwanted excitations of the out-putrsquos amplitude quadrature needs to be reduced to a minimum Noise arisesdue to power fluctuations of the input laser light back-scattered laser lightinside the interferometer thermally driven displacements of mirror surfacesand in many more ways The important measure of the sensitivity of aninterferometer obviously is its signal-to-noise-ratio The most useful measureis given in terms of the noise spectral density S(f=Ω2π) that is normalizedto the physical unit of the signal S(f) is in fact a lsquonoise-to-signal-ratiorsquoand can be seen as the signal-normalized variance of the photo diode out-put decomposed into spectral components versus sideband frequency f withthe resolution bandwidth of 1 Hz As an example S(100 Hz) = 10minus39 m2Hzmeans that the instrumental noise in the one hertz band around 100 Hz equalsa signal that would be produced if the mirror of one interferometer arm os-cillates with an amplitude of just

radic10 middot 10minus20 m in the very same band Such

small spectral densities are achieved by gravitational-wave detectors [Abbott(2016)]

57

52 Quantum measurement noise and shot noise

The most fundamental noise source in laser interferometers is due to thequantum noise of light which is in fact two-fold [Caves et al (1980)] Firstof all there is lsquoquantum measurement noisersquo which arises in the processof photo-electric detection For coherent states the quantum measurementnoise is the lsquophoton counting noisersquo of mutually independent photons andusually simply called lsquoshot-noisersquo Fig 2 (b i) shows a time series of suchnoise hiding the actual signal The frequency components of the shot noiseare well described by the quantum uncertainty of the output fieldrsquos amplitudequadratures XΩ∆Ω see Subsec 22 (Recall this quantity corresponds to the

differential phase quadrature YΩ∆Ω of the light beams in the interferometerarms) The photon counting noise has a white Fourier spectrum howeverthe lsquoshot noisersquo of an interferometer is usually normalized to the signal whosetransfer function is usually not white for instance due to the presence of armcavities or a signal-recycling cavity

All current and planned gravitational-wave detectors are Michelson-typelaser interferometers with operating points very close to a dark fringe Thelight power in the output port is just a couple of tens of mW which canbe handled by a single photo diode In this configuration the signal-to-shot-noise-ratio is actually maximized which can be shown in three steps [Bachorand Ralph (2004)] For the first step we use plane waves to describe theelectric field in the output port of a Michelson interferometer For perfectinterference contrast at the balanced beam splitter ie for perfect modematching and for defining φ = 0 as the dark port condition we get

Eout(t φ) =1

2E0 sin(ωt+ φ)minus 1

2E0 sin(ωt) (31)

where E0 is the amplitude of the total internal field whose two parts hasaccumulated a differential phase It directly follows for the squared fields

E2out(t φ) =

(sin

φ

2

)2

(E0 cos(ωt+ φ2))2 (32)

We now turn to a light beam with a localized transversal mode that can befocussed onto a photo-electric detector The photo diode has perfect quantumefficiency ie the rate of photo electrons is not only proportional to the rateof output field photons but also has a unity slope efficiency Since the optical

58

frequency is too high to be resolved we consider the averaged light power

P out(φ) =

(sin

φ

2

)2

P (33)

The next step is a formulation of the signal being the derivative of detectedphoton number versus phase Let n be the average value of the photonnumber per measuring time interval Eq (33) can then be rewritten as

nout(φ) =

(sin

φ

2

)2

n (34)

rArr dnout(φ) = n sinφ

2cos

φ

2dφ (35)

The final step is the calculation of the signal-to-shot-noise-ratio Shot noiserefers to coherent states which have a standard deviation of the photonnumber of σ(n) =

radicn

dnout(φ)

σ(nout)=n sinφ

2cosφ

2dφ

radicn sinφ

2

(36)

and find for a signal-to-noise ratio of unity for coherent states and for anon-zero but still small phase difference ∆φCoh 2π

1 =radicn cos

φ

2∆φCoh with φ 6= 0 (37)

In this equation the smallest measurable phase difference is given for φrarr 0

∆φCohmin =

1radicn (38)

This is the well-known shot-noise limit of high-precision phase sensing ∆φCohmin

is the smallest phase shift that can be measured with a signal-to-noise ra-tio of one when using n mutually independent photons per measuring time(those of a coherent state) when the loss of photons is assumed to be zeroThe typical purpose of a laser interferometer is the continuous sensing (moni-toring) of a continuously changing phase An illustrative example is the phasesignal produced by the black hole merger measured by Advanced LIGO onSept 14 in 2015 (Fig 1 in [Abbott (2016)]) The measuring interval should

59

be short to be able to resolve the time-evolution of the signal Generally themeasurement of an arbitrary signal that lasts for a finite time thus needs tobe understood as l subsequent measurement intervals using n photons eachIt can be shown that Eq (38) is also valid for interferometers operated athalf fringe ie when each output port contains the same light power In thiscase photo diodes need to be placed in both output ports and the actualsignal is provided by their difference voltage

Due to its importance the shot-noise limit deserves some remarksThe phase φ in Eq (31) is the phase difference of two mode-matched fieldsand might be accumulated by a single pass along the length L such as ina Mach-Zehnder interferometer or in a double pass such as in a (simple)Michelson interferometer or in four passes as realized in a Michelson inter-ferometer with folded arms [Grote (2005)] The shot-noise limit in Eq (38)and its scaling therefore holds independent of the number of passes Theclaim in Ref [Higgins et al (2007)] that the scaling according to Eq (38) canbe surpassed by multiple passes is not justifiedThe fact that Eq (38) is derived by approaching φ rarr 0 correctly describesthe actual operation point of gravitational-wave detectors which is close tobut not exactly at a dark port In practice a tiny offset from dark port ischosen at which the shot noise is well above the photo diodersquos electronic darknoiseEq (38) solely depends on the number of quanta but not on the lightrsquos wave-length λ Of course the shot-noise limit for the change of an optical pathlength ∆L does depend on the wavelength and ∆φCoh

min needs to be replacedby ∆φCoh

min = 2π∆LCohminλ

Finally an essential result of the shot-noise limit is that the ideal precisemeasurement should use lsquoas much quanta as possible per measuring inter-valrsquo which translates to lsquoas much light power in the interferometer armsas possiblersquo Eq (38) is indeed the one and only reason why gravitational-wave detectors use high power lasers power-recycling and arm resonatorsExtending the measuring time for a given light power can also improve thesensitivity but only if the signal repeats ie is periodic Let us assume thatone period of the signal is resolved by l intervals using n photons each Inthis case repeating the overall measurement k times improves Eq (38) by1radick The fundamental statement of Eq (38) however does not change

since the actual photon number n may then simply incorporate the factor k

60

For a given average photon number the shot-noise limit in Eq (38) canonly be surpassed by using photons that are quantum correlated ie by usingnonclassical states of light How is the shot-noise limit surpassed with thehelp of squeezed states A Michelson interferometer that is operated closeto a dark fringe acts like an almost perfect mirror for both input ports Allthe input light is back-reflected towards the laser source This also accountsfor the quantum uncertainty of the input light The quantum uncertaintythat impinges onto the photo diode thus (mainly) enters the interferometerthrough its (almost) dark port An interferometer that uses displaced co-herent states entering from one port can thus be improved by replacing theordinary vacuum entering the signal output port by a squeezed vacuum stateThis was the proposal by CM Caves in 1981 [Caves (1981)] which is labeledhere with lsquoCSVrsquo If the differential phase quadrature of the interferometer issqueezed Eq (38) then within the limit of large coherent state displacementα sinh2r improves to

∆φCSVmin asymp

eminusrradicn (39)

(The above expression is an approximation since the squeezing operationproduces a small number of photons that are not accounted for here) Ofcourse the mode of the squeezed vacuum needs to be precisely matchedto the mode of the interferometer The first experimental demonstrations ofsqueezed phase measurements used a Mach-Zehnder [Xiao et al (1987)] and apolarization interferometer [Grangier et al (1987)] Fig 2 shows how spatialdegeneracy between an externally generated squeezed mode and the signalmode in a Michelson interferometer is achieved using a polarizing beam split-ter and a Faraday rotator Again the limit in Eq (39) can only be achievedif optical loss is zero Optical loss not only reduces the signal but here alsoreduces the squeeze parameter see Eqs (15) and (16)

Let us consider an example The sensitivity of a laser interferometer thatuses coherent states with an excitation of 1023 photons per second can beimproved by a factor of

radic10 by either adding 09 middot 1024 photons per second

or by adding about just 2 photons per second and bandwidth in hertz thatbelong to the 10 dB squeezed vacuum confer Eq (30) Since the full sig-nal band of ground-based GW detectors covers sideband frequencies up to10 kHz just 2 middot 104 photons per second are necessary At a wavelength ofλ = 1064 nm these values correspond to a power increase by 168 kW and

61

37 fW respectively

The question arises whether a scaling of the sensitivity better than propradic1n is possible It was theoretically shown that in principle the scaling

can indeed considerably be improved yielding the so-called Heisenberg limitor Heisenberg scaling [Bondurant and Shapiro (1984) Yurke et al (1986)Braunstein (1992) Holland and Burnett (1993)]

∆φHLmin prop

1

n (40)

The Heisenberg scaling requires nonclassical states that have a certain num-ber of quanta similar to Fock states ie n = n The theoretically optimalstates describe a superposition of n (N) indistinguishable photons in oneinterferometer arm while having zero (0) photons in the second arm and viceversa and were named lsquoN00Nrsquo-states [Dowling (2008)] A specific propertyof these states is lsquosuper-resolutionrsquo The output ports of the interferometershow an n-times faster oscillation of the interference fringes when changingthe phase between the two interferometer arms Super-resolution correspondsto an n-times improved signal transfer function and was demonstrated forinstance in Refs [Rarity et al (1990) Kuzmich and Mandel (1998) Mitchellet al (2004) Afek et al (2010)] The presence of this nonclassical phe-nomenon however does not prove a sensitivity better than the semi-classicalbound according to Eq (38) Sensitivity is rather related to the signal-to-noise-ratio and needs to take into account all imperfections as well as theprobability of a successful detection of the sensing state [Thomas-Peter et al(2011)] All experiments so far used post-selection on particular measure-ment outcomes and neglected the typically large probability that nothingwas detected

Super-resolution was demonstrated with up to n = 5 [Afek et al (2010)]In addition to the fact that super-resolution does not prove a sensitivity bet-ter than the semi-classical bound photon numbers in state-of-the-art super-resolution experiments are extremely small compared to the photon numberof about 1023 (within a measuring interval of one second) of coherent statesused in Ref [Abbott (2016)] and of about 1022 using coherent states plussqueezed vacuum states used in Ref [Abadie (2011)]( which did prove a sen-sitivity better than the semi-classical bound)

Another interesting and related question is what the smallest phase is

62

that can be estimated in a single measurement again using a given numberof quanta Taking into account that no prior information about the phaseshift exists still a scaling proportional to 1n is possible In the limit of largen however an additional factor of π is required in the nominator of Eq (40)[Sanders and Milburn (1995) Berry and Wiseman (2000)] yielding

∆φHLmin asymp πn (41)

The state that can actually achieve this bound is different from the N00Nstate and was found in [Summy and Pegg (1990) Luis and Perina (1996)Berry and Wiseman (2000)] A N00N state is not the optimum state forphase estimation (via a single measurement) since it only provides one bit ofinformation A recent review on generell aspects on phase measurements isgiven by Ref [Demkowicz-Dobrzanski et al (2015)]

It is important to note that Eqs (38-40) do not consider photon lossExperiments that demonstrated super-resolution and aimed for achievingthe scaling in Eq (40) were conditioned on zero photon loss Let η gt 0 bethe average efficiency of detecting (all) photons Eq (38) then reads

∆φηCohmin =

radic1

η n (42)

Eq (39) turns into

∆φηCSVmin asymp

radicηeminus2r + 1minus η

η n(43)

and Eq (40) turns into [Demkowicz-Dobrzanski et al (2012 2013)]

∆φηmin =

radic1minus ηη n

for 0 lt η lt 1 (44)

For non-zero photon loss most interestingly the ultimate sensitivity of aphase measurement for a given photon number also shows a 1

radicn -scaling

The difference between the CSV strategy of using bright coherent states incombination with squeezed vacuum states which is bounded by Eq (43) andthe strategy of using the optimal nonclassical state which is bounded byEq (44) is marginal in practice [Demkowicz-Dobrzanski et al (2013)] Forgravitational-wave detectors and for any other laser interferometer using in-tense light there is no need for an alternative to the CSV strategy

63

We now turn back to the shot noise according to Eq (38) Generally noisecan be decomposed into its spectral contributions For a simple Michelsoninterferometer without arm resonators and without a signal-recycling cavitythe square-root of the single-sided shot-noise spectral density normalized tothe differential arm length change x in units of m

radicHz is given by [Saulson

(1994)] radicSMI

SNx =

radic~c2

2ωPprop 1radic

P (45)

where ω is the optical angular frequency of the quasi-monochromatic carrierlight and P the total light power in both arms including the built-ups fromcavities In combination with a squeezed vacuum whose relative phase gen-erates squeezing of the output lightrsquos amplitude quadrature the right handside reduces according to the factor eminusr Note that the single-sided spectraldensity is only defined for positive sideband frequencies and thus twice aslarge as the double-sided spectral density

The spectral density of the measurement of a GW induced strain is givenby the same expression but normalized to h = xL (If the gravitational waveis oriented in an optimal way with respect to the Michelson interferometerone arm is squeezed while the other is expanded by the same amount of ∆L =x2 and h then corresponds to the actual gravitational-wave amplitude)The square-root of the single-sided shot-noise spectral density normalized tostrain in units 1

radicHz is given by

radicSMI

SNh =

radic~c2

2L2ωP (46)

Equations (45) and (46) show that the smallest measurable signal (corre-sponding to unity signal-to-shot-noisendashratio) is inversely proportional to thesquare root of the laser power and has a white spectrum for sideband frequen-cies much smaller than the carrier frequency see horizontal line in Fig 23

All first- and second-generation GW detectors use power-recycling andadditional cavities to improve their sensitivities Fabry-Perot arm resonatorsdo not only increase the light power but additionally also increase the signalfor signal frequencies inside the resonator linewidth For lossless Fabry-Perot

64

arm resonators the spectral densities in Equations (45) and (46) need to bemultiplied by the following factor [Kimble et al (2001)]

HFP =

radicL2(γ2

FP + Ω2)

c2 (47)

where γFP = cTFP(4L) is the Fabry-Perot arm resonatorrsquos half bandwidthand TFP is the light power transmission of the input mirror The end mirrorsare assumed to have perfect reflectivity A similar expression can be derivedfor describing the improvement due to signal-recycling [Buonanno and Chen(2001)]

In summary shorter laser wavelengths higher light powers and squeezingof the amplitude quadrature of the interferometer output reduce shot noise ina broadband way ie for all signal frequencies Fabry-Perot arm resonatorsas well as signal-recycling provide improvements mainly for frequencies insidethe resonator linewidths

53 Quantum back-action and quantum radiation pressure noise

In laser interferometers quantum back-action noise results from the un-certainty of the lightrsquos radiation pressure force on the interferometer mirrorsand is also called lsquo(quantum) radiation pressure noisersquo (RPN) Its origin isthe quantum uncertainty of the differential amplitude quadrature XΩ∆Ω ofthe fields in the interferometer arms It results in an uncertain momentumtransfer to the mirrors and thus in an position uncertainty of the mirrors atfuture times with respect to their differential mode of motion [Caves et al(1980)] The physical mechanism of radiation pressure corresponds to anintensity dependent phase shift [Pace et al (1993)]The higher the light power in the arms of a laser interferometer the loweris its shot-noise spectral density see Eq (45) Unfortunately the spectraldensity of quantum back-action noise increases with light power The single-sided force noise spectral density reads

radicSRPNF =

radic8~ωPc2

(48)

Whereas the force noise of the quantum radiation pressure has a white spec-trum the RPN does not since the mirrorrsquos reaction to external periodicforces depends on frequency The link between the Fourier component of an

65

external force F (Ω) and the Fourier component of the displacement x(Ω) isgiven be the mechanical susceptibility HM It reads for an harmonic oscillatorwith mass M

HM(Ω) =1

M | minus Ω2 + Ω2M + iΩΩMQ|

(49)

where ΩM is the oscillatorrsquos resonance frequency and Q its quality factorThe square root of the single-sided spectral density of the RPN normalized

to the displacement of an harmonic oscillator with mass M is then given by

radicSRPNx = HM(Ω)

radic8~ωPc2

(50)

In GW detectors the test mass mirrors are suspended as pendula with highmechanical Q-factors and their centre of mass motion corresponds to that ofa harmonic oscillator The resonance frequencies of the pendula are lowerthan the detection band of interest The mechanical susceptibility is thereforeoften approximated for the so-called free-mass regime as H fm

M (Ω) = (mΩ2)minus1The square root of the single-sided spectral density of the RPN normalizedto differential displacement of two mirrors with each of mass M in a simpleMichelson interferometer is given by [Saulson (1994)]radic

SfmMIRPNx =

radic2~ωPc2m2Ω4

propradicP (Ω ΩM) (51)

where m = M2 is the mirrorsrsquo reduced mass In case of a simple Michel-son interferometer that is enhanced with arm cavities the spectral density inEq (51) needs to be multiplied with the expression given in Eq (47) In com-bination with a squeezed vacuum whose relative phase generates squeezing ofthe output lightrsquos phase quadrature the right hand side reduces according tothe factor eminusr Note if the radiation pressure noise is squeezed the shot noisemust be anti-squeezed or vice versa The radiation pressure noise calibratedto strain of space time is given by the right side of Eq (51) divided by theinterferometer arm length LIn summary heavier masses longer laser wavelengths lower light powersand squeezing of the amplitude quadrature in the interferometer arms re-duce radiation pressure noise

66

1 10 100 1000Frequency [Hz]

Radiation pressure noise

Shot noise

SQL

10-21

10-18

10-15

10-12

radicSx

[mradic

Hz]

ndashndash

ndashndash

Figure 23 Displacement-normalized quantum noise spectral densities ndash Consid-ered is a simple Michelson interferometer with neither arm cavities nor signal recyclingThe two end mirrors (m = 100 g) of the interferometer arms are suspended as pendulahaving a resonance frequency of ΩM2π = 1 Hz and a Q-factor of 107 The interferometeruses quasi-monochromatic light (in coherent states) with a total power of 4 kW Opticalloss and the offset from a dark output fringe is assumed to be negligible Wavelengthλ = 1550 nm The standard quantum limit (SQL) corresponds to the lowest noise achiev-able at a given sideband frequency when varying the light power without using quantumcorrelations

54 Interferometer total quantum noise and the standard quantum limit

Both shot noise and radiation pressure noise contribute to the total quan-tum noise of a given interferometer If they are not quantum correlatedwhich is the case for a conventional Michelson interferometer when detectingthe output lightrsquos amplitude quadrature their variances add up (The resultis not shown in Fig 23) It can easily be deduced from the previous sectionsthat changing the laser power will shift the two quantum noise contributionsHowever the total quantum noise never goes below the standard quantumlimit (SQL) [Braginsky and Manukin (1967)]

Let us consider Fig 23 for sideband frequencies much greater than thependulum resonance Here the test mass mirrors react as free masses whenexerted to external forces The SQL in this free-mass regime is calculated

67

by minimizing the sum of the squares of Eqs (45) and (51) [Saulson (1994)]

Sfmtotx =

~c2

2ω

[1

P+

4ω2

c4m2Ω4P

] (52)

Its derivative reads

dSfmtotx

dP=minus1

P 2+

4ω2

c4m2Ω4 (53)

Setting the above equation to zero provides the optimum laser power versussideband frequency in order to achieve the lowest total quantum noise

P fmopt =

c2mΩ2

2ω (54)

Inserting the optimal light power into Eq (57) provides the square root ofthe single-sided noise spectral density of the free-mass SQL in m

radicHzradic

SfmSQLx =

radic2~mΩ2

(55)

Again m is the reduced mass and dividing by the interferometer arm lengthL yields normalization to the GW-induced strain h Eq (55) shows that theSQL falls off with sideband frequency The corresponding equation for aMichelson interferometer that uses arm cavities readsradic

SfmFPSQLx =

radic~

mΩ2

(1

HFP

+HFP

) (56)

with HFP according to Eq (47)Using the expression for the SQL the square root of the total quantum

noise spectral density of a Michelson interferometer in the free-mass approx-imation can be written asradic

SfmFPtotx =

radicSfmFP

SQLx

2

[1

k+ k

] (57)

with the radiation pressure coupling parameter

k(Ω) =2ωP

mc2Ω2 (58)

68

For a fixed light power and fixed reduced mass of the mirrors the quantumnoise limited interferometer reaches the SQL when k = 1 which is realizedat the angular sideband frequency ΩSQL =

radic2ωP(mc2)

Note that neither squeezing the phase quadrature nor squeezing the am-plitude quadrature of the interferometer light leads to sub-SQL performance[Caves (1981)] also confer Ref [Schnabel (2005)] Fig 3 (left) As we willsee in the next sections the standard quantum limit can be surpassed if shotand radiation pressure noise are correlated Then the total quantum noise isnot given by the sum of the variances ie the sum of the squares in Eqs (45)and (51)

55 Squeezed light for surpassing the standard quantum limit

A measurement with sensitivity better than the standard quantum limit(SQL) is also called a lsquoquantum non-demolition (QND)rsquo measurement [Bra-ginsky and Khalili (1995 1996) Kimble et al (2001)] Several QND tech-niques for laser interferometers were proposed in recent decades [Jaekel andReynaud (1990) Kimble et al (2001) Purdue and Chen (2002) Chen (2003)McClelland et al (2011) Danilishin and Khalili (2012) Graf et al (2014)]What they have all in common is they exploit quantum correlations betweenobservable uncertainties

Arguably the most extensive way of introducing quantum correlationsand surpassing the SQL is the injection of squeezed states of light [Jaekeland Reynaud (1990)] If the squeezed quadrature angle of the injected statesis neither 0 nor 90 the quantum uncertainties of the amplitude and phasequadrature amplitudes that describe the differential field in the two interfe-rometer arms become correlatedLet us consider a very simplified setup that just consists of a quasi-mono-chromatic light field that is back-reflected from a quasi-free mirror Thelight power and the mass be such that reflected light in a coherent stateresults in a measurement of the mirror position with a noise spectral den-sity at the SQL at sideband angular frequency ΩSQL At this frequencyquantum measurement noise and back-action noise are of the same sizeie the uncertainty in X produces an equally large additional uncertaintyin Y Upon reflection the quadrature amplitude variances change from∆2X = ∆2Y = 14 to 2∆2X = ∆2Y = 12 This result corresponds tothe situation in Fig 23 at the crossing frequency of shot noise and radiation

69

pressure noise The coupling of the uncertainty variances can be describedby the matrix K = (1 minusk 0 1) where k = 1 at the SQL If the modulationstate at ΩSQL is the ground state its variances are transferred according to

KT

(1 00 1

)K =

(1 0minus1 1

)(1 00 1

)(1 minus10 1

)=

(1 minus1minus1 2

) (59)

In accordance with Fig 23 the variance of YΩSQLis twice as large as the vac-

uum noise varianceNow let the quantum noise of the light field be 10 dB squeezed at 45

(Eq (19)) The projection of the quantum uncertainty onto the X-observableproduces the radiation pressure noise by being transferred with the couplingfactor k = 1 at the SQL into the Y -observable in fact with negative signsince a larger value of X produces a larger optical path length and thus aretardation of the phase Due to the squeezing at 45 the initial uncertaintyin Y cancels with the additional uncertainty that originates from the onein X The following calculation shows that the strength of the cancellationcorresponds to the initial squeezing strength Upon reflection the quantumuncertainties transform in the following way(

1 0minus1 1

)(505 495495 505

)(1 minus10 1

)=

(505 minus01minus01 02

) (60)

The state of light after reflection has a squeezed phase quadrature amplitudeThe improvement in comparison to Eq (59) is exactly 10 dB The quantumnoise improvement corresponds to the input squeeze factor and is also a mea-sure by what factor the SQL is surpassed Squeezed vacuum injection thusallows surpassing the SQL upon measuring the conventional Y -quadrature(which is realized by a single photo diode in the interferometerrsquos output port)as first realized by [Unruh (1983) Yuen (1983) Jaekel and Reynaud (1990)]

In the example above the input squeeze angle is optimized for a sin-gle sideband frequency Injecting a broadband squeezed vacuum field withfrequency-independent squeeze angle of 45 would result in a rather bad inter-ferometer quantum-noise performance at frequencies far smaller or largerthan ΩSQL Fig 24 shows the quantum-noise performance if the input fieldhas squeeze angles that are optimized for every k(Ω) as given in Eq (58)Injected squeezing can thus lead to a broadband sub-SQL performance ifthe quantum measurement noise (shot noise) and the quantum back-actionnoise (radiation pressure noise) are correlated in an optimal way Due to the

70

10 50 100 500 1000 5000Frequency [Hz]

Dis

plac

emen

t noi

se sp

ectra

l den

sity

[mradic

Hz]

10-20

10-19

10-18

Total quantum noise 0dB

Total quantum noise -10dB

SQL

Y

X

Y

X

Y

X

Figure 24 Surpassing the SQL with squeezed-light injection ndash At shot-noise lim-ited sideband frequencies squeezing of the Y -quadrature amplitude improves the noisespectral density of the interferometer At radiation-pressure-noise limited sideband fre-quencies squeezing of the X-quadrature amplitude improves the noise spectral density ofthe interferometer If both kinds of quantum noise contribute equally (at the SQL markedwith a dot) a squeeze angle of 45 results in surpassing the SQL by the full squeeze factorsee Eq (60) In the graph here the squeeze angle is optimized for all frequencies result-ing in a broadband quantum noise reduction [Jaekel and Reynaud (1990)] Measurementsensitivities beyond the SQL (shaded area) are in the so-called quantum non-demolition(QND) regime [Kimble et al (2001)] Dashed horizontal lines represent the (squeezed)shot noise Dashed straight lines with negative slope represent the (squeezed) radiationpressure noise The calculations use 10 dB of squeezing a conventional Michelson inter-ferometer with neither arm resonators nor signal recycling a light power at the beamsplitter of 1 MW at a wavelength of λ = 1550 nm and mirror masses of 1 kg

correlation shot noise and radiation pressure noise can be squeezed simulta-neously

71

Light with a frequency-dependent squeeze angle

The discovery that shot noise and radiation pressure noise can be squeezedsimultaneously and thus a broadband reduction of quantum noise beyond theSQL be achieved required the insight that the spectral analysis of light definesa spectrum of many lsquosideband modulation modesrsquo that all can be in differ-ent quantum states An ordinary squeezing resonator which is on resonancefor light at twice the pump wavelength produces a spectrum of modulationmodes that all have the same squeeze angle A frequency-dependent squeezeangle can be introduced by reflecting such a field from a detuned single-ended filter cavity which was suggested by Kimble and coworkers [Kimbleet al (2001)] They showed that the optimal frequency dependence thatleads to the broadband improvement shown in Fig 24 can be realized byusing altogether two filter cavities as shown in Fig 25 Motivated by this re-sult research and development on filter cavities for optimizing the frequencydependence of broadband squeezed fields has been very active in recent years[Corbitt et al (2004) Chelkowski et al (2005) Dwyer et al (2013) Kweeet al (2014) Straniero et al (2015) Oelker et al (2016)]

Photo diode Squeezed vacuum

Faraday Rotator

Coherent light

YΩΔΩ

Figure 25 Frequency dependent squeezing injection ndash A broadband squeezed fieldwith a frequency-dependent squeeze angle that is optimal for gravitational-wave detectorsis produced by reflecting off an ordinary broadband squeezed field from two detuned opticalfilters [Kimble et al (2001)]

A light field with a frequency-dependent squeeze angle was first demon-strated in Ref [Chelkowski et al (2005)] see Figs 26 and 27 The experi-

72

Detuned filter cavityLockingphoto diode

Isolator

EOM R asymp 1

SHG

EOM

DBSOPA

LaserIsolator

Mode cleaner

LO

Homodyne detector

Figure 26 Generation of a frequency-dependent squeezing ndash A frequency-dependent orientation of the squeeze ellipse was first demonstrated in Ref [Chelkowskiet al (2005)] Initially a conventional spectrum of squeezed vacuum states of light was gen-erated in a squeezing resonator (lsquoOPArsquo) The squeezed vacuum was transmitted throughan optical isolator to a detuned filter cavity After reflection the squeezed vacuum stateswere absorbed in a balanced homodyne detector (BHD) The phase of the BHDrsquos localoscillator (LO) was changed for quantum state tomography of the squeezed states in diffe-rent regions of the spectrum The result showed a frequency-dependent orientation of thesqueeze ellipse see Fig 27 SHG second harmonic generation EOM electro-optical mod-ulator for applying phase modulation sidebands for cavity length control DBS dichroicbeam splitter R mirror reflectivity λ4 quarter wave plate for turning linear polarizedlight into circular polarised light and vice versa

ment consisted of a standing-wave squeezing resonator which produced ans-polarized broadband amplitude quadrature squeezed field accompanied bya dim continuous-wave DC control field with a wavelength of λ = 2πcω =1064 nm The squeeze bandwidth covered sideband frequencies up to aboutΩ(2π) = 30 MHz which corresponded to the linewidth of the squeezing res-onator The optical cavity for producing the frequency dependence of thesqueeze angle was a standing-wave cavity composed of a plane incouplingmirror of reflectivity r1 =

radic097 and a concave end mirror of reflectivity

r2 =radic

09995 The cavity length was L = 50 cm resulting in a linewidthof 147 MHz The squeezed field first passed a Faraday isolator to preventinterference effects between the filter cavity and the squeezing resonator A

73

Phase quadrature

min

max

-2 -1 0 1 2

141 MHz

-2

-1

0

1

2

Am

plit

ud

e q

uad

ratu

re

Figure 27 Frequency-dependent squeezing ndash Picture top right Reconstructed con-tour plot of the Wigner function of the sideband modulation at Ω(2π) = 141 MHz afterreflection from a 1515 MHz detuned filter cavity The state shows quantum correlationsbetween phase and amplitude quadratures ie squeezing at an angle of here about 40The white circle visualizes the standard deviation of the vacuum state uncertainty Thewhite ellipse represents the standard deviation of the squeezed uncertainty Small picturesMeasurement results on the same continuous-wave laser beam at various sideband frequen-cies around 15 MHz For each tomographic picture noise histograms of 100 equidistantquadrature angles were measured In each case the laser beam was phase locked to a ref-erence beam and the quadrature angle stably controlled and stepwise rotated The phasereference was given by a phase modulation at 198 MHz [Chelkowski et al (2005)] Thepicture was first published in Ref [Schnabel (2005)] (copyright ccopy2007 by Imperial CollegePress)

λ4-waveplate turned the s-polarized field into a circularly polarized beamwhich was then mode matched into the detuned cavity The retro-reflectedfield was analyzed by a balanced homodyne detector (BHD) for quantumstate tomography The filter cavity was electro-optically controlled to be de-tuned by 1515 MHz with respect to the DC control field The cavity length

74

control was achieved by the Pound-Drever-Hall (PDH) locking techniqueutilizing a circularly polarized laser beam that carried 15 MHz phase mod-ulation sidebands and was coupled into the filter cavity from the back Theoutput voltage of the BHD was characterized by a spectrum analyser as wellas used to perform quantum state tomography In the latter case the BHDoutput voltage was mixed down with an electronic local oscillator at differentradio-frequencies around 15 MHz and low-pass filtered to set the resolutionbandwidth to ∆Ω(2π) = 100 kHz The final electric signal corresponds to atime series of quadrature amplitude measurements XθΩi∆Ω Quantum statetomography is a method to reconstruct the phase space quasi-probability dis-tribution (Wigner function) of quadrature amplitudes from sets of measuredXθΩ∆Ω distributions when varying the angle θ [U Leonhardt (1997)] Forevery sideband frequency Ωi 100000 quadrature values were measured di-vided up on 100 equidistant quadrature angles Each quadrature angle wasstably controlled with a precision of plusmn1 Fig 27 shows the reconstructedWigner functions which were all measured on the same laser beam but atdifferent sideband frequencies For these measurements the detuned filtercavity was locked to the lower sideband at minus1515 MHz The result clearlyshows the frequency-dependent orientation of the squeeze ellipse In a morerecent experiment a frequency-dependent squeeze angle was also realized inthe kHz regime [Oelker et al (2016)]

56 Optomechanically induced (ponderomotive) squeezing

The radiation pressure of light when acting on a movable mirror re-sults in an intensity dependent phase shift [Pace et al (1993)] The cou-pling produces a so-called lsquoponderomotive effectrsquo [Braginsky and Manukin(1967)] which is of third order optical nonlinearity and which transforms abright coherent state inside an interferometer into a squeezed state of light[Vyatchanin and Matsko (1993)] This type of squeezed-light generationis usually called lsquoponderomotive squeezingrsquo or lsquooptomechanical squeezingrsquoConsequently even if no squeezed field is injected into the interferometercorrelations between the quadrature amplitudes are generated that allow forsurpassing the SQLPonderomotive squeezing as produced by the interferometer itself can onlybe exploited for evading back-action (radiation pressure noise) It can notbe used to squeeze the interferometer shot noise This is why ponderomotivesqueezing is fundamentally less extensive than injecting externally producedsqueezed states of light [Corbitt et al (2006)] suggested an external pon-

75

deromotive squeezing source for gravitational-wave detectors In this casedue to its external generation also the interferometerrsquos shot noise can besqueezed Recently ponderomotive squeezing was observed for the first time[Brooks et al (2012) Purdy et al (2013)] The achieved squeeze factors aremuch smaller than those produced by optical-parametric down-conversion[Vahlbruch et al (2016)]

Let us have a look again at Eq (59) Rotating the covariance matrix onthe right by arctan(minus

radic54minus 12) asymp minus58 indeed reveals squeezing

(cos 58 minussin 58

sin 58 cos 58

)(1 minus1minus1 2

)(cos 58 sin 58

minussin 58 cos 58

)asymp(

262 00 038

) (61)

The vacuum-noise normalized variance of 038 corresponds to about 42 dBof ponderomotive squeezing This is the general value that is produced atthe angular sideband frequency ΩSQL At higher frequencies the squeezefactor gets smaller at lower frequencies higher The squeezing strength of42 dB can be observed if the photo diode in the interferometer output portis replaced by a balanced homodyne detector using a local oscillator phase ofabout minus58 It can be shown however that the optimal signal-to-quantum-noise-ratio at the SQL is achieved for a local oscillator phase of exactly 45At this angle back-action is fully evaded

Full evasion of radiation pressure noise at all frequencies requires an op-timized frequency dependence of the relative local oscillator phase This canbe achieved by reflecting off the interferometer output field from two detunedfilter cavities [Kimble et al (2001)] The scheme was called lsquovariational out-putrsquo In the case of zero optical loss this scheme can fully evade radiationpressure noise just leaving the shot noise as the only quantum noise contri-bution

The variational-output scheme can be used to enhance the frequency-dependent squeezed input scheme The right site of Eq (60) shows thatthe output statersquos squeezing is not optimally detected in the Y -quadratureRather than with a single photo diode the detection should be done witha balanced homodyne detector with optimized phase of its local oscillatorIn this case the output lightrsquos quantum noise is solely given by squeezedshot noise The total quantum noise in Fig 24 would then be given by thelowest (dashed) horizontal line This combined scheme was called lsquosqueezed

76

variationalrsquo [Kimble et al (2001)] It can be realized by reflecting off theinterferometer output light from in total two optical filter cavities placed infront of the balanced homodyne detector

57 Conclusions

The highest quantum-noise-limited sensitivities of high-precision laserinterferometers are achieved by employing a large number of quanta tomaximize the signal strength in combination with strongly squeezed statesto minimize the quantum noise From this perspective it is clear that thequantum-noise-limited sensitivity of future gravitational-wave detectors willbe further improved ndash by increasing the light power and the squeeze factorTo be able to do so the optical loss in these devices needs to be reduced

In principle the optical loss in laser interferometers can be made smallbut never zero Recent theoretical research has shown that for any non-zero loss the sensitivity scales proportional to 1

radicn at best where n is the

average photon number per measurement This scaling is efficiently achievedby combining strongly displaced coherent states with squeezed vacuum statesof light

If a repeated measurement is not only limited by quantum measurementnoise but also by quantum back-action noise squeezed states of light can beused to simultaneously reduce both ie in the case of an interferometer shotnoise and radiation pressure noise

6 The first application of squeezed light in an operating gravita-tional-wave detector

Squeezed states of light have been successfully used to improve the sensi-tivity of the gravitational-wave detector GEO 600 from 2010 up to the pointwhen this Review was written [Abadie (2011) Grote et al (2013)] Afterdecades of proof-of-principle experiments [Xiao et al (1987) Grangier et al(1987) McKenzie et al (2002 2004) Vahlbruch et al (2005 2006 20072008) Goda et al (2008)] the implementation of a squeezed-light source inGEO 600 has resulted in the first sensitivity improvement beyond shot noiseof a measurement device that targets new observations in nature The im-plementation of squeezed states in GEO 600 was not done to provide anotherproof-of-principle demonstration but was realized because it offered a rela-tively cheap way of further improving the measurement sensitivity Of course

77

the sensitivity of GEO 600 can also be further increased by purely classicalmeans however the implementation of arm resonators to enable higher lightpowers without increasing the thermal load on the beam splitter or eventhe realization of longer interferometer arms are much more expensive Inthis respect the sensitivity improvement of GEO 600 with squeezed light canarguably be regarded as the first lsquotruersquo application that developed out of thefield of lsquononclassical (quantum) metrologyrsquo (Note that the term rsquoquantummetrologyrsquo is currently defined in different ways [Giovannetti et al (2006)Gobel and Siegner (2015)] and the term rsquononclassicalrsquo referring to a non-classical P-function gives a distinct description)

61 Gravitational waves

Einsteinrsquos General Theory of Relativity [Einstein (1916)] or simply lsquoGen-eral Relativityrsquo (GR) predicts that accelerating mass distributions producegravitational radiation analogous to electromagnetic radiation from accel-erating charges Experimental evidence of their existence is given by theobservation of the slow spiraling together of two neutron stars caused bythe loss of orbital energy to gravitational waves The inspiral rate exactlymatches the predictions of Einsteinrsquos theory [Weisberg and Taylor (2005)]Recently Advanced LIGO observed gravitational waves for the first time[Abbott (2016)] thereby giving the go-ahead for gravitational-wave astron-omy The gravitational-wave source was the final inspiraling and the mergerof two black holes 13 billion light years away from earth

Gravitational-waves evolve in the far field of the source propagate withthe speed of light and are measurable on earth with laser interferometersFig 28 displays a gravitational wave propagating along a certain directionGravitational waves are dynamical changes of space-time They are transver-sal and quadrupolar in nature and have two polarization states

A variety of known astrophysical and cosmological sources are predictedto emit gravitational radiation that should reach the Earth with a measur-able strength [Sathyaprakash and Schutz (2009)] The first gravitational waveevent detected was produced by two black holes of 36 and 29 solar massesDuring the final 02 seconds of their inspiraling they produced a peak gravi-tational strain in our solar system of 10minus21 covering frequencies up to 250 Hz[Abbott (2016)] Other predicted sources are mergers of neutron stars supernovae and background signals from the Big Bang According to GR GWsfrom complex astrophysical sources carry a plethora of information that willhave a major impact on gravitational physics astrophysics and cosmology

78

L +Δ L

Binary system

fBS

Figure 28 Space-time oscillation ndash Gravitational waves are dynamical deformations ofspace-time that form in the plane perpendicular to the direction of wave propagation Asa result distances between free-falling test masses in a transverse plane will change witha strain h = ∆LL For black hole or neutron star binary systems with orbital frequencyfBS distances will oscillate at frequency fGW = 2fBS The wavelength of this oscillation isgiven by λGW = cfGW where c is the speed of light The wave of orthogonal polarizationwith respect to the one shown is rotated by 45 around the propagation axis

62 Interferometric detection of gravitational waves

Current gravitational wave detectors are kilometre-scale laser interfero-meters [Dooley et al (2016) Aasi (2015) Acernese (2015) Aso et al (2013)]Continuous-wave laser light is split into two beams traveling in orthogonaldirections Both beams are reflected back towards the central beam splitterwhere they interfere Gravitational waves change the optical path lengthdifference and thus the light power directed towards the photo-diode thatis positioned in the signal output port of the beam splitter A gravitationalwave at frequency fGW = ΩGW(2π) reveals itself as a light-power modula-tion at the same frequency The spectral decomposition of the output signalis described by a spectrum of the quadrature amplitude YΩ∆Ω introduced inSec 3 It corresponds to the amplitude quadrature amplitude of the outputlight and relates to the differential phase quadrature of the interferometerarms

79

Photo diode

Squeezed vacuum

Faraday Rotator

Signal

Quantum noise

Coherent light XΩΔΩ

YΩΔΩ

(a) (b)

YΩΔΩFigure 29 Squeezed-light-enhanced interferometric measurement ndash (a) Michelsoninterferometer with arm cavities power recycling and signal recycling (see main text forexplanation) The interferometer is operated close to a dark fringe such that the quantumnoise entering from the dark port is back-reflected The squeezed field is mode-matched tothe signal output field (b) Phase space diagram of the gravitational-wave signal outputat sideband frequency Ω(2π) The quantum noise is squeezed below the ground stateuncertainty and thus the signal to quantum noise ratio improved

The first key ingredient of an interferometric gravitational-wave detectorare suspended heavy mirrors that can be regarded as quasi-free in the di-rection of laser light propagation thereby acting as test masses that probespacetime Being on ground current detectors are located in rather noisy en-vironments that allow the realization of undisturbed quasi-free mirrors onlyabove a sideband frequency of the order of 10 Hz Since sufficiently strongGW signals are expected up to a frequency of 10 kHz todayrsquos gravitationalwave detectors target at signals in the acoustic band from 10 Hz to 10 kHzThe quasi-free motion of the test mass mirrors in this frequency regime isachieved by suspending the mirrors as sophisticated multiple-stage pendulain vacuum chambers [Aasi (2015)] Far above the pendularsquos resonant frequen-cies which are typically around 1 Hz the centre of masses of the mirrors areisolated from vibrations of the ground and they react on frequency compo-nents of small external forces approximately as free masses The mirrors andtheir suspensions are built from materials having exquisitely high mechani-cal quality factors This helps to concentrate the thermal energy that causes

80

displacements of the mirror surface into well-defined vibrational frequencymodes At these particular very sharp frequencies no gravitational wavescan be detected

The second key ingredient of an interferometric gravitational-wave de-tector is laser light with a power of up to hundreds of kilowatts or evenmegawatts The light is quasi-monochromatic and needs to show very lowamplitude (quadrature) noise and phase (quadrature) noise at sideband fre-quencies within the detection band Low amplitude noise is necessary toprovide a shot noise limited output field It is also necessary to avoid classi-cal radiation pressure noise which becomes an issue if the light power or themirror masses in the two arms are not identical Low phase noise is requiredif the storage time of the light in the two arms is not identical This mightaccidentally occur due to different linewidths of the arm cavities or mightbe part of the interferometer design to allow for the length control schemeproposed by Schnupp [Heinzel et al (1998)] To maximize the light power in-side the interferometerrsquos cavities it should be produced in an almost perfecttransversal spatial distribution of a Gaussian TEM00 mode

Light sources of gravitational-wave detectors are ultra-stable NdYAGmaster-slave systems that provide up to 200 W of light at 1064 nm [Winkel-mann et al (2011) Kwee et al (2012)] The high power in the interferometerarms is achieved by cavity built-ups in the so-called power-recycling cavityand in the arm cavities Power recycling uses a partially reflective mirror thatis located between the light source and the interferometer beam splitter Itssurface is matched to the lightrsquos wave front and forms an optical cavity to-gether with the rest of the interferometer Since gravitational-wave detectorsare operated close to a dark fringe large power built-ups can be achievedThe highest power built-up is achieved for a mirror transmission equal to the(given) interferometer round trip loss In this case an impedance-matchedcavity is achieved The power-recycling cavity as well as the arm cavitiesare stabilized on resonance for the input light The difference between theirfunctionality is that the power-recycling cavity does not limit the detectionbandwidth of the interferometer GEO 600 as well as Advanced LIGO em-ploy a third type of cavity the so-called signal-recycling cavity Similarlyto power recycling a partially reflecting mirror that is placed between theoutput port of the beam splitter and the photodiode is used to resonantly en-hance the GW signal [Meers (1988)] The signal-recycling cavity resonantlyenhances the signal modulation fields within its linewidth without furtherenhancing the carrier light power In combination with low linewidth Fabry-

81

Perot arm resonators it can also be used to extract the signal by reducing theeffective finesse of the arm resonators for the signal sidebands This schemeis called resonant sideband extraction [Heinzel et al (1996)] The signal-recycling cavity has also been tested in a detuned setting in which just theupper or lower sideband is extracted or resonantly enhanced respectively[Heinzel et al (2002)] Current gravitational-wave detectors however usecarrier-tuned signal recycling

All these techniques are lsquoclassicalrsquo approaches for maximizing the signal-to-shot-noise ratio At frequencies above a few hundred Hertz howevershot-noise is still the limiting noise source in gravitational-wave detectorsFuture gravitational-wave detectors will therefore use even higher light pow-ers but further increasing the light power becomes more and more challeng-ing Optical materials with less light absorption need to be found to coun-teract an increasing thermal load inside the mirrors Mirror masses need tobe further increased to counteract the increasing radiation pressure noiseNonclassical approaches are superior and become more and more attractivethe farther classical approaches are pushed to the extremes Nonclassicalapproaches allow for simultaneously increasing the signal-to-shot-noise ratioand the signal-to-radiation-pressure-noise ratio without changing light poweror mirror masses see Fig 24 They also allow for a complete evasion of radia-tion pressure noise [Braginsky and Khalili (1995 1996) Kimble et al (2001)]see Subsec 56

63 Squeezed-light enhancement of the gravitational-wave detector GEO 600

In 2010 GEO 600 was equipped with the squeezed-light source shownin Fig 18 The location of the squeezed-light source close to the outputport is shown in Fig 30 It was known that GEO 600 was shot-noise limitedat sideband frequencies above about 700 Hz In this frequency regime thereplacement of the ordinary vacuum states that entered the interferometerfrom the output port by a spectrum of squeezed vacuum states was expectedto reduce the noise spectral density into the nonclassical regime It was notprecisely clear what squeezing factors could be expected since the opticalloss upon mode-matching an external field into the output port propagationalong the arms and the final photo-electric detection was not determined

Critical components were the quantum efficiency of the photo-diode aswell as the optical loss of Faraday rotator for overlapping the squeezed fieldwith the output mode Also the transversal output mode of GEO 600 was an

82

issue since it contained significant contributions from higher order modesto which a good mode matching of the squeezed mode was not possible Theimplementation of the squeezed-light source thus was accompanied with anew custom made InGaAs photo-diode with 3 mm diameter The goal was aquantum efficiency of greater 99 [Vahlbruch et al (2016)] Also the Fara-day rotator was custom-made and optimized for lowest optical loss which

Figure 30 GEO 600 ndash View into the central building of the British-German GW detectorlocated close to Hannover Germany The vacuum chambers contain the suspended beamsplitter power- and signal recycling mirrors additional input and output optics as well asmirrors to realize a double pass of the laser light through the 600 m long interferometerarms By courtesy of the AEI

83

involved a rather precise rotation of the polarisation of 45 plusmn 05 over anaperture of more than 15 mm Finally a ring cavity (output mode cleaner)was placed in front of the photo-diode which acted as a passive filter forhigher transversal modes Since GEO 600 was not limited by radiation pres-sure noise and since it used a carrier-tuned signal-recycling cavity a frequencyindependent orientation of the squeezing angle was optimum After several

100 200 300 500 700 1000 2000 3000 500010-22

10-21

10-20

Obs

erva

tory

noi

se c

alib

rate

d to

str

ain

[1radic

Hz]

10-19

10-18

10-17

Noise calibrated to test m

ass displacement [m

radicHz]

Sideband frequency f

Figure 31 Nonclassical reduction of the GEO 600 instrumental noise ndash Shownare the square roots of the noise spectral densities without squeezed vacuum states (top)and with squeezed vacuum states (bottom) [Abadie (2011)] Their calibrations [Affeldtet al (2014)] to gravitational-wave strain and differential arm length change are shownon the left and right y-axes respectively Note that both traces increase towards higherfrequencies due to the frequency-dependent signal enhancement of the signal-recyclingcavity The injection of squeezed vacuum states leads to a broadband noise reductionof up to 35 dB at shot-noise limited frequencies The spectral features are for instancecaused by excited violin modes of the mirror suspensions (600ndash700 Hz and harmonics)Data by courtesy of the AEI

months the combination of the squeezed-light source and the gravitational-wave detector succeeded The main laser of the squeezed-light source wasphase locked to the main laser of GEO 600 and a stable mode matching be-

84

tween the squeezed field and the interferometer output field was achieved[Abadie (2011)] The effect on GEO 600rsquos (strain normalized) noise spectraldensity is shown in Fig 31 At frequencies above about 700 Hz the squareroot of noise spectral density was reduced by up to one third This value cor-responds to a quantum noise variance of 045 (minus35 dB) with the shot noisevariance normalized to unity For isotropically distributed gravitational-wavesources this factor produces a detection rate increase by a factor of 153 asymp 34After its integration into GEO 600 the squeezed-light source was used in allscientific runs seeking for gravitational waves for instance in the observa-tional run S6eVSR4 that was undertaken from June 3rd to September 5th

in 2011 [Grote et al (2013)]Towards the end of 2011 right before the start of the detector upgrade

to Advanced LIGO a nonclassical sensitivity improvement was also demon-strated in one of the LIGO detectors [Aasi (2013)] In this experiment anonclassical sensitivity improvement corresponding to up to 215 dB abovefrequencies of about 150 Hz was achieved The successful test is a strongmotivation for a squeezed-light upgrade of Advanced LIGO Note that thedesign of Advanced LIGO was completed in 1999 and squeezed-light sourceswere not mature at those times

In the past years the squeezing enhanced GEO 600 detector was not onlyused for observations but also was the control of the injected squeezed modefurther improved Stabilizing the overlap between squeezed mode and brightmode of the interferometer to close to perfect is necessary to reduce theeffective optical loss and to maximize the measurable squeezing factor Re-cently superior methods for stabilizing the longitudinal phase of squeezedvacuum mode were found [Dooley et al (2015)] and the first automatic align-ment system for stabilizing and optimizing the transversal mode overlap wasdemonstrated [Schreiber et al (2016)]

64 Are squeezed states the optimal nonclassical resource ingravitational-wave detectors

For a given number of photons Eq (41) quotes the ultimately smallestphase change that can be measured with a signal-to-noise-ratio of one Thescaling with number of photons per measuring time of this Heisenberg limitseems appealing compared to the scaling achievable with coherent states orsqueezed states according to Eqs (38) and (39) respectively The Heisenberglimit however is only valid for precisely zero photon loss Since the non-classical states required to achieve Eqs (40) and (41) show an exponentially

85

increasing sensitiveness to loss when increasing the photon number the ac-tual scaling can not be deduced from Eq (40) Proposals to use Fock statesand the so-called N00N states for optimizing interferometer sensitivities [Hol-land and Burnett (1993) Dowling (1998) Mitchell et al (2004) Afek et al(2010)] are thus only applicable when the experiment is conditioned on zerophoton loss As discussed in recent publications the correct expression forthe fundamental sensitivity limit needs to consider not only the total photonnumber inside the interferometer but also the total photon loss [Dorner et al(2009) Ko lodynski and Demkowicz-Dobrzanski (2010) Knysh et al (2011)Escher et al (2011) Demkowicz-Dobrzanski et al (2012)]

Based on these earlier works Ref [Demkowicz-Dobrzanski et al (2013)]proved that the nonclassical sensitivity enhancement of GEO 600 reportedin Ref [Abadie (2011)] has been exceedingly close to fundamental quantuminterferometry bound under given energy constraints and photon loss lev-els More than that it was generally proven that the approach of com-bining displaced coherent states and squeezed vacuum states is optimal forgravitational-wave detectors

In Ref [Abadie (2011)] the gravitational-wave detector GEO 600 used aneffective number of photons per second of approximately n = 2 middot 1022 whichcorresponded to a total optical power inside the interferometer arms of P asymp37 kW at a wavelength of 1064 nm The total optical loss was 1minus η asymp 038The injected squeezing factor was eminus2r asymp 01 For these numbers the ratioof Eqs (43) and (44) is calculated to

∆φCSVmin

∆φgenmin

asymp

radicηeminus2r + 1minus η

1minus ηasymp 108 (62)

which is a good approximation within the limit of large coherent state dis-placements α sinh2r The quantum noise of GEO 600 including thesqueezed-light source was just 8 above the fundamental quantum inter-ferometry bound An increased squeezing strength of 16 dB (eminus2r asymp 0025)which is in reach would bring the approach based on coherent states andsqueezed vacuum states to within just 2 above the fundamental bound

Future GW detectors will have significantly reduced optical loss values(1 minus η) lsquoLossrsquo includes scattering and absorption at mirrors non-perfectfringe contrasts and the non-perfect quantum efficiency of the photo detec-tor Optical loss reduction is important for at least four reasons First itleads to an increased signal second it leads to a reduced quantum noise when

86

employing squeezed states third less absorption reduces the thermal load onthe test mass mirrors and fourth less scattering reduces the probability ofback-scattered light which produces disturbance signals [Billing et al (1979)Vahlbruch et al (2007) Punturo et al (2014)] The higher the finesse valuesof the arm and signal-recycling cavities are the more significant is opticalloss at mirror test masses the beam splitter and the signal-recycling mirrorThe finesse value of the power-recycling cavity and the loss of mirrors andlenses that guide the output field to the photo-diode are less critical Suitablephoto detectors of 995 quantum efficiency are available today [Vahlbruchet al (2016)] but achieving a total optical loss of 10 is still challengingThe reason for that is that first of all a measurement device aiming for bestabsolute sensitivity should use as much quanta (photons) as possible Highfinesse values for the enhancement cavities are thus essential but results inan unavoidable scaling-up of the effect of mirror losses A realistic exampleof future gravitational wave detectors thus considers η = 09 with a squeezingfactor of 20 dB (eminus2r = 001) In this case the quantum noise will be about4 above the ultimate fundamental bound for a given photon number

From Eq (62) it can be concluded that there is no need for any moresophisticated nonclassical states than squeezed states In particular non-classical states with a defined photon number such as N00N states are notrequired Within the approximation quoted this result is independent ofthe photon number This result is also independent of the physical systemused for interferometric phase estimation and can also be made for quantum-enhanced atomic clock calibration in the presence of dephasing Here the-oretical results indicate that the precision of Ramsey interferometry withspin-squeezed states is close to the optimal one in the asymptotic regime ofa large number of atoms [Huelga et al (1997) Ulam-Orgikh and Kitagawa(2001) Escher et al (2011)] as already stated in Ref [Demkowicz-Dobrzanskiet al (2013)] More sophisticated nonclassical states with fixed number ofn quanta might still be useful for the exceptional case when the absorptionof one quantum already results in zero measurement sensitivity anyway Anexample is an ensemble measurement where the absorption of a single photondemolishes the source of the phase change to be characterized A typicallyused approach of conditioning the measurement result on n clicks of n singlephoton counters conditions on precisely zero loss and is thus able to use theadvantage of Eq (40) over Eq (39)

87

65 Conclusions

Squeezed states of light will contribute to realizing gravitational-wave ob-servatories with much higher sensitivities than existing or planned ones Tobenefit from squeezed states in a most efficient way optical loss in terms ofabsorption and scattering must be minimized In particular the optical lossof mirror coatings and mirror substrates need to be minimized The rele-vant mirrors include the test masses the balanced beam splitter the signalrecyclingextraction mirror and all optical components between the latterand the photo diode Excellent spatial mode matching between the brightinterferometer field and the squeezed vacuum field is also of great impor-tance Achieving this requires further improvement of the surface figures ofall reflective optical components of the interferometer as well as improvedhomogeneity of all optical components that the light passes through

The quantum noise reduction achieved in a gravitational-wave detectoris of course always smaller than the highest squeeze factor provided by thesqueezed-light source As an example let us consider the observation of 15 dBof nonclassical noise suppression directly at the source If the squeezed fieldsenses an additional loss of 5 when propagating through the interferometerwhich is a very challenging number from todayrsquos point of view the remainingsqueezing level is about 11 dB see Eq (16)

7 The application of 2-mode-squeezed light in laser interferome-ters

71 Quantum Dense Metrology

At first glance the application of bi-partite (two-mode) squeezed statesto a device whose goal is measuring a single observable seems meaninglessSqueezing the uncertainty of that observable should be the optimum one cando This is indeed true when concerning just quantum noise but recentlyit was discovered that in the presence of classical disturbances bi-partitesqueezing can improve such measuring devices [Steinlechner et al (2013)]The concept was named quantum dense metrology (QDM) The potentialimprovement of a gravitational-wave detector with bi-partite squeezed statesis shown in Fig 32 (a) A description is given in the caption The pre-condition for a potential improvement can be best understood within a phasespace diagram Fig 32 (b) contains two different kinds of lsquosignalsrsquo The firstis the actual signal which always shows up as a phase space displacement

88

along the Y axis The second is a disturbance signal that can produce adisplacement in arbitrary direction in phase space A prominent example

Faraday Rotator

Coherent light

Photo detectors Balanced homodyne detectors

Squeezed vacuum

Squeezed vacuum

Quantum noise in

Y (A) and X (B)

XΩΔΩ(B)

YΩΔΩ(A)

(a) (b)

Disturbance projected

onto Y

and X

XΩΔΩ(B)

YΩΔΩ(A)

Example of (unknown) disturbance

Signal

Figure 32 Bi-partite-squeezed-light-enhanced measurement ndash (a) Setup for theapplication of bi-partite (two-mode) squeezed light in a laser interferometer on the basisof QDM Two squeezed vacuum fields are overlapped on a balanced beam splitter withprogrammable squeeze angles for instance with a relative angle of 90 which produces abi-partite state as shown in Fig 12 The beam splitter outputs are entangled for any rela-tive angle greater than zero One part is matched to the interferometer mode The secondpart is kept outside as a reference beam The interference of the interferometer outputand the reference beam is arranged with such a phase difference that it reproduces the twosqueezed inputs on the photo detectors The two squeezed beams are photo-electricallydetected measuring the respective squeezed quadrature (using balanced homodyne detec-tors) Both beams carry half of all interferometer induced modulations which includesignals as well as disturbances A single readout as shown in Fig 21 cannot distinguishbetween the two kinds The double readout shown here provides additional informationand allows for recognition of the disturbance [Steinlechner et al (2013)] as well as in prin-ciple a modeling of the disturbance and with a correct model an improvement of thenoise spectral density of the interferometer [Ast et al (2016)] (b) Phase space diagramdescribing phase quadrature readout A as well as amplitude quadrature readout B Bothshow squeezed quantum noise The amplitude quadrature readout does not contain anygravitational-wave signal ie any feature in this channel must be due to disturbancesThis information can be used to improve the interferometer

89

for such a disturbance is parasitic interference due to back-scattered laserlight [Vahlbruch et al (2007)] Back-scattering is a limiting noise at lowsignal frequencies of gravitational-wave detectors [Billing et al (1979) Vinetet al (1997) Hild (2007) Ottaway et al (2012) Punturo et al (2014)] Notethat all noise that couples in via unwanted motions of the test mass mirrorsso-called lsquodisplacement noisersquo always produces a phase space displacementalong the Y axis and cannot be tackled with QDM

Fig 33 shows measurement results obtained in Ref [Steinlechner et al(2013)] In a table-top experiment one part of a bi-partite squeezed state ofa continuous-wave mode at 1064 nm was mode-matched into the output portof a Michelson laser interferometer operated at its dark fringe in full analogyto Fig 32(a) A lsquosignalrsquo was produced by driving the piezo behind one of theend mirrors at a frequency of 555 MHz The lsquodisturbancersquo was introducedby re-injecting a small amount of light that leaked through the second endmirror with an additional piezo-mounted mirror The piezo was driven ata frequency of 517 MHz to produce a phase modulation An additionalDC voltage defined an arbitrary and unknown optical path length of thelight before being re-injected and as such the phase space orientation of thedisturbance signal This mechanism of a parasitic interference is realizednaturally in any interferometric device due to back-scattering of quanta frommoving surfaces in the environment

The interferometer output consisted of the signal as well as the distur-bance with a quantum uncertainty given by one subsystem of the bi-partiteentanglement It was overlapped with the second subsystem of the entan-gled state on a balanced beam splitter and the two outputs were analysedwith balanced homodyne detectors The phases of the bi-partite entangle-ment and the BHD local oscillators were controlled to resemble Fig 32(b)ie both BHDs measured a squeezed uncertainty regardless of the phase ofthe (generally unknown) disturbance

The beam splitter that combines interferometer output and the entangledreference beam unavoidably splits the signal as well as the disturbance intotwo paths For a balanced beam splitter this generally reduces the signaland disturbance power by 3 dB for both quadrature measurements Fig 33shows however that both BHDs performed about 6 dB below shot noisewhich demonstrates the usefulness of the scheme The squeeze factor can inprinciple be infinite which thus qualifies the lsquo3 dB penaltyrsquo In the abovefigure the additional information from the second BHD output was used torecognize the parasitic interference in the first BHD output providing a lsquovetorsquo

90

50 51 52 53 54 55 56 57

50 51 52 53 54 55 56 57-70

-75

-80

-85

-90

-70

-75

-80

-85

-90

Y (A) (Ω2π)Ω∆Ω

Frequency ( f )

Noi

se p

ower

(dB

m)

Noi

se p

ower

(dB

m)

X (B) (Ω2π)Ω∆Ω

~ ndash6 dB

~ ndash6 dB

Shot noise reference

Shot noise reference

Signal Projected disturbance

Projected disturbance

Vet

o

No signal

Figure 33 Bi-partite-squeezed-light-enhanced measurement ndash The result wasachieved in a table-top setup [Steinlechner et al (2013)] In the two panels the lower

(blue) traces show the squeezed quadrature noise-power spectra ∆2Y(A)Ω∆Ω(Ω2π) (top) and

∆2X(B)Ω∆Ω(Ω2π) (bottom) as simultaneously measured with balanced homodyne detectors

lsquoArsquo and lsquoBrsquo respectively The conventional Y -measurement (top) cannot distinguish be-tween signal and disturbances The additional X-measurement (bottom) does not detectany phase quadrature signal thus any feature in this measurement is a parasitic signal dueto a disturbance The respective projection onto the Y -measurement can thus be lsquovetoedrsquoIn a more sophisticated approach the X-data might be used to model and then to elim-inate the disturbance as well as its projections on both quadrature measurements Theresult is a reduced spectral density of the actual phase quadrature measurement [Ast et al(2016)] Traces shown here are slightly sloped due to the decreasing transfer functions ofthe balanced homodyne detectors The resolution bandwidth was ∆Ω(2π) = 10 kHz thevideo bandwidth was 100 Hz All traces were averaged three times

signal to trigger its removal from the data streamThe question arose whether the additional information can be used to

reduce the actual noise spectral density of the first measurement ie to re-cover signals that were buried by parasitic interferences Very recently it wasshown that this is indeed possible Ref [Ast et al (2016)] reports a table-top proof-of-principle experiment in which the additional information in the

91

QDM approach could be used for improving the sensitivity of an interfero-meter The measurement sensitivity was improved from above-shot-noise tosub-shot-noise (sub-Poissonian) performance This result was possible notbecause the way the parasitic interference arose was known but because theadditional information provided by QDM allowed for fitting a model of theexcess noise to the readout data

Quantum dense metrology (QDM) as shown in Fig 32 improves a mea-surement by simultaneously reading out two conjugate observables Bothreadout observables show a squeezed quantum noise and act as estimatorsof independent physical quantities This situation was recently described aslsquoquantum-mechanics freersquo [Tsang and Caves (2012)] QDM is based on anEinstein-Podolsky-Rosen (EPR) entangled [Einstein et al (1935)] bi-partitesystem as described in Subsec 34 EPR entanglement was previously con-sidered for the quantum-informational task of dense coding which doublesthe capacity of quantum communication channels [Bennett et al (1992)Braunstein and Kimble (2000)] The application of EPR entanglement inmetrology was first proposed by DrsquoAriano et al [DrsquoAriano et al (2001)]

72 Conclusions

A single beam that carries an optimized spectrum of squeezed vacuumstates and that is injected into the interferometerrsquos dark port provides themost efficient and practically optimal approach to reduce the quantum noisein laser interferometers by means of nonclassical states (see Section 5) Theconclusion of the section here is that two entangled beams provide a superiorapproach if the interferometerrsquos sensitivity is limited by classical noise thatis not exclusively restricted to the actual observable which is the phasequadrature amplitude Y Parasitic interferences due to laser light that isbackscattered from vibrating surfaces are an example Current gravitational-wave detectors use light fluxes of about 1024 photons per second [Abbott(2016)] Just a single photon per second and hertz which leaves the mainlight beam and is backscattered from a vibrating surface and in this way getsfrequency shifted into the detection band produces a significant disturbancesignal The lsquoquantum-dense metrologyrsquo approach might provide a powerfultechnique to tackle this problem

Very recently it turned out that QDM is not the only technique that mayexploit EPR entanglement to improve phase measurements Ref [Ma et al(2017)] proposes to use EPR entanglement to simultaneously suppress shotnoise and radiation pressure noise in a gravitational-wave detector without

92

the need for an additional filter cavity (confer subsection 55) In this caseEPR entanglement is exploited that is carried by one broadband squeezedbeam and that is present between quadrature amplitudes defined with re-spect to different optical frequencies ω and ωprime as investigated in Ref [Hageet al (2010)] Such lsquofrequency multiplexedrsquo EPR entanglement might resultin considerably lower costs of building a gravitational-wave detector with abroadband simultaneous squeezing of shot noise and radiation pressure noiseAlso this proposal does not lead to a fundamentally lower quantum noise butrather improves on classical aspects of an interferometer

8 Summary and Outlook

In many cases experiments that involve interference of quantum statescan be described in a semi-classical way This description uses the classicalwave picture for the interference part of the experiment and subsequently theclassical particle picture when the states transfer their energy to a detectoror more generally to a thermal bath This semi-classical description is notpossible when using the specific class of lsquononclassicalrsquo states Squeezed statesof light are a prominent example of these Squeezed states and other nonclas-sical states allow for observations that made Einstein Podolsky and Rosenformulate their critical and seminal paper on quantum theory [Einstein et al(1935)]

In the review here it is argued that after many successful proof-of-principle experiments with nonclassical states in the past decades the routineuse of squeezed-light in observational runs of the gravitational-wave detectorGEO 600 goes beyond proof-of-principle and is a true application of nonclassi-cal light Since 2010 the squeezed-light source has improved the measurementsensitivity of GEO 600 in basically every observational run [Abadie (2011)Grote et al (2013) Dooley et al (2016)] At quantum noise limited frequen-cies ie above a few hundreds of hertz the sensitivity has been improvedcorresponding to a squeezing strength in the noise spectral density of up to37 dB which corresponds to an increase of the average gravitational-wavedetection rate by a factor of 043minus32 = 36 This success is a strong motiva-tion to also equip the Advanced LIGO Virgo and Kagra gravitational-wavedetectors with squeezed light Similar improvement factors even down tolower signal frequencies are expected [LSC (2013)] The achievable improve-ment factors are mainly limited by the optical loss on the squeezed states

93

and much higher factors are achievable in principleUp to now squeezed states have not been used to reduce the radiation-

pressure noise in gravitational-wave detectors The reason is that so far othernoise sources are larger than radiation pressure noise and such an effect can-not be observed It is expected however that future gravitational-wavedetectors will eventually be partly limited by radiation pressure noise Fromthis point on squeezed light will be used to simultaneously reduce shot noiseand radiation pressure noiseSqueezed states are the optimum nonclassical states for gravitational-wavedetectors or more generally for all laser interferometers operating with largeaverage photon numbers per measuring interval [Demkowicz-Dobrzanski et al(2013)] In addition to using higher light powers and heavier test mass mir-rors higher squeeze factors will thus contribute to mitigate the lightrsquos quan-tum noise in laser interferometers

Two-mode (bi-partite) squeezed light has not been used in gravitational-wave detectors so far They are not capable of further reducing the quantumnoise in laser interferometers but they can be used to mitigate classicalnoise that originates from fluctuating phase space displacements A well-known such noise source is back-scattered light Proof-of-principle experi-ments were performed recently [Steinlechner et al (2013) Ast et al (2016)]This new technique could turn out to be valuable in next generations ofgravitational-wave detectors in particular in those targeting high sensitivi-ties at low sub-audio signal frequencies and using high light powers Suchan implementation in gravitational-wave detectors does not require any newtechnology Compared to a squeezed-light enhanced interferometer just asecond squeezed-light source is required

It is certainly remarkable that those quantum states that made EinsteinPodolsky and Rosen falsely think quantum theory incomplete are now ex-ploited as new technologies in measurement devices targeting new observa-tions in nature

Acknowledgements

RS thanks M Ast J Bauchrowitz C Baune S Chelkowski J DiGugliel-mo A Franzen B Hage J Harms A Khalaidovski L Kleybolte NLastzka M Mehmet S Steinlechner and H Vahlbruch for their contri-butions many fruitful discussions and their support with the figures and JFiurasek for many valuable comments on the manuscript Thanks are also

94

due to Y Chen F Khalili and H Miao for fruitful discussions within thequantum noise working group of the LIGO Scientific Collaboration (LSC)Special thanks are due to H Vahlbruch and H Grote together with theGEO 600 team for their pioneering work on the squeezed-light implementa-tion in GEO 600 RS is supported by the Deutsche Forschungsgemeinschaft(Grant No SCHN 757-6) and by the European Research Council (ERC)project lsquoMassQrsquo (Grant No 339897)

References

References

Aasi J et al Jul 2013 Enhanced sensitivity of the LIGO gravitationalwave detector by using squeezed states of light Nature Photonics 7 (8)613ndash619URL httpwwwnaturecomdoifinder101038nphoton2013177

Aasi J et al Apr 2015 Advanced LIGO Classical and Quantum Gravity32 (7) 074001URL httparxivorgabs14114547httpstacks

ioporg0264-938132i=7a=074001key=crossref

20895763c84bce3f8929251031b2475c

Abadie J et al Sep 2011 A gravitational wave observatory operatingbeyond the quantum shot-noise limit Nature Physics 7 (12) 962ndash965URL httparxivorgabs11092295httpwwwnaturecom

doifinder101038nphys2083

Abbott B P et al Feb 2016 Observation of gravitational waves from abinary black hole merger Phys Rev Lett 116 061102URL httplinkapsorgdoi101103PhysRevLett116061102

Acernese F et al 2015 Advanced virgo a second-generation interferome-tric gravitational wave detector Classical and Quantum Gravity 32 (2)024001URL httpstacksioporg0264-938132i=2a=024001

95

53 Quantum back-action and quantum radiation pressure noise 6554 Interferometer total quantum noise and the standard quantum

limit 6755 Squeezed light for surpassing the standard quantum limit 6956 Optomechanically induced (ponderomotive) squeezing 7557 Conclusions 77

6 The first application of squeezed light in an operating gravita-tional-wave detector 7761 Gravitational waves 7862 Interferometric detection of gravitational waves 7963 Squeezed-light enhancement of the gravitational-wave detector

GEO 600 8264 Are squeezed states the optimal nonclassical resource in

gravitational-wave detectors 8565 Conclusions 88

7 The application of 2-mode-squeezed light in laser interfero-meters 8871 Quantum Dense Metrology 8872 Conclusions 92

8 Summary and Outlook 93

3

1 Introduction

Laser interferometers are used to monitor small changes in refractiveindices rotations or surface displacements such as mechanical vibrationsThey transfer a differential phase change between two light beams into achanging power of the output light which is photo-electrically detected forexample by a photo diode The light is produced in a lasing process thatusually aims for a coherent (Glauber) state In practice laser light is oftenin a mixture of coherent states producing excess noise in the interferometricmeasurement But even if the laser light is in a (pure) coherent state its de-tection is associated with noise usually called lsquoshot-noisersquo This arises fromthe quantisation of the electro-magnetic field which for a coherent stateresults in Poissonian counting statistics of mutually independent photons

1000099009800 10100 10200

Photon number n

0000

0002

0004

0006

0008

0010

0012

Pro

babi

lity Squeezing

Poisson distribution

Figure 1 Poissonian and squeezed photon statistics ndash The upper boundary of eacharea represents the probability distribution of detected photon number n when perform-ing a large number of measurements on an ensemble of identical states having an averagephoton number of n = |α|2 = 10000 where α is the coherent field excitation or lsquodis-placementrsquo The broader curve shows the lsquoPoissonianrsquo distribution which describes thecounting statistic of mutually independent particles ie those of the coherent state Dueto the large value of α the distribution is almost Gaussian with a standard deviation ofplusmnradicn The narrow curve corresponds to the equally displaced 10 dB squeezed state which

obviously has a lsquosub-Poissonianrsquo photon statistic Note that squeezed states with smallor even without any coherent excitation (squeezed vacuum states) exhibit quite differentphoton statistics ndash see Fig 13 for example

4

If the coherent state is highly excited and thus the average number ofphotons n per detection interval is large the Poissonian distribution canbe approximated by a Gaussian distribution with a standard deviation ofplusmnradicn During the past decades squeezed states of light have attracted a lot

of attention because they can exhibit less quantum noise than a coherent stateof the same coherent excitation ie they can show sub-Poissonian countingstatistic see Fig 1

θ

Shot noise Squeezed noise

(a) (b)

Brightlaser input

Squeezedvacuum input

Faradayrotator

Photo diode

Michelsoninterferometer

Signal5050

(i)Ph

oto

curr

ent [

rel

units

]

Time [ms]

(ii)

0 5 10

Figure 2 Squeezed-light enhanced Michelson interferometer ndash (a) In addition tothe conventional operation of a Michelson laser interferometer with bright coherent lighta broadband squeezed-vacuum field is injected into the signal output port and overlappedwith the bright interferometer mode The interferometer is operated close to a dark fringesuch that most of the bright coherent light as well as most of the squeezed vacuum areback-reflected from the Michelson interferometer respectively Due to interference withthe broadband squeezed vacuum the interferometerrsquos output light on the photo diodeshows reduced variance in the photon number statistic as shown in Fig 1 Overlappingthe two light fields is possible with theoretically zero loss by the combination of a Faradayrotator and a polarizing beam splitter (PBS) A signal is produced by modulating therelative arm length (b) Simulated data for photo diode measurements Without squeezing(i) the signal of the laser interferometer is not visible With squeezing (ii) the shot noiseis reduced and here a sinusoidal signal visible

Squeezed states belong to the class of lsquonon-classicalrsquo states which areconsidered to be at the heart of quantum mechanics These states are de-fined as those that cannot be described as a mixture of coherent states Inthis case their Glauber-Sudarshan P -functions [Sudarshan (1963) Glauber(1963)] do not correspond to (classical) probability density functions ie theyare not positive-valued functions As a lsquoclassicalrsquo example the P -function ofa coherent state corresponds to a δ-function

5

But the question remains what property of coherent states justifies thename lsquoclassicalrsquo even though coherent states are quantum states and showquantum uncertainties My answer to this question is the following Allexperiments which only involve coherent states and mixtures of them allowfor a description that uses a combination of classical pictures As we will seebelow this description swaps between two different classical pictures and isthus not truly classical but semi-classical (A more precise description of thenature of coherent states uses the term lsquosemi-classicalrsquo)

Let us consider a laser interferometer that uses light in a coherent stateFirstly the light beam is split in two halves by a beam splitter The twobeams travel along different paths and are subsequently overlapped on abeam splitter where they interfere exactly as classical waves would do Theelectric fields superimpose thereby producing the phenomenon of interfer-ence Up to this point there is no reason to argue light might be composedof particlesSecondly the new (still coherent) beams that result from the interference areabsorbed for instance by a photo-electric detector In the case of coherentstates the detection process can be perfectly described in the classical parti-cle picture in which the particles appear independently from each other in atruly random fashion yielding the aforementioned Poisson statistic Duringthe detection process no wave feature of the light is present Let us havea closer look A truly random (lsquospontaneousrsquo) event is an event that hasnot been triggered by anything in the past This allows us to make a clearcut between the first part of the experiment described by the classical wavepicture and the second part of the experiment described by the classicalparticle picture Both lsquoworldsrsquo are disconnected The subsequent applicationof two classical pictures is not truly classical but lsquosemi-classicalrsquo It is in-deed the observation that the photons occur individually with truly randomstatistics that allows this semi-classical description In the case of a mixtureof coherent states the photon statistics are super-Poissonian which can beunderstood as a mixture of different Poissonian distributions In the caseof a slowly changing coherent state the mean value n depends on time Inall these cases the semi-classical description is appropriate Let me pointout that in this very reasonable description photons do not exist before theyare detected eg absorbed Further note that the famous double-slit exper-iment with coherent states also allows for the same semi-classical description

For squeezed states [Yuen (1976) Walls (1983)] the situation is different

6

As before the interference can be fully described by the classical wave pic-ture The result of the detection process however is different from that ofmutually independent random events It is also different from any super-Poissonian statistics that could be produced by mixing an arbitrary num-ber of different andor time-dependent Poissonian distributions Insteadthe squeezed probability distribution in Fig 1 suggests that the probabil-ity of detecting a photon decreases with the more photons that are alreadydetected in the same time interval over which a single measurement is in-tegrated From this observation one must conclude that the photons donot individually appear in a random fashion upon detection There mustbe lsquoquantumrsquo correlations between the photons These correlations mustexisted before detection since there is no interaction between the photonsduring their detection Pre-existing correlations between detected photonsseem to imply that the photons themselves existed before detection ie attimes when interference occurred In a semi-classical description howeverphotons are classical particles and cannot interfere for instance on a beamsplitter At this point the semi-classical picture breaks down Squeezedstates are therefor lsquononclassicalrsquoThe failure of the semi-classical model described above generally certifiesnonclassicality

Squeezed states are usually not characterized by counting their pho-tons but by measuring canonical continuous-variable phase-space observ-ables Measurements are performed as usual on an ensemble of identicalstates and quasi-probability density functions are calculated from the dataThe Glauber-Sudarshan P -function is the quasi-probability density distribu-tion over coherent states If the P -function of a state is entirely positivethe state is a coherent state or a (classical) mixture of coherent states Thestate is considered as semi-classical If the P -function is not a positive-valuedfunction the state cannot be expressed as a (classical) mixture of coherentstates and is thus nonclassical [Gerry and Knight (2005) Vogel and Welsch(2006)] A non-positive-valued P -function is the sufficient and necessary con-dition for the failure of the semi-classical model The Wigner function is thequasi-probability phase-space representation over the canonical continuous-variable phase-space observables themselves [Gerry and Knight (2005)] TheWigner functions of squeezed states are entirely positive Although subject todiscussion this fact does not mean that squeezed states are less nonclassicalthan Fock states or cat states which not only have a nonclassical P -function

7

but also a partially negative Wigner function (A cat state is a quantum su-perposition of two macroscopically distinct states [Monroe (2002)] referringto Schrodingerrsquos-cat gedanken experiment [Schrodinger (1935)]) In practicesqueezed states can even be regarded as superior nonclassical states becausethey represent the only nonclassical state that has been produced in a steadystate fashionIn almost all experiments so far the generation of Fock states and cat statesinvolves a probabilistic event such as the detection of a photon in anotherbeam path to herald these states In fact squeezed states provide the non-classical resource for the probabilistic preparation of Fock states as well as catstates But only the squeezed states themselves show a nonclassical effect in astationary way Limited only by the time duration and the frequency span ofthe mode that is in a squeezed state the squeezing effect can be continuouslyobserved independently of the time when the measurement is performed andalso independently of the measurement integration time This fact is of greatimportance for applications of squeezed states in measurement devices sincea squeezed-light-enhanced measurement remains unconditional and the ef-fective measurement time is not reduced

In past decades squeezed states of light were used in many proof-of-principle experiments to research their potential for improving the sensitivityof laser interferometers [Grangier et al (1987) Xiao et al (1987) McKen-zie et al (2002) Vahlbruch et al (2005) Goda et al (2008) Taylor et al(2013)] or the performance of imaging beyond the shot-noise limit [Lugiatoet al (2002) Treps et al (2003)] both accompanied by a huge number oftheoretical works Potential applications in secure optical communication(quantum key distribution) were also proposed and proof-of-principle ex-periments demonstrated [Ralph (1999) Furrer et al (2012) Gehring et al(2015)] This review restricts itself to the improvement of laser interfero-meters since only here has the application of squeezed light gone beyondproof-of-principle The gravitational-wave detector (GWD) GEO 600 hasoperated with squeezed light now for more than seven years starting in 2010[Abadie (2011) Grote et al (2013)] GEO 600 is a 600 m long Michelsonlaser interferometer built for the detection of gravitational waves Thesewaves are audio-band and sub-audio-band changes of space-time curvatureoriginating from cosmic events such as the merger of neutron stars or blackholes as detected recently [Abbott (2016)] In GWDs such as GEO 600 [Doo-ley et al (2016)] Advanced LIGO [Aasi (2015)] Advanced Virgo [Acernese

8

(2015)] and KAGRA [Aso et al (2013)] conventional laser technology hasbeen pushed to extremes over the past decades Noise spectral densities nor-malized to space-time strain of less than 10minus23 Hzminus12 have been measured[Abbott (2016)] Progress will continue and based on the successful appli-cation in GEO 600 squeezed light is now widely accepted to provide a newadditional technology to contribute to the new field of gravitational-waveastronomy It was also successfully tested in one of the LIGO detectors in2013 [LSC (2013)] and is an integral part of the European design study forthe 10 km Einstein-Telescope [Punturo et al (2010)]

GEO 600 has already taken several years of lsquosqueezedrsquo observational datawhich has increased its sensitivity at signal frequencies above 500 Hz Withthe implementation of a squeezed light source in GEO 600 the applicationof nonclassical states in metrology has been pushed beyond merely proof-of-principle

lsquoTwo-mode squeezed statesrsquo show a squeezed uncertainty in at least onejoint continuous variable of two subsystems lsquoArsquo and lsquoBrsquo Examples of jointvariables are differences and sums of phase-space observables of A and BTwo-mode squeezed states not only belong to the class of nonclassical statesbut due to their bi-partite character also to the class of lsquoinseparablersquo orlsquoentangledrsquo states They are the ideal states to demonstrate the Einstein-Podolsky-Rosen paradox [Einstein et al (1935)] as first achieved in [Ouet al (1992)] Apart from fundamental research on quantum mechanicsrecent proof-of-principle experiments demonstrated their usefulness in inter-ferometric measurements that go beyond the application of simple squeezedstates [Steinlechner et al (2013) Ast et al (2016)] This experiment is thefinal topic of this review

2 Observations on light fields in squeezed states

Generally there are two different kinds of observables that can be subjectof a measurement performed on a quantum system The first kind is associ-ated with the systemrsquos wave property In optics it corresponds to the electricfield strength at a given phase angle ϑ The according (dimensionless) opera-tors are called the quadrature amplitudes Xϑ and have a continuous spectrumof eigenvalues Quadrature amplitudes are measured in very good approx-imation with a balanced homodyne detector using the interference with a

9

bright local oscillator beam see Fig 3 (a) In practice any measurement ofXϑ integrates over some sideband (Fourier) spectrum within the angular fre-quencies Ω plusmn ∆Ω2 The sideband information always needs to be quotedA straight forward but rather untypical way is by adding subscripts whichleads to Xϑ

Ω∆Ω The classical analogue of the quadrature amplitude operator

XϑΩ∆Ω is the modulation depth of the optical field at modulation phase angle

ϑ and at angular modulation frequency Ω measured over the band ∆Ω lt 2ΩThe uncertainties of the statersquos quadrature amplitudes at different phases ϑare limited by a Heisenberg uncertainty relation see section 3 The secondkind of measurement is associated with the systemrsquos particle property andis given by the photon number operator n associated with a measuring timeinterval ∆t Its precise measurement requires a photon counter ideally withsingle photon resolution The measurement result obviously has a discretespectrum Continuous as well as discrete observables are usually subject toquantum uncertainties and thus quantum noiseUsually the measurementrsquos integration time and frequency band actuallydefine the physical system that is characterized In quantum optics experi-ments the interrogated physical system is called a lsquomodersquo

21 Definition of a lsquosingle modersquo

Let us define a light field or generally any quantum system to be asingle mode if it corresponds to the lsquosmallest entity of a waversquo In this caseits spectral and temporal distributions as well as waist size and divergenceare at their Fourier limits and all other properties such as optical axis waistposition and polarization are well defined For instance a linearly polarizedlongitudinal resonance of an optical standing-wave cavity defines such a singlemode if the cavity finesse is high and transversal modes are non-degenerateThe complete photo-electrical detection of a cavity mode however is notstraight forward Most quantum optical experiments are instead performedon propagating light In this case single modes are defined by spatial filtersand by temporal-spectral measurement windows both being at the Fourierlimit Examples for single modes are a laser pulse and a spectraltemporalcutout from a continuous observation of a quasi-monochromatic continuous-wave light beam in the spatial TEM00 mode both at the Fourier limits

In classical physics the only remaining free parameter of a given singlemode is its excitation energy In quantum physics the situation is differentFor a given energy a single mode can be in many different quantum states

10

which differ in their quantum statistics Examples are coherent states num-ber (Fock) states and squeezed states

22 Observations on squeezed states using a single PIN photo-diode

An ideal PIN photo-diode absorbs the full energy of a light mode andproduces one photo electron for every absorbed photon energy It uses theinternal photo-electric effect inside a semiconductor such as silicon or In-GaAs In contrast to avalanche photo-diodes PIN photo-diodes operatewith unity gain lsquoPINrsquo stands for lsquopositiversquo lsquointrinsicrsquo and lsquonegativersquo and isdescribing the doping of the semiconductor layers A PIN photo-diode is op-timally suited for the continuous monitoring of a rather bright light field ofup to several tens of milliwatts An example is the photo-diode in the outputport of a gravitational-wave detector as shown in Fig 2 (a) The prominentwavelength of 1064 nm which is emitted by NdYAG lasers has an opticalfrequency of ν = ω(2π) = 282 middot 1014 Hz The period of the field oscillationis a few femtoseconds and cannot be directly resolved with photo-electric de-tectors However variations of the electric field around the averaged opticalfield oscillation on longer time-scales can be resolved Applying an electronicbandpass filter at the sideband angular frequency Ω plusmn ∆Ω2 to the photovoltage provides information about the lsquodepth of the lightrsquos amplitude mod-ulationrsquo which is also called the lsquoamplitude of the amplitude quadraturersquo Itcan also slowly vary in time and reads

Xϑ=0

Ω∆Ω(t) equiv XΩ∆Ω(t) equiv X (1)

The subscript is usually skipped as it is done with the time dependenceas indicated on the right Applying the electronic bandpass filter in factdefines the mode of the light being detected The structure of the defini-tion in Eq (1) forms the basis of interferometric signals and quantum noisealso in the semi-classical case of coherent states Lets take an example Inthe recent observation of gravitational waves [Fig 1 bottom row in Abbott(2016)] the time-frequency representation of the gravitational-wave signalcorresponded to the amplitude quadrature amplitude XΩ∆Ω(t) of the inter-ferometer output light Note that a larger value of ∆Ω allows for changes ofthe quadrature amplitude on shorter time scales

If the light fieldrsquos lsquomodulation modersquo does not contain any quanta simplybecause there are no photons that have a frequency difference of plusmnΩ with

11

respect to the carrier it is in its ground state In this case lsquovacuum noisersquois observed which originates from the ground state uncertainty Since thevacuum noise only becomes measurable as a beat with a bright light field itcan also be seen as the carrierrsquos band-path filtered shot noise A modulationmode in a displaced vacuum state (a coherent state) corresponds to nonzerocoherent modulationThe measured level of the vacuum noise generally depends on the power ofthe bright carrier light and on the electronic amplification In any case itprovides the reference for certifying lsquosqueezingrsquo Observations using a singlePIN photo-diode require an independent measurement to quantify vacuumnoise A necessary condition is that attenuating the total fieldrsquos light powerresults in the same attenuation of the measured XΩ∆Ω values If they showa stronger attenuation a coherent modulation or thermal noise might bepresent If they show a weaker attenuation the photo-diode and its electron-ics might be saturated

Fig 2 (b) illustrates how a broadband squeezed field improves the mea-surement of an amplitude modulation in time domain based on a PIN photo-diode Shown is a simulated time sequence of XΩ∆Ω-data sampled from thephotoelectric voltage In this simulation all sideband frequencies from zero(DC) to the cutoff frequency of the detector electronics (Ωcut) are included(Ω = ∆Ω2 = Ωcut2) No additional band pass filter is applied making it amaximally broadband detection Although the data in Fig 2 (bi) contains aclassical amplitude modulation of the detected light this signal is not visibledue to random noise here representing shot noise Fig 2 (bii) shows thesame situation but with shot noise that is squeezed over the full detectionband The quantum uncertainty of the modulation depth is squeezed andthe classical signal becomes visible

It needs to be noted that a single PIN photo-diode can only measure theamplitude of the amplitude quadrature XΩ∆Ω(t) but not the non-commutingobservable the lsquoamplitude of the phase quadraturersquo

Xϑ=90

Ω∆Ω (t) equiv YΩ∆Ω(t) equiv Y (2)

For values that are small compared to the field strength of the bright fieldthe quantity Y approximately describes the bright fieldrsquos lsquophase modulationdepthrsquo

12

23 Observations on squeezed states using a balanced homodyne detector

-72

-69

-66

-63(i)

(ii)

(iii)

(a)

LO

(Squeezed)signal input

Phaseshifter

PD1

PD2

5050

01 0200 03

-75

-60

-57Balanced homodyne detector

(b)

Time [s]

Noi

se p

ower

[dB

m]

Figure 3 Balanced homodyne detection (BHD) ndash (a) Setup The quadrature atchoosable angle ϑ of the signal field is measured by overlapping the latter with a localoscillator (LO) field of the same mode parameters on a balanced beam splitter and record-ing the difference voltage from two PIN photo-diodes as shown In order to meet theBHD approximation the LO needs to be much more intense than the signal field A closeto perfect mode overlap between LO and signal input field is crucial For a non-perfectoverlap the detector measures the input state with unwanted contributions of the vacuumstate (b) Noise power measurements (i) on an electronically amplified and band-passfiltered quadrature amplitude of the vacuum field (Xvac

Ω∆Ω) (signal input blocked) (ii) on

a squeezed quadrature (XsqzΩ∆Ω) of a squeezed vacuum state (ϑ = 0) and (iii) on re-

spective quadratures of the same squeezed state where the phase angle ϑ was continuouslyshifted by changing the optical path length of the LO The measurement data shows about5 dB of squeezing and was first published in [Chelkowski et al (2007)] Ω2π = 5 MHz∆Ω2π = 100 kHz

In contrast to a single PIN photo diode a balanced homodyne detector(BHD) is suitable to measure the quantum statistic of all types of modu-lations ie for all angles ϑ Such a detector consists of two identical PINphoto-diodes a balanced beam splitter and an external homodyne local os-cillator field that is much brighter than the signal beam and that has anadjustable phase The signal beam corresponds to the squeezed field whichin many experiments is in a squeezed vacuum field having an optical powerthat usually corresponds to just a few photons per mode The two beams areoverlapped on the balanced beam splitter with close to perfect mode match-ing and the two interference outputs are focussed onto the photo diodes seeFig 3 (left) The electric output signal of the BHD is the difference of the

13

photo diode voltages The LO takes over the role of the carrier light fieldbut with the possibility to choose the phase shift ϑ This way eigenvaluesof X Y or Xϑ can be measured where the latter is given by the followinglinear combination of the first two

Xϑ(t) = cos(ϑ) X(t) + sin(ϑ) Y(t) (3)

If the modulation depths of signal and local oscillator beams are weak com-pared to their coherent amplitudes |α| and |αLO| the output voltage of aBHD corresponds to eigenvalues of the following operator

V (t) prop 2cos(ϑ) |αLO||α|+ |αLO| Xϑ(t) + |α| XϑLO(t) (4)

The lsquohomodyne approximationrsquo further involves |αLO| |α| such that theterm on the right can be neglected even if the local oscillator shows someclassical quadrature excitation The output voltage of a BHD is usually spec-trally analysed or at least spectrally filtered which removes the DC part infull analogy to a single photo diode (see previous subsection) Sampling thefiltered voltage provides eigenvalues proportional to the generalized quadra-ture amplitude in Eq (3)

V BHDΩ∆Ω(t) prop |αLO| Xϑ

Ω∆Ω(t) (5)

Fig 3 (a) shows the setup of a balanced homodyne detector for the char-acterization of squeezed states Setting ϑ = 0 eigenvalues of the ampli-tude modulation depths can be sampled from the photo voltage accordingto Eq (5) Setting ϑ = 90 eigenvalues of the phase modulation depths aremeasured The datarsquos expectation values 〈Xϑ〉 provide the coherent displace-ment of the squeezed state The datarsquos variances

∆2Xϑ equiv 〈(Xϑ)2〉 minus 〈Xϑ〉2 (6)

provide the statersquos (quantum) noise A pure squeezed state as well as asqueezed state that experienced photon loss have Gaussian quantum statisticsand are thus fully described by the expectation values and variances (first andsecond moments) of two orthogonal quadratures but only if one quadraturereflects the lowest quadrature variance

14

In most experiments with squeezed light the photo electric voltage ac-cording to Eq (5) is not sampled with a data aquisition system but the signalis directly fed into a spectrum analyser measuring the noise power of the volt-age If the expectation value 〈Xϑ〉 is zero the noise power is proportionalto the variance ∆2Xϑ in Eq (6) The reference for quantifying the squeezefactor is measured by blocking the (squeezed) signal field in Fig 3 (a) Themeasured vacuum noise level corresponds to the LOrsquos (electronically ampli-fied) shot noise level

Traces (ii) and (iii) in Fig 3 (b) show measured noise powers of the mod-ulation mode (Ω2π = 5 MHz ∆Ω2π = 100 kHz) being in a squeezed vac-uum state (i) is proportional to the variance of the ground state uncertainty∆2Xvac

Ω∆Ω (ii) is proportional to the quantum noise variance of the squeezed

quadrature amplitude ∆2XsqzΩ∆Ω (iii) is proportional to the quantum noise

variance of the quadrature amplitude with scanned phase ∆2XΩ∆Ω(ϑ(t))

To fully characterize a quantum state ie to do quantum state tomogra-phy [Vogel and Risken (1989)] a BHD is a prerequisite But also interfero-metric measurements with balanced homodyne detectors instead of singlePIN photo-diodes have several advantages A correctly implemented BHDreadily provides the vacuum noise level when the signal beam is blockedWith a BHD the optimum operating point of the interferometer is preciselyat a dark fringe If a perfect dark fringe can practically be achieved ampli-tude noise of the laser does not couple into the signal port If the interfero-meter has balanced arm length also frequency noise of the laser then does notcouple into the signal port Some quantum non-demolition schemes with theprospect of evading quantum radiation pressure noise require the detection ofa non-canonical quadrature angle [Jaekel and Reynaud (1990) Kimble et al(2001)] Here the adjustable phase of a BHD provides a straight forwardapproach The experimental exploration of BHDs for gravitational-wave de-tectors only has started recently [Steinlechner et al (2015)]

A light field can be analysed with respect to many different modulationfrequencies Ω The result constitutes a spectrum [Breitenbach et al (1998)]where in principle every modulation mode can be in a different quantumstate Fig 4 shows spectra of squeezed states from 5 MHz to 100 MHz with∆Ω2π = 1 MHz The lower curve shows the spectrum of the most strongly

15

-14-12-10-8-6- 4-20 2 4 6 8

10 12 14 16 18

6 8 10 20 40 60 80 1005

Squeezed noise

Anti-squeezed noise

Vacuum noise

Frequency [MHz]

Nor

mal

ized

noi

se v

aria

nce

[dB

]

Figure 4 Spectrum of quadrature amplitude variances ndash Shown are the quantumnoise properties of a large number of modulation modes having a resolution bandwidth of∆Ω(2π) = 1 MHz For all traces the balanced homodyne detector output was analysedwith a spectrum analyser Squeezing of XΩ∆Ω (bottom trace) and anti-squeezing of

YΩ∆Ω (top trace) versus f = Ω(2π) are shown relative to the vacuum noise variance Thespectrum below 5 MHz is not shown since it contained less squeezing due to laser relaxationoscillation in the carrier field as well as disturbances from back-scattered light [Vahlbruchet al (2007)] Disturbances at frequencies above 70 MHz originated from relatively largedetector dark noise which was subtracted from all traces shown here The thin linerepresents a theoretical model that takes into account for the linewidth of the squeezingcavity The data was first presented in in Ref [Mehmet et al (2010)]

squeezed variances in this case the variances of XΩ∆Ω The upper spectrum

shows the variance in the orthogonal quadrature amplitude (YΩ∆Ω) Allvariances are normalized to those of the corresponding vacuum state Thesqueeze factor reduces towards higher frequencies due to the linewidth ofthe squeezing cavity The anti-squeezing is always higher than the absolutevalue of the squeezing due to Heisenbergrsquos uncertainty relation and due to thepresence of optical loss The curves do not represent pure squeezed states butmixed squeezed states with a significant contribution from vacuum statesdue to optical loss Pure squeezed states can only be produced by making

16

the influence of all decoherence processes negligibleThe choice of the resolution bandwidth (RBW ∆Ω) during data taking

and processing defines the spectral-temporal modulation modes includingtheir number within the detected spectrum For any setting of the RBWthe quantum mechanical properties of the quadrature amplitudes XΩ∆Ω and

YΩ∆Ω [Caves (1985)] fully correspond to those introduced for quadratures instandard text books and which are reviewed in Sec 3

24 Observations on two-mode squeezed states using balanced homodyne de-tectors

Two-mode squeezed states are composed of two subsystems lsquoArsquo and lsquoBrsquoand are bi-partite entangled states with a Gaussian quantum statistic Toavoid conflicts with different usage of the term lsquomodersquo they can synony-mously be named lsquobipartite Gaussian entangled statesrsquo or lsquobipartite squeezedstatesrsquo which will be mainly used in this Review In the same way multi-partite Gaussian entangled states correspond to multi-partite squeezed states

The measurement observables that prove or disprove the bi-partite squeez-ing property are XA

Ω∆Ω minus XBΩ∆Ω and Y A

Ω∆Ω + Y BΩ∆Ω where the minus and

plus signs may be swapped Bi-partite squeezed states are precisely thosestates that were discussed by Einstein Podolsky and Rosen (EPR) in theirseminal paper [Einstein et al (1935)] Fig 5 shows a measurement resulton bi-partite squeezed light [Eberle et al (2013)] The variances of bothjoined observables are squeezed as shown in the two lower traces They wererecorded consecutively by adding or subtracting the outputs of two balancedhomodyne detectors But by interfering the subsystems on a beam splitterone could even measure both joined observables simultaneously This pos-sibility is correctly described in quantum theory since their commutator iszero

The so-called EPR paradox arises as follows If we either measure XAΩ∆Ω

and XBΩ∆Ω or Y A

Ω∆Ω and Y BΩ∆Ω it is obvious from the data in Fig 5 that

we can always predict the measurement result at subsystem lsquoBrsquo when know-ing the result at subsystem lsquoArsquo This seems to suggest that both quantitiesat lsquoBrsquo are precisely defined simultaneously before the measurement on lsquoArsquowhich contradicts the rigorous (and correct) interpretation of their non-zerocommutator that they are not precisely defined simultaneously

To solve this paradox EPR conjectured that the wavefunction as definedby quantum theory does not provide the full information This led to a

17

discussion of whether hidden variables existed that needed to be included ina complete theory of quantum mechanics (see also Bell [Bell (1966)]) Theexperimentally observed violation of Bellrsquos inequality [Bell (1964) Aspectet al (1981) Giustina et al (2013) Hensen et al (2015)] however ruled outthe existence of (local) hidden variables

Based on that the EPR paradox needs to be solved in a different wayContrary to what EPR assumed it is in fact possible to predict the valueof an arbitrary observable of a physical system A with certainty via a mea-surement on system B although this observable was not defined before themeasurement Without any interaction a measurement on subsystem lsquoArsquo notonly creates lsquorealityrsquo of eg XA

Ω∆Ω simultaneously lsquorealityrsquo is also created

regarding the observable XBΩ∆Ω describing subsystem lsquoBrsquo Here the term

lsquorealityrsquo has the meaning as defined by EPR [Einstein et al (1935)] Simi-larly the detection of one photon of a two photon entangled number statenot only produces the reality of this photon but also that of a second oneA discussion of Einstein-Podolsky-Rosen entanglement can also be found in[Schnabel (2015)] Note that the EPR paradox can also be described aslsquoquantum steeringrsquo [Schrodinger (1935) Cavalcanti et al (2009) Handchenet al (2012)] It should also be mentioned that two-mode squeezing beingdetected with BHDs and not with photon counters cannot be used to violatea Bell inequality The latter topic is outside the scope of this Review

Bi-partite squeezed states were first characterized with balanced homo-dyne detectors by the group of J Kimble in 1992 [Ou et al (1992)] Gener-ally the EPR paradox becomes more pronounced the stronger the bi-partitesqueezing is A measure of the strength of EPR entanglement was introducedby M Reid [Reid and Walls (1985)] According to this measure the resultin Fig 5 can be quantified to ε2 = 00309 where the critical value is one Itcorresponds to the strongest Gaussian EPR entangled state generated so far

For a long time it looked like that two-mode squeezed states are notuseful for laser interferometers The reason for that belief was that a laserinterferometer as any other measurement device too is built to measureone observable It seems to be ideal already if the quantum noise in thissingle observable is squeezed The increased quantum noise in the orthog-onal observable is not harmful in this case and squeezing in two differentobservables useless Only recently realistic scenarios were discussed in whichtwo-mode squeezing in fact does improve the performance of a laser inter-

18

-12

-10

-8

-6

-4

-2

0

0 1 2 3 4 5 6 7 8 9 10

100

dB

109

dB

Δ (XAvac + X B

vac ) = Δ (Y

Δ (XA + X B)

Avacminus Y B

vac )2 2

2

Δ (YAminus Y B)2

Nor

mal

ized

noi

se v

aria

nce

[dB

]

Times [s]

Figure 5 Two-mode squeezing measurement ndash For this measurement the outputs oftwo balanced homodyne detectors are added or subtracted and the variances (noise pow-ers) of the results recorded The upper trace was measured with modes lsquoArsquo and lsquoBrsquo beingin their ground states This measurement served as a reference level Strong two-modesqueezing was observed as shown by the lower two traces The sideband frequency wasΩ(2π) = 8 MHz and the resolution bandwidth was ∆Ω(2π) = 200 kHz The measure-ment results were first published in Ref [Eberle et al (2013)]

ferometer [Steinlechner et al (2013)] The proof-of-principle experiment isreviewed in Sec 7

25 Observations using photon counters

Alternatively to field quadratures an optical mode in a squeezed statecan also be characterized at least partly by detecting its photon numberdistribution For a pure squeezed vacuum state such a measurement wouldreveal the existence of solely even photon numbers including a large prob-ability for zero photons The average photon numbers of squeezed vacuumstates with feasible squeeze factors are very small of the order of one persecond and bandwidth in hertz see Fig 13 (a) ndash (c) A distribution with closeto zero probability of odd photon numbers however has not been measuredso far The reason is the lack of ideal photon counters First of all theefficiency of these detectors ie their probability of converting one photoninto one click and no photon into no click must be almost perfect lsquoLostrsquophotons as well as dark counts wash out the oddeven oscillations Further-more most detectors available can only distinguish between zero and one

19

photon This problem can be solved by distributing the squeezed mode ontoa large number of single photon detectors using an array of beam splitterssuch that all paths have a low probability of carrying more than one pho-ton Photon number measurements on squeezed vacuum states neverthelessplay an extremely important role in quantum optics When the squeezingstrength is very low the probability of detecting more than 2 photons can beneglected and the detection of a photon heralds the existence of a second one

0

300

600

900

1200

-40 -20 0 20 400

5

10

15

20

25

Two-

fold

coi

ncid

ence

s [1

(4s

)](a) (b)

Non-degeneratetwo-mode squeezing

5050

APDA1

APDA2

APDHerald

Photon coincidencedetection

Delay [ns]

Thr

ee-f

old

coin

cide

nces

[1

(4s)

]1500

Figure 6 Coincidence clicks from non-degenerate photon pairs ndash The first suchexperiment was reported in Ref [Hong and Mandel (1986)] (a) shows a setup with threeavalanche photo-diodes (APDs) for proving the successful heralding of a single photonnumber state (b) Histograms of the two-fold coincidence detections at APDHerald andAPDA1 (red) and at APDHerald and APDA2 (yellow) with theoretical models (solid lines)If the two-mode squeezing just carried one photon in each spatial subsystem the three-foldcoincidence detection should be zero Indeed the according histogram (grey points righty axis) shows only a few events These are produced by false (dark) counts of the APDsThe delay for the three-fold coincidences is defined as the time between counts at lsquoA1rsquo andlsquoA2rsquo given that the trigger APDHerald detected a photon (within a 100 ns time window)The data was taken on photons that were up-converted from 1550 nm to 532 nm and itwas first published in Ref [Baune et al (2014)]

If a mode of light is always excited by either zero or two photons lsquocondi-tionalrsquo or lsquoheraldedrsquo one-photon Fock states can be realized (Measurementson an ensemble of the n-photon Fock state would always produce the mea-surement result n ie Fock states have a zero photon number uncertaintyThey are also called lsquonumber statesrsquo) The above concept of producing aone-photon Fock state obviously requires the deterministic and balanced dis-tribution of the down-converted signal and idler fields into two different paths

20

In order to achieve this the signal and idler fields need to be non-degenerateUsually a mode in a squeezed state is composed of degenerate signal andidler fields and this degeneracy thus needs to be removed Possible waysare producing the down-converted fields at well separated wavelengths [Vil-lar et al (2005) Su et al (2006) Li et al (2010) Samblowski et al (2011)]separating the upper and lower sidebands belonging to an ordinary squeezedmode by frequency filters [Schori et al (2002) Hage et al (2010)] and us-ing spatial filters [Hong et al (1987)] A frequently used approach is usingtype II parametric down-conversion where the photons within a pair are al-ways orthogonally polarized [Ou et al (1992) Kiess et al (1993) Kwiat et al(1995)]The list of experiments with conditional or heralded photon number states islong They showed for instance nonclassical g(2)-functions [Hong et al (1987)]and violations of Bell inequalities [Weihs et al (1998)] Fig (6) shows a re-sult from a more recent experiment in which a bipartite-squeezed state withsubsystems at 1550 nm and 810 nm was produced the subsystem at 1550 nmsubsequently up-converted to 532 nm and the lsquoquantum non-Gaussianityrsquo ofheralded up-converted single photons demonstrated [Baune et al (2014)]Squeezed states are also the resource for the conditional generation of super-positions of coherent states [Ourjoumtsev et al (2006) Neergaard-Nielsenet al (2006)] and so-called N00N-states [Afek et al (2010)]

The generation of nonclassical states mentioned in the paragraph aboveis not stationary but relies on a probabilistic trigger event The produc-tion of squeezed states themselves usually happens in a stationary fashionThis distinction has an important consequence for applications of nonclassi-cal states in measurement devices Only (stationary) squeezed states allowfor a continuous improvement of a measurement Avoiding any loss of mea-suring time is generally of high relevance for the detection of short-livedsignals with unknown arrival time as well as for the detection of long-livedquasi-monochromatic signals since the signal-to-noise-ratio (SN) improveswith measuring time

26 Conclusions

The detection of squeezed light produces measurement results that canbe considered as remarkable Let us focus on experiments where a mode in abright coherent state is overlapped with a mode in a squeezed vacuum state

21

as shown in Figs (1) and (3) In both setups the squeezed vacuum field caneasily be blocked which allows us to compare the measurement results ona bright coherent state with and without the interference with the squeezedvacuum state Without squeezing the photo-electric detectors measure alarge number of photon events with a large quantization noise (shot noise)The large noise reflects the fact that all photon events were independent fromeach other as shown in Fig 2 (bi) With squeezing the photo-electric de-tectors again measure a large number of photon events with an expectationvalue that is even slightly higher but nevertheless the quantization noise ofall detected photons is significantly reduced Fig 2 (bii)Based on the discussion of EPR entanglement in Subsec 24 the photo-electric detection of the output light of a squeezing-enhanced laser inter-ferometer (with αlowastα1) produces the reality of photons This way we cankeep the lsquowave picturersquo in which no photons exist when light travels alongthe interferometer arms and when it interferes at the beam splitter Whenthe energy of the beam is elevating electrons to the conductance band ofthe photo-diodersquos semi-conductor n photon events simultaneously appearwithin the measuring interval with probability P (n) What conclusion hasto be drawn if the probabilities resemble a sub-poissonian statistic ndash Theoccurrence of photon events is still truly random but in this case not forindividual photons The occurrence of photons is correlated in such a waythat the probability of detecting an additional photon in the same time in-terval reduces the larger the number of already detected photons is Whatfollows from the discussion of EPR entanglement for a photon counting ex-periment with pure squeezed vacuum and ideal photon counters Here theprobabilistic detection of one photon entails the detection of a second onewith certainty With some smaller probability a third photon is detectedwhich entails the detection of a fourth photon with certainty and so on

If a photon of a mode that was not interrogated by the environment beforeis absorbed its reality is created in this very moment If the photon belongsto a squeezed state this process instantaneously influences the probability ofother photons becoming reality

Of course a more general statement can be made based on the insightthat interaction with the environment creates the reality of any kind ofquanta including electrons atoms and molecules

22

3 Theoretical description of squeezed states

31 The quadrature amplitude operators

Consider a single mode of light at optical frequency ω Its Hamiltonoperator reads

Hω = ~ω(n+

1

2

)= ~ω

(adaggerωaω +

1

2

)= ~ω

(X2ω + Y 2

ω

) (7)

where n is the photon number operator and aω and adaggerω are the annihilationand creation operators which obey the commutation rule

[aω a

daggerω

]= 1 The

operator aω has a complex-valued dimensionless eigenvalue spectrum andcorresponds to the complex amplitude αω in classical optics Xω and Yω arethe hermitian amplitude and phase quadrature operators The eigenvaluesof the quadrature operators are also dimensionless and proportional to theelectric fields at the oscillationrsquos antinode and at the oscillationrsquos node Inthe above equation they are defined such that their variances are ∆2Xω =∆2Yω = 14 if the oscillator is in its ground state ie if 〈n〉 = 0

Although Eq (7) simply describes the energy of an harmonic oscillator itis the essence of quantum theory since it mathematically describes the wave-particle dualism Whereas the eigenvalues of n have a discrete spectrum theeigenvalues of Xω and Yω have a continuous spectrum In classical opticsthe phase quadrature is zero In quantum optics its expectation value is alsozero but its uncertainty contributes to the overall energy

Eq (7) describes a cavity mode as well as a section that is cut from apropagating quasi-monochromatic light beam The latter example is of highrelevance in actual experiments By setting the sectionrsquos time window ie themeasuring time interval the time-frequency (lsquomodulationrsquo) mode is defined

The quadrature operators introduced in Eq (7) and displayed in Fig 7 donot correspond to lsquoXrsquo and lsquoY rsquo that are of relevance in laser interferometry andin optical communication and which were already discussed in Subsec 22and 23 The optical frequency of visible and near-infrared light is far too highto be transferred to an oscillation of photoelectric voltage Quite general alaser interferometer targets signals at audio or radio band frequencies Ωi ωSuch a measurement is achieved as stated before by decomposing the photo-electric voltage from the photo diode at the interferometer output into asingle-sided spectrum (positive frequencies only) of intervals of Ωplusmn∆Ω2

23

(a)

(b)

(c)

(d)

2π 4π

2π 4π

2π 4π

2π 4π

Figure 7 Phase spaces and electric field oscillations of monochromatic lightndash Top Left Monochromatic light in a coherent state is represented by a phasor (whitearrow) including its quantum uncertainty (white dashed circle and fuzzy area) located inthe phase-space spanned by the quadratures Xω and Yω When the phase space rotateswith optical frequency ω2π the projection of the quantum phasor onto a fixed (vertical)axis corresponds to the electric field E(t) as shown on the right side (a) Weakly displacedcoherent state (b) Corresponding amplitude squeezed state The electric field uncertaintyaround the zero average field region is anti-squeezed (c) Vacuum state at the same opticalfrequency (d) Corresponding squeezed vacuum state The meaning of the uncertaintycould be carved out by supplementing them with monochromatic waves all having theoptical frequency ω2π Changing amplitudes then display amplitude quadrature noiseChanging shifts along the time axis model the electric field uncertainty at the expectedzero crossing They are not implemented in the graphics here however since any of thosewaves does not exist due to Heisenbergrsquos uncertainty relation

24

The signals as well as the quantum uncertainties carried by a beam oflight are thus described by a spectrum of pairs of non-commuting quadratureoperators Mathematically every such operator is defined by an integral overthe Fourier components within the bandwidth The spectral weighting of theFourier components is called the lsquowindow functionrsquo By going to sidebandintervals a spectrum of a new type of optical mode is defined which describesthe modulation of the electric field in the respective frequency interval Ω plusmn∆Ω2 In this Review we call it a lsquomodulation modersquo

The quadrature operators that are defined around a modulation frequencyΩ with a bandwidth of ∆Ω are the quadrature amplitude operators thatare relevant in laser interferometry Whenever they are not related to aspecific band we use the short form XΩ∆Ω(t) equiv X and YΩ∆Ω(t) equiv Y cfEqs (1) and (2) These operators can slowly vary with time where thetime dependence is limited by ∆Ω (The time dependence is not due toquantum uncertainty which usually is time independent but for instancedue to the time dependence of the signal eg a passing gravitational wave)Let us consider now a pair of quadrature operators for a particular sidebandΩ plusmn ∆Ω2 The Hamilton operator of the corresponding modulation modeis found by switching to the frame rotating at optical frequency ω Thetransition is done by applying the unitary transformation U = exp(iωadaggerat)generating a new Hamiltonian H = U daggerHωU minus i~UpartU daggerpartt The Hamiltonianof the modulation mode reads

H = ~Ω

(nΩ +

1

2

)= ~Ω

(adaggera+

1

2

)= ~Ω

(X2 + Y 2

) (8)

where nΩ is the (occupation) number operator for the modulation modeand a and adagger its annihilation and creation operators The commutation rule[a adagger

]= 1 is unchanged X and Y are the amplitude and phase quadra-

ture amplitude operators respectively They correspond to the depth of theamplitude modulation and for weak excitations to the depth of the phasemodulation respectively They are the conventional hermitian field operatorsin experimental quantum optics Note that modulation modes at angularfrequency Ω can be described by a superposition of three optical frequenciesa carrier at ω an upper sideband at ω + Ω and a lower sideband at ω minus ΩThe quantum mechanical description of modulation states in connection tooptical carrier and upper and lower sidebands is known as the lsquoTwo-PhotonFormalismrsquo [Caves and Schumaker (1985) Schumaker and Caves (1985)]

The quadrature amplitude operators in Eq (8) are again defined such

25

that the variances of the uncertainty of a modulation field in its ground stateor in a coherent state are

∆2Xvac = ∆2Yvac = 14 (9)

Generally quadrature operators X and Y as defined in Eqs (7) and (8) arethe real and imaginary parts of the annihilation operator

a = X + iY hArr adagger = X minus iY (10)

hArr X =1

2

(a+ adagger

) Y =

1

2i

(aminus adagger

) (11)

They satisfy the commutation relation[X Y

]=i

2 (12)

and their variances are limited by a Heisenberg uncertainty relation of thefollowing form

∆2X∆2Y ge 1

16 (13)

A quantum state is called a ldquosqueezed staterdquo [Bachor and Ralph (2004)]if ∆2Xϑ lt 14 for an arbitrary field quadrature Xϑ = X cosϑ + Y sinϑ seeEq (3) The angle of the lowest variance below 14 is called the squeeze angleθ The largest factor by which the variance is below 14 is called the squeezefactor often given on a decibel (dB) scale using the following transformation

minus 10 middot log10

(∆2Xθ

∆2Xvac

) (14)

The squeeze factor can also be described by the squeeze parameter r

eminus2r =∆2Xθ

∆2Xvac

(15)

When a squeezed state experiences optical loss it remains squeezed butthe squeeze factor is reduced Also the statersquos purity is reduced ie theproduct of the quadrature uncertainties increases above the minimum valueOptical loss corresponds to mixing the state with the vacuum state Let∆2Xϑ be the variance of a quadrature amplitude ∆2Xvac the variance of the

26

(quadrature angle independent) ground state uncertainty and (1 minus η2) therelative energy loss Then the resulting quadrature variance reads

∆2Xprime

ϑ = η2∆2Xϑ + (1minus η2)∆2Xvac (16)

To maximize the benefit from squeezed states in applications stronglysqueezed states need to be generated and optical loss minimized Opticalloss occurs due to absorption and scattering in the optical components inthe path of the squeezed beam including the squeezing resonator itself anddue to non-perfect matching to the interferometer mode non-perfect inter-ference contrast of the interferometer and non-perfect quantum efficiency ofthe photo diodes The sum of all losses including those outside the inter-ferometer need to be less then 10 to allow a nonclassical quantum noisesuppression of a factor of 10 in power ie 10 dB

32 Phase space representations of squeezed states

The Wigner function ndash The properties of squeezed states are nicely dis-played by the Wigner function W (X Y ) [Wigner (1932)] An example interms of a squeezed vacuum state is shown in Fig 8 It is a quasi-probabilitydistribution which contains the statersquos full information including its quan-tum statistic There are two ways how a Wigner function provides a sufficientcriterion for nonclassicality First by containing negative values second byfeatures that have a smaller (squeezed) width compared with the Wignerfunction of the ground state Integrating the Wigner function over Y pro-vides the probability density of measurement results ie of the eigenvaluesof the observable X and vice versa

infinintminusinfin

W (X Y )dY = p(X)

infinintminusinfin

W (X Y )dX = p(Y ) (17)

where p(X) and p(Y ) are the observed probability distributions also exem-plarily shown in Fig 8

The ground state coherent states as well as (quadrature) squeezed stateshave quadrature eigenvalue probability densities that are Gaussian TheirWigner functions are also Gaussian and thus entirely positive Wigner func-tions of other nonclassical states for instance Fock states exhibit negativevalues For this reason the Wigner function is called a quasi -probabilityfunction

27

X Y XY

Figure 8 Wigner function and its projections ndash Displayed is the full information ofa squeezed vacuum state The quasi probability density of the Wigner function (centre)is plotted along a third dimension and is color-coded Also shown are the actual mea-surement results from which the Wigner function is reconstructed They are representedby the squeezed and anti-squeezed Gaussian projections onto the X and Y axes Theirprobability densities p(X) and p(Y ) are given on the vertical axes The Gaussian measure-ment statistic of the first clearly shows squeezing compared to the ground state statistic(dashed) The squeeze factor is 116 dB and the squeeze parameter r = 1335 [Mehmetet al (2010)]

Fig 9 shows the Wigner functions for (a) a coherent state (b) the ground(vacuum) state (c) a displaced squeezed state and (d) a squeezed vacuumstate All Wigner functions describe a modulation of the carrier light atsideband frequency Ω integrated over the frequency interval ∆Ω The car-rier light is not part of these Wigner functions The displacement in (a)represents a classical amplitude modulation (b) corresponds to the absenceof any photons with a frequency offset of plusmnΩ from the local oscillator field(c) and (d) represent states whose amplitude modulation depth is more pre-cisely defined than that of the ground state Fig 10 shows Wigner functionspectrum for a broadband squeezed vacuum field Every Wigner functiondescribes the modulation field at some modulation frequency Ωi integrated

28

X

Y

X

(a) (b)

X X

(c) (d)

Y

Y Y

Figure 9 Simplified representation of Wigner functions ndash The darker the shadedareas the larger is the phase-space quasi-probability Shown are four different (time-independent) states of a modulation mode at frequency Ω for a specific resolution bandwidth ∆Ω Panel (a) represents a coherent state the displacement (α) corresponds to aclassical amplitude modulation Panel (b) represents the ground (vacuum) state (c) adisplaced squeezed state and (d) a squeezed vacuum state both with squeeze angle θ = 0The latter is in analogy to Fig 8 Again the light field that carries the modulation is notpart of the pictures

over the resolution bandwidth (RBW) of ∆Ω

The Glauber-Sudarshan P -function ndash The P -function [Glauber (1963)Sudarshan (1963)] is calculated by de-convoluting the Wigner function fromthe ground state uncertainty [Gerry and Knight (2005)] For displaced vac-uum states (coherent states) the P -function corresponds to a displaced δ-function The mathematical expression of the P -function of a squeezed statecontains infinitely high orders of derivatives of the δ-function [Vogel andWelsch (2006)] Such a function contains negativities but cannot be dis-played It is possible however to define a phase-space quasi probabilityfunction for squeezed states that can be displayed and that does show neg-

29

0

Ω1

Ω2

Ω

Single-sided spectrum

Y

X

Figure 10 Hint of a Wigner function spectrum ndash A single-sided spectrum (positivefrequencies only) with respect to the carrier field can be used to visualize a broadbandsqueezed field Shown are two examples displaying a squeezed vacuum state at Ω1 anda displaced amplitude squeezed state at Ω2 The individual Wigner functions cover theresolution bandwidth ∆Ω gt 0 (not shown) In general the squeezing strength as well asthe squeeze angle and the displacement are a function of sideband frequency

ativities as a sufficient and necessary condition for certifying the squeezingeffect This lsquononclassicality functionrsquo is calculated by de-convoluting theWigner function from an uncertainty distribution that is steeper than theGaussian distribution A pronounced negativity of a squeezed vacuum stateof up to 69 standard deviations was found [Kiesel et al (2011)]

The double-sided phasor picture ndash This phasor picture links quantumstates of modulations with the quantum states of the contributing opticalfields [Bachor and Ralph (2004)] and is mathematically described by thetwo-photon-formalism [Caves and Schumaker (1985) Schumaker and Caves(1985)] Generally a weak amplitude or phase modulation at frequency Ω ofa carrier field at optical frequency ω can be understood as the carrierrsquos beatwith two optical frequencies at ωplusmnΩ The double-sided phasor picture is ableto display a spectrum of different and independent modulation frequenciesin the rotating frame of the carrier field The carrier light field is time-independent but the upper and lower sidebands are not They rotate withplusmnΩi(2π) respectively around the frequency axis

30

ω0

Yω

Xω

Upper sideband

Lower sideband

ω0+Ω1

ω0+Ω2

ω0-Ω2

ω0-Ω1

ω

Upper sideband

Lower sideband

Yω

Xω

ω0+Ω1

ω0+Ω2

ω0-Ω2

ω0-Ω1

ω

ω0

Figure 11 Double-sided phasor pictures ndash Phase spaces at optical frequency ω0 plusmnΩirotate around the frequency axis with sideband frequency plusmnΩi Its sign determines thedirection of rotation with respect to the local oscillator in the rotating frame at ω0 Apair of phase spaces need to be superposed to provide a description of a modulation fieldat |Ωi| Top Amplitude quadrature squeezed field with a coherent displacement at |Ω2|The displacement corresponds to a classical amplitude modulation The uncertaintiesof all optical frequencies are circular and larger than that of the ground state (dashed)Quantum correlations are indicated by additional symbols More details are given in themain text Bottom Corresponding spectrum of (displaced) vacuum states which do nothave any quantum correlations

31

Fig 11 shows such a double-sided phase space picture where the carrierrsquosmodulation at Ω1 is in a squeezed vacuum state and where the modulationat Ω2 is in a displaced squeezed state The picture shows how a classicalamplitude modulation as well as the quantum statistic of a modulation fieldis decomposed into contributions from upper and lower sidebands For asqueezed modulation field the upper and lower sidebands show no squeezedbut circular thermally excited quantum uncertainties The uncertainties of apair of sidebands however show correlations as well as anti-correlations InFig 11 these (anti-) correlations are marked with times and + for the modulationfrequency Ω1 and with N and bull for the modulation frequency Ω2

33 Covariance matrix representation of (single-party) squeezed states

Since squeezed states have a Gaussian quantum statistic four numbersare sufficient for their full description These numbers are the second mo-ment of the quadrature amplitude showing the strongest squeezing and thesecond moment of its orthogonal quadrature amplitude as well as their firstmoments describing the displacement These four numbers are sufficientto calculate the Wigner function shown in Fig 8 In general the quadra-ture of strongest squeezing is not perfectly aligned with one of the axesof the measurementrsquos coordinate system The so-called covariance matrix(VXXVXY VY XVY Y ) [Simon et al (1994)] accounts for phase space rotationsand enables the calculation of how these states evolve within an interfero-metric arrangement Their components are normalized to the vacuum noisevariance ∆2Xvac = ∆2Yvac and read

VXY =

langXY + Y X

rangminus 2

langXranglang

Yrang

2∆2Xvac

(18)

The following examples represent the ground state a pure 10 dB am-plitude quadrature squeezed state and a pure 10 dB squeezed state with asqueeze angle of 45

Vvac =

(1 00 1

) V0

01 =

(01 00 10

) V45

01 =

(505 495495 505

) (19)

with V4501 =RT

45V001R45 where Rα = (cosα minussinα sinα cosα) is the rota-

tion matrix

32

34 Phase space representation of two-mode (bi-partite) squeezed states

XA

YA

XB

YB

Figure 12 Bi-partite squeezed vacuum state ndash Shown is a Wigner-function-basedphase space representation in close analogy to the single party version in Fig 9 (d) Thepicture describes a single modulation at frequency Ω with bandwidth ∆Ω Measurementsat party A and B reveal local Wigner functions that correspond to thermal states sincethe uncertainties (indicated by the color and by the large circles) are symmetric and largerthan that of the ground state (indicated by the small dashed circles) The uncertaintieshowever show correlations and anti-correlations here indicated by + and times respectivelyThe strength of these (anti-)correlations are indicated by ellipses Bi-partite squeezingie entanglement is present if the short axes of the ellipses are shorter than the diameterof the ground state uncertainty The picture in fact represents Einstein-Podolsky-Rosenentanglement [Einstein et al (1935)] From a measurement of XA or YA the correspondingmeasurement result at party B can always be inferred with an uncertainty that is smallerthan the ground state uncertainty

A bi-partite state enables a measurement on subsystem A and simulta-neous a measurement on subsystem B For a large number of simultaneousensemble measurements of the same quadrature amplitude Xϑ the followingtwo joint quadrature variance can be calculated

∆2(XAϑ plusmn XB

ϑ ) (20)

A state that is symmetrically shared between two parties (A and B) is calleda two-mode squeezed state if the variances of joint quadrature measurementsfulfill the following inequality [Duan et al (2000)] ie

∆2(XA minus XB)

∆2(XAvac minus XB

vac)+

∆2(Y A + Y B)

∆2(Y Avac + Y B

vac)lt 2 (21)

33

with ∆2(XAvac plusmn XB

vac) = ∆2(Y Avac plusmn Y B

vac) = 2∆2Xvac A lsquotwo-mode squeezedstatersquo reveals entanglement in the second moments of the measurement statis-tics It is thus a lsquobi-partite Gaussian entangled statersquo

Fig 12 displays a (pure) bi-partite squeezed vacuum state distributedbetween A and B The state shows full symmetry regarding its subsystemsat the two sites The large circles and the colored area represent Wignerfunctions of the subsystems Measurements of the quadrature amplitudesXA YA XB and YB show identical variances and the correlations and anti-correlations have identical strength since ∆2(XA minus XB) = ∆2(YA + YB) lt12 for our normalization of quadrature amplitudes having a ground statevariance of 14

Generally a symmetric bi-partite squeezed state fulfills another quantita-tive (Gaussian) entanglement criterion if less than 50 of the vacuum state issymmetrically mixed into the initially pure state Bi-partite squeezed statesare always entangled but in this case they are even Einstein-Podolsky-Rosen(EPR) entangled [Reid (1989)] allowing the demonstration of the quantumsteering effect [Einstein et al (1935) Schrodinger (1935) Reid (1989) Cav-alcanti et al (2009)] The first such experiment was performed by Ou etal [Ou et al (1992)] using type II parametric down-conversion (PDC) Laterexperiments produced bi-partite squeezed vacuum states by overlapping twosqueezed vacuum states each produced with type I PDC on a balanced beamsplitter and used the entangled output for the demonstration of quantumteleportation [Furusawa et al (1998) Bowen et al (2003ca)] The criterionin Eq (21) and the EPR criterion from [Reid (1989)] was experimentallycompared in Ref [Bowen et al (2003b)] The steering effect in asymmet-ric bi-partite squeezed states were recently experimentally characterized inRef [Handchen et al (2012)]

Fig 12 shows features similar to those in the top part of Fig 11 Thisis not a coincidence and shows that a bi-partite squeezed state can also begenerated by spatially splitting the upper and lower sideband of a (single-party) squeezed state This was first experimentally demonstrated by thegroup of E Polzik [Schori et al (2002)] and later used for EPR multiplex-ing of a single longitudinal mode of a squeezing resonator [Hage et al (2010)]

35 Covariance matrix representation of bi-partite squeezed states

Also the full information of bi-partite states including the entanglementcan be cast by the covariance matrix [Simon et al (1994)] which can be used

34

to calculate the propagation of these states in laser interferometers Againall variances are normalized to the vacuum noise variance in full analogy toEq (18) The generic bi-partite covariance matrix has dimension 4times4 andreads

Vbp =

VXAXA

VXAYA VXAXBVXAYB

VYAXAVYAYA VYAXB

VYAYBVXBXA

VXBYA VXBXBVXBYB

VYBXAVYBYA VYBXB

VYBYB

with

VXAYB =

langXAYB + YBXA

rangminus 2

langXA

ranglangYB

rang2∆2Xvac

ϑ

(22)

Due to the symmetry in Eq (22) the 4times4 covariance matrix is fully spec-ified by just ten independent coefficients If the phase spaces at A and Bare aligned along the strongest correlations and anti-correlations the ma-trix components referring to different quadrature amplitudes eg VXAYA arezero Such entangled states can be produced by overlapping two squeezedfields with a squeeze angle difference of 90 on a balanced beam splitter

A symmetric bi-partite squeezed vacuum state which is also called anlsquoS-classrsquo [DiGuglielmo et al (2007)] bi-partite squeezed vacuum state shows(anti-)correlations in two joint quadratures as defined in Eq (21) For a puresuch state of 10 dB squeezing the covariance matrix reads

Vbp10|10 =

505 0 495 0

0 505 0 minus495495 0 505 0

0 minus495 0 505

The following covariance matrix describes a so-called lsquoV-classrsquo 10 dB bi-partite squeezed vacuum state Here only one joint quadrature shows 10 dBsqueezing whereas the orthogonal joint quadrature shows vacuum noise Thestate is obtained by overlapping one 10 dB squeezed state with a vacuumstate on a balanced beam splitter

35

Vbp0|10 =

055 0 045 0

0 55 0 minus45045 0 055 0

0 minus45 0 55

The first measurement of all elements of such a covariance matrix was achievedin [DiGuglielmo et al (2007)]

36 Photon numbers of squeezed states

In contrast to the ground state squeezed vacuum states do have photonexcitations As said earlier quantum theory links the wave and the particlepictures Indeed the squeeze factor of a modulation mode is directly con-nected to a certain photon number excitation Squeezed states of light areproduced via spontaneous photon pair generation eg by parametric down-conversion The following operator S is called the lsquosqueeze operatorrsquo [Gerryand Knight (2005)] It creates and annihilates photon pairs

|r θ〉 = S(r θ) |0〉 (23)

where |r θ〉 is a squeezed vacuum state with squeeze parameter r and squeezeangle θ and |0〉 is the vacuum state The definition of the squeeze operatoris

S(r θ) = exp

[1

2

(reminusiθa2 minus reiθadagger 2

)] (24)

The following shows that this definition indeed results in a state with squeezedquadrature amplitude variances Lets set θ = 0

〈0| Sdagger(r 0) X S(r 0) |0〉 =1

2〈0| Sdagger(r 0)

(a+ adagger

)S(r 0) |0〉 (25)

〈0| Sdagger(r 0) Y S(r 0) |0〉 =i

2〈0| Sdagger(r 0)

(aminus adagger

)S(r 0) |0〉 (26)

Using the Baker-Hausdorff formula we get

Sdagger(r θ) a S(r θ) = a cosh r minus adaggereiθsinh r (27)

Sdagger(r θ) adagger S(r θ) = adaggercosh r minus aeminusiθsinh r (28)

36

Since 〈0| X |0〉 = 〈0| Y |0〉 = 0 also Eqs (25) and (26) are zero To finallycalculate the variances we need

〈0| Sdagger(r 0) X2 S(r 0) |0〉 =1

4〈0| Sdagger(r 0)

(a2 + adaggera+ aadagger + adagger2

)S(r 0) |0〉

〈0| Sdagger(r 0) Y 2 S(r 0) |0〉 = minus1

4〈0| Sdagger(r 0)

(a2 minus adaggeraminus aadagger + adagger2

)S(r 0) |0〉

Given that SSdagger is the identity and using again Eqs (27) and (28) we get theexpected variances

∆2X =1

4

[cosh2 r minus 2cosh r sinh r + sinh2 r

]=

1

4eminus2r

∆2Y =1

4

[cosh2 r + 2cosh r sinh r + sinh2 r

]=

1

4e2r

Since the squeeze operator can only create and annihilate photon pairsa squeezed vacuum state without photon loss must correspond to an evennumber of photons But not only photon loss also a coherent displacementleads to flattening out the odd-even oscillations The probability of detectingN photons in a pure displaced squeezed state are derived for instance in[Gerry and Knight (2005)] and read

P (N) = |〈N |α r θ〉|2 =(05 tanh r)N

N cosh rexp

[minus|α|2 minus 1

2(αlowast2eiθ + α2eminusiθ)tanh r

]times∣∣∣HN

[(αcosh r + αlowasteiθsinh r)

radiceiθsinh(2r)

]∣∣∣2 (29)

where HN is the N th Hermite polynomialFig 13 shows the photon number distributions for 5 different pure squeezed

states according to Eq (29) Panels (a) to (c) show squeezed vacuum stateswith 43 dB 86 dB and 172 dB of squeezing Panel (d) shows the moregeneral case of a squeezed state with a coherent displacement α 6= 0 Dueto θ = 0 the state is amplitude quadrature squeezed Panel (e) refers to thecorresponding phase quadrature squeezed state For comparison panel (f)shows the photon number distribution of the coherent state with the samedisplacement

The panels in Fig 13 represent the diagonal elements of the statersquos den-sity matrix in number basis Only the latter also contains the coherences

37

002

004

006

000 000

002

004

006

008

403020100 Photon number n

403020100Photon number n

Pro

babi

lity

403020100000

005

010

015

100 2 4 6 8100 2 4 6 800

04

06

02

100 2 4 6 800

03

02

01Pro

babi

lity

Pro

babi

lity

08

00

04

06

02

08

(a) (b)

(c) (d)

(e) (f)

Figure 13 Photon number distributions ndash All panels represent pure states (a) 43 dBsqueezed vacuum state (r = 05 α = 0) (b) 86 dB squeezed vacuum state (r = 1 α = 0)(c) 172 dB squeezed vacuum state (r = 2 α = 0) (d) displaced 86 dB squeezed state(r = 1 θ = 0 α = 4) (e) displaced 86 dB squeezed state (r = 1 θ = π2 α = 4) (f)Coherent state (r = 0 α = 4) The average photon numbers are about 027 14 131174 174 and exactly 16 see Eq (30)

between photon numbers [Gerry and Knight (2005)] Figures as shown heregenerally do not give full descriptions of the states

38

A squeezed vacuum state (r 6= 0) always has a non-zero photon numberand can not be the ground state The average photon number of a puresqueezed vacuum state can be calculated using Eq (8) With the maximallysqueezed quadrature variance ∆2Xsqz the average photon number is given by

n = 〈n〉|α=0θr〉 = ∆2Xsqz +(∆2Xsqz)

minus1

16minus 1

2=eminus2r

4+e2r

4minus 1

2 (30)

with the vacuum noise variance normalized to one quarter A coherent dis-placement further adds |α|2 photons on average

4 Squeezed-light generation

41 Overview

Squeezed light was first produced in 1985 by Slusher et al using four-wave-mixing in sodium atoms in an optical cavity [Slusher et al (1985)]Shortly after squeezed light also was generated by four-wave-mixing in an op-tical fibre [Shelby et al (1986)] and by degenerate parametric down-conversion(PDC) in a 2nd-order nonlinear crystal placed in an optical cavity [Wu et al(1986)] The pumped cavity was operated below its oscillation thresholdie the parametric gain did not fully compensate the round trip losses whichis also called lsquocavity-enhanced optical-parametric amplification (OPA)rsquo

The early day experiments achieved squeeze factors of a few percent up toabout 3 dB Today squeeze factors of more than 10 dB are directly observedin several experiments [Vahlbruch et al (2008) Eberle et al (2010) Stefszkyet al (2012) Vahlbruch et al (2016)] All of them are based on cavity-enhanced OPA (below threshold) The parametrically amplified mode isdegenerate ie signal and idler modes are identical In particular the down-conversion process is of lsquotype Irsquo which means that the amplified mode has awell-defined polarization Squeezed states can also be generated above oscil-lation threshold In Refs [Villar et al (2006) Jing et al (2006)] bi-partitesqueezing was generated with above-threshold PDC Both experiments usedtype II PDC which provides orthogonally polarized signal and idler fieldsType II PDC below threshold was also used to generate squeezed and bi-partite squeezed fields [Grangier et al (1987) Ou et al (1992)] All these

39

experiments were performed in the continuous-wave regime which is also thefocus of this Review Squeezed states of modulations of trains of laser pulseshowever have been also generated since the 1980s using either PDC or theoptical Kerr effect [Slusher et al (1987) Bergman and Haus (1991) Our-joumtsev et al (2006) Dong et al (2008)] For an overview of the develop-ments in squeezed-light generation in the continuous-wave as well as pulsedregime see Ref [Bachor and Ralph (2004)] Squeezed-light generation inopto-mechanical setups [Aspelmeyer et al (2014)] which use the intensitydependent phase shift from radiation pressure was discussed in Refs [Paceet al (1993) Rehbein et al (2005) Corbitt et al (2006)] and recently ex-perimentally achieved by several groups [Brooks et al (2012) Safavi-Naeiniet al (2013) Purdy et al (2013)]

42 Degenerate type I optical-parametric amplification (OPA)

This section provides a graphical description of how degenerate type IOPAPDC turns a vacuum state into a squeezed vacuum state and a coher-ent state into a displaced squeezed state The process requires a bright pumpfield and a 2nd-order nonlinear crystal For simplicity we set all nonlinearitiesabove 2nd-order to zero

Let us consider a short segment of the second-order nonlinear crystalpumped with light of optical frequency 2ν All other modes that enter thecrystal shall not contain any photons ie are in their vacuum states Ofthese the only mode of interest is that at optical frequency ν which spatiallyoverlaps with the pump mode Fig 14 shows the total electric field of theoptical input E in and the 2nd-order nonlinear dielectric polarisation of thecrystal P The latter is proportional to the total electric field of the outputEout The pump field at 2ν periodically drives the vacuum field at ν betweenregions of low and high polarisation This process transforms the vacuumstate into a squeezed vacuum state in the output [Bauchrowitz et al (2013)]The output further contains the hardly depleted pump field and frequencydoubled parts of the pump field at 4ν It is again emphasized that Fig 14displays OPA in a small segment of the crystal In reality the nonlinear effectaccumulates over the crystal length or even over several passages since thecrystal is usually put into an optical resonator A noticeable effect is achievedif all infinitesimal contributions constructively interfere This is achieved incase of phase matching ie if the wave fronts of the modes at 2ν and νpropagate with the same speed and thus do not run out of phase Note that

40

t

t

Squeezing

Figure 14 Optical parametric amplification of a vacuum state ndash The upper leftcorner shows the crystal polarization P(E) = ε0

(χ(1)E + χ(2)E2

) ie the separation of

charge carriers by the electric component of an optical field E The graph illustrates howan input quantum field (from below) is projected into an output quantum field (towardsthe right) In the example shown here the input field is composed of a classical pumpfield E in

2ν at frequency 2ν and zero-point fluctuations E inν of a field at frequency ν cf

Fig 7(c) The superposition E in of these two fields is transferred into a time-dependentdielectric polarization that is the source of (and thus directly proportional to) the elec-tric component of the output field Eout The quantum uncertainty of the output fieldshows a phase-dependent (parametric) amplification at frequency 2ν Spectral decompo-sition of the output field Eout reveals coherent amplitudes at frequencies 2ν and 4ν and asqueezed vacuum state Eout

sqzν cf to Fig 7(d) The concept of this figure was published inRef [Bauchrowitz et al (2013)]

41

t

t Amplitudequadraturesqueezing

Figure 15 Optical parametric amplification of a coherent state ndash The pictureshows how a displaced vacuum state is transformed into a displaced squeezed state Thepumprsquos electric field is maximal when the input field at fundamental frequency ν shows azero crossing The phase relation described results in an output state that is amplitudequadrature squeezed If the pump field at the input in phase was shifted by half of itswavelength the squeezed output field were a phase quadrature squeezed The squeezinggeneration displayed here corresponds to the transition from Fig 7(a) to Fig 7(b) but withan additional reduction of the displacement The concept of this figure was published inRef [Bauchrowitz et al (2013)]

in actual squeezing experiments the 4ν component is usually suppressed byphase miss-matching

Fig 15 shows the same process but now for an input field at frequencyν in a coherent state In this case the relative phase between the two inputstates ∆ϕ = ϕ2ν minus 2ϕν is relevant In Fig 15 the relative phase is set suchthat the expectation value of the field at frequency ν is zero when the pump

42

(a) (b)

(c) (d)

X

Y

X

YΔϕ = ndash50deg

X

Y

Δϕ = 0deg

X

Y

Δϕ = 90deg

Figure 16 Phase-space illustration of degenerate OPA ndash The (displaced) dashedcircle in each diagram represents the uncertainty of the initial state at optical frequencyν The (displaced) shaded area represents the state after degenerate optical parametricamplification All quantum uncertainties shown correspond to pure states The boldgreen arrow describes the bright second-harmonic pump field whose uncertainty can beneglected The phase between the 2nd harmonic pump and the initial state (∆ϕ = ϕ2ν minus2ϕν) determines the result of the parametric amplification

field reaches its maximum (∆ϕ = 90) The output at the fundamentalfrequency is then an amplitude squeezed state with a deamplified coherentamplitude

Fig 16 summarizes the squeezing operation on the vacuum state as wellas on displaced vacuum states for different phase relations ∆ϕ between thetwo input fields

43 Cavity-enhanced OPA

Placing the nonlinear crystal inside a cavity can greatly enhance thedown-conversion efficiency but not only that A cavity introduces a thresholdfor the pump power above which the parametric gain is infinite just limited

43

by the finite pump power In this case the vacuum uncertainty of the inputfield at frequency ν is amplified to a bright laser field at frequency ν Thedevice is then called an optical-parametric oscillator (OPO) For the gener-ation of squeezed states however the pump power is usually kept (slightly)below threshold Due to nonzero optical loss there exists a pump powersmaller than the threshold above which the tiny improvement of squeezingis not noticeable anymore Getting the pump power closer to the thresholdcould even reduce the observed squeeze factor if a fluctuating squeeze an-gle projects anti-squeezing into the observed quadrature amplitude [Franzenet al (2006) Suzuki et al (2006) Dwyer et al (2013)] The cavity has an-other important purpose It confines the transverse spatial mode usuallyto TEM00 This mode confinement is crucial for any efficient application ofthe squeezed state in laser interferometry since it allows the suppression ofanti-squeezing from other transversal modes The squeezing process requiresa nonlinear material that should show negligible absorption at both opticalfrequencies involved in particular at the wavelength of the squeezed modeIn Refs [Vahlbruch et al (2008) Mehmet et al (2009)] 10 dB and 116 dBof squeezing were achieved using MgOLiNbO3 The highest squeeze factorstoday are produced in (quasi phase matched) periodically poled KTP [Eberleet al (2010) Mehmet et al (2011) Stefszky et al (2012) Vahlbruch et al(2016)]

The optical cavity that is built around the nonlinear crystal is vital forsqueezed-light generation and it deserves a detailed consideration Gener-ally the mode propagating away from a cavity is the result of interferenceat the cavity coupling mirror One contribution is given by the intra-cavityfield attenuated by the amplitude transmission coefficient t of the outcouplingmirror The second contribution is given by the outside field that is reflectedby the same mirror with amplitude reflectivity r =

radic1minus t2 and spatially

overlapped with the first Also the mode from a squeezing resonator is suchan interference product

The impedance matched resonator

Let us consider first an empty optically stable and loss-less Fabry-Perotresonator built from two identical mirrors each with amplitude reflectivityr = r1 = r2 lt 1 A propagating field be perfectly mode-matched to one ofthe cavity resonances In this setup the resonator shows zero reflection andthe resonator is said to be impedance matched (for all such input fields)

44

|α〉

r1α

r12

= +

|α〉 r22 = r1

2

minusr1α equiv t1 αcav = 1minus r12 minusα sdot r1

1minus r12

0

XY

Figure 17 Empty impedance-matched resonator ndash Mode-matched and resonant lightthat enters the cavity from the left here displayed by a quantum phasor for a coherentstate is fully transmitted including its quantum uncertainty The back-reflected statedestructively interferes with itself for all frequencies well within the cavity linewidth The180 phase shift of the transmitted cavity field amplitude minusr1α is a necessary conditionin order to fulfill energy conservation on cavity resonance Impedance matching is forinstance achieved for a resonant cavity with matched mirror reflectivities (r2

2 = r21) and

zero optical loss The complex amplitude of the field traveling towards left inside the cavityis readily deduced from the figure and reads αcav = minusαr1(1minus r2

1)minus12 (highlighted by thedashed circle) In the displayed setup zero field uncertainties are reflected however alsothe vacuum state that enters the cavity from the right is fully transmitted (not shown)

r1α

XY

r12 r2

2 =1

= + |α〉

minus 1+ r1( )α equiv t1 αcav = 1minus r12 minus 1+ r1( )α

1minus r12

|α〉

Figure 18 Empty maximally overcoupled resonator ndash Maximal overcoupling isachieved for a resonant cavity with a perfect end-mirror reflectivity (r2

2 = 1) and zerooptical loss For a given input-mirror reflectivity r2

1 the intracavity light power is maximalMode-matched and resonating light entering the resonator from the left is fully reflectedThe complex amplitude of the field traveling towards left inside the cavity is readilydeduced from energy conservation to αcav = minusα(1 + r1)(1 minus r2

1)minus12 (dashed circle) Inthis setup no uncertainty from the right couples to the left side of the cavity

45

Obviously the interference described in the previous paragraph is fullydestructive The same resonator also shows zero reflection of the input fieldrsquosquantum uncertainty since the interference happens between parts of thesame quantum state The mode propagating away from such a resonatorhowever is not in a nonclassical but in a vacuum state because the vacuumstate that enters the cavity through the opposite site is also fully transmittedThe interference at the coupling mirror of an impedance matched resonatoris displayed in Fig 17

r1α

r12

+

=

OPA PDC

Xcavg r1(1+r1) Xcav

Δ2Xg 0 Δ2Yg

XY

r22 =1

|α〉

Figure 19 Squeezing resonator ndash Shown is the interference at the zero-loss squeezingresonator operated at threshold The lower line represents the perfectly squeezed modepropagating away from the cavity towards the left The parametric gain medium inside thecavity deamplifies the X quadrature of the cavity mode (Xcav) by the factor r1(1 + r1)which is the ratio of the intra-cavity field amplitudes of the two previous figures The Xquadrature of the field that is back-reflected towards the left destructively interferes withitself similar to the situation of the impedance matched cavity in Fig 17 The parametricpower gain per resonator round-trip (G) needs to mimic the effect of an end mirror withreflectivity r2

2 = r21 For this reason the deamplification of Xcav corresponds to a round-

trip deamplification factor of r1 equivradic

1G The round-trip amplification factor for Ycav

then is 1r1 equivradicG which exactly compensates for the outcoupling and thus determines

the parametric oscillation threshold (threshold for bright lasing) The variances of thequantum uncertainties ∆2Ycav and ∆2Y are thus infinite In this setup no field uncertaintyfrom the right couples to the left of the cavity and a perfectly X-quadrature-squeezed fieldoutside the squeezing resonator is produced

46

The perfectly over-coupled single-ended resonator

We now increase the reflectivity of the far mirror lsquo2rsquo to being perfect (r2 = 1)This way the counter-propagating vacuum state can not enter the cavityAgain a propagating field be perfectly mode-matched through mirror lsquo1rsquo toone of the cavity resonances For frequencies well inside the cavity linewidththe situation is displayed in Fig 18 The setup protects the left side of thecavity against vacuum fluctuations entering through mirror lsquo2rsquo but of coursedoes not squeeze quantum noise The intra-cavity built-up factor is too highfor achieving destructive interference below the vacuum uncertainty on theleft side of the resonator

The impedanced-matched single-ended squeezing resonator

Building on the two previous concepts the straight forward approach now isto start from the perfectly over-coupled single-ended resonator and insert anattenuator into the cavity that does not couple the cavity mode to any bathbut still results in a roundtrip efficiency of precisely r1(lt 1) in amplitudeOptical loss is not appropriate since it increases the coupling of the cavitymode to a thermal bath neither would any phase-insensitive atenuator beappropriate It is easy to show that a phase-insensitive attenuator adds ad-ditional uncertainty since otherwise the commutation relation [a adagger] = 1 isviolated The amplification process that matches our requirement is OPA Toachieve infinite squeezing in X on cavity resonance a second-order nonlinearcrystal needs to be put into the cavity and pumped such that the intra-cavityamplitude quadrature is attenuated by the factor (1 + r1)r1 (on cavity reso-nance) with respect to the empty cavity This factor is readily deduced fromFigs 17 and 18 Due to the symmetry in parametric amplification the intra-cavity phase quadrature is then amplified by (1 + r1)r1 and the round-tripgain has a value of 1r1 in amplitude In this situation not only infinitesqueezing but also the (laser) threshold of the resonator is achieved sincethe round-trip gain of the intra-cavity phase quadrature equals its roundtriploss here fully given by the incoupling mirror

The physical descriptions in Figs 17 to 19 are fully consistent with ob-servations in squeezing experiments The consideration above in particularshows that the intra-cavity field shows a finite squeezing strength while theexternal field shows infinite squeezing The strongest intra-cavity squeezefactor possible is (1 + r1)2r2

1 In the high reflectivity limit this factor corre-

47

sponds to 6 dB Averaged over the full cavity mode the squeeze factor of thecavity mode is in this limit even limited to 3 dB [Walls and Milburn (2008)]Higher intra-cavity squeeze factors are possible for lower mirror reflectivities

44 The generation of squeezed light for laser interferometry

With the insights gained in the previous subsection we now turn to ac-tual experiments The application of squeezed states in laser interferometrycertainly requires large squeeze factors (idealy accompanied with the highestpossible purity) to maximize the impact in terms of sensitivity improvementIn cavity-enhanced OPA the highest parametric gain is achieved on cavityresonance ie at zero sideband frequency But this is not the main reasonwhy this Subsection focusses on the generation of squeezed states at lowsideband frequencies The application of squeezed states in a laser inter-ferometer requires that their sideband frequencies cover the devicersquos signalband Ground-based gravitational wave (GW) detectors have a detectionband from about 10 Hz to 10 kHz frequencies which can be considered aslsquolowrsquo compared to typical frequencies in quantum optics experiments

Squeezing at MHz sideband frequencies is easier to observe than at acous-tic frequencies because the latter are often polluted with excess noise fromlight beams that serve as control beams [Bowen et al (2002) McKenzie et al(2004)] and parasitic interferences from back-scattered light [Vahlbruch et al(2007)] Furthermore the observation of squeezing at low sideband frequen-cies requires a more stable setup since larger measuring times are necessaryThe observation of strong squeezing at MHz frequencies however alreadysets an upper limit to the optical loss of the setup At least the same squeezefactor can be observed at lower frequencies

There are two different main topologies for squeezing resonators TheFabry-Perot-type standing-wave resonator consists of a minimum number ofmirror surfaces and has the advantage of being compact and thus robustagainst mechanical vibrations Usually one or even two mirror coatings aredirectly placed on the spherical and polished surfaces of the nonlinear crys-tal itself [Wu et al (1986) Grangier et al (1987) Breitenbach et al (1998)Vahlbruch et al (2008) Eberle et al (2010) Vahlbruch et al (2016)] TheBowtie traveling-wave resonator has the advantage of providing a separa-tely accessible counter propagating mode for cavity length control [Ou et al(1992) Takeno et al (2007)] It shows no direct back-reflection of incoupledlight which helps reducing parasitic interferences [Stefszky et al (2012)]

48

(a)

(b)

(c)

(d)Squeezingresonator

To inter-ferometer

OPA

SHG

BHD

Mode cleaner

LO

Laser

DBS

DBS

Figure 20 Generation of squeezed light ndash (a) Example of a 2nd-order nonlinearcrystal for the squeezed-light generation at 1064 nm Shown is a bi-convex 65 mm long7MgOLiNbO3 crystal whose polished surfaces also carry the mirror coatings of the res-onator The crystal thus realizes a monolithic squeezing resonator as it was used forthe first demonstration of 10 dB squeezing [Vahlbruch et al (2008)] (b) Optical con-figuration of a half-monolithic (hemilithic) standing-wave squeezing resonator Here thecavity length can be adjusted by displacing the coupling mirror The crystal surface insidethe cavity is anti-reflection coated The photograph shows a 10 mm long PPKTP crystalsqueezing resonator as used for the GEO 600 squeezed-light source [Abadie (2011)] (c)Mechanically stable housing of a standing-wave squeezing resonator The crystalrsquos temper-ature is stabilized at its phase matching condition using Peltier elements (d) Schematicfor the squeezed-light generation After spatial filtering of continuous-wave laser lighttwo hemilithic standing-wave resonators are employed The first generates second har-monic pump light (SHG) The second (OPA) generates a squeezed vacuum field at theinitial wavelength The squeezed states are observed by a balanced homodyne detector(BHD) or alternatively sent and mode-matched to the optical mode of an interferometerbeforehand LO local oscillator DBS dichroic beam splitter

49

Fig 20 (a) and (b) show photographs of typical nonlinear crystals used forsqueezed-light generation at near infra-red wavelengths The crystals shownhere form a monolithic standing-wave squeezing resonator (a) or are partof a half-monolithic standing-wave squeezing cavity (c) shows a tempera-ture stabilized and mechanically stable housing of the squeezing resonator(d) shows a schematic of a full setup for the generation of squeezed vacuumstates of light for an application in a laser interferometer The only brightinput required for the squeezing resonator (OPA) is the second-harmonicpump field The resonator mode at fundamental frequency is thus initiallynot excited by photons ie it is in its ground state characterized by vacuumfluctuations due to the zero point energy see Fig 7 (c) [Gerry and Knight(2005)] The pump field spontaneously decays in the degenerate pair of sig-nal and idler fields The combined down-converted field leaving the resonatorexhibits quantum correlations which give rise to a squeezed photon countingnoise when overlapped with a bright coherent local oscillator beam Thedetection is done either in a balanced homodyne detector (BHD) or with asingle photo diode The squeeze factor increases the closer the pump powerof the squeezing resonator gets to the oscillation threshold and the lower theoptical loss on down-converted photon pairs is

441 High squeeze factors ndash minimizing decoherence

Squeezed states of light have significant impact on the sensitivity of laserinterferometers if large squeeze factors can be produced Squeezing of 3 dBimproves the signal-normalized quantum-noise spectral density by a factorof 2 This factor corresponds to doubling the (coherent state) light powercirculating inside the interferometer Squeezing of 10 dB corresponds to aten-fold power increase The experimentally demonstrated squeeze factorswere considerably improved in recent years [Takeno et al (2007) Vahlbruchet al (2008) Polzik (2008) Eberle et al (2010) Stefszky et al (2012)]culminating in a value of as large as 150 dB [Vahlbruch et al (2016)] Thisvalue corresponds to the same reduction of signal-normalized quantum noisethat is achieved by increasing the light power by a factor of 32 (At this pointit is already noted that squeezing the quantum noise can simultaneouslyreduce quantum measurement noise (shot noise) as well as quantum backaction noise (radiation pressure noise) This is not possible with scaling thelight power of coherent states see Subsec 55)

50

Ideally a parametric squeezed-light source can produce an infinite squeez-ing level see Fig 19 fundamentally just limited by the energy provided bythe pump field In practice the limit is set by decoherence mechanismsThe by far most important one is optical loss Optical loss occurs duringsqueezed-light generation its propagation through the interferometric setupincluding imperfect mode matchings and finally the photo-electric detec-tion Also detector dark noise [Schneider et al (1998)] phase noise [Takenoet al (2007)] and excess noise [Bowen et al (2002)] impair the observablesqueezing strength

Optical loss is usually understood as coupling the squeezed mode to a zerotemperature bath ie overlapping it with a vacuum mode For any amountof loss the resulting state is still squeezed But to be able to directly observesay 10 dB of squeezing the total loss on the state needs to be less than 10in this example cf Eq (16) To minimize optical loss the nonlinear crys-tal as well as lenses and beam splitters in the interferometric path need toshow very low absorption and scattering at the wavelength of the squeezedlight PPKTP shows absorption of about 10minus4cm and below at near-infraredwavelengths Low OH content fused silica is a suitable material for all otheroptics Absorptions of less than 10minus6cm were measured [Hild (2007)] Coat-ings on crystal surfaces and on all other optical components should also showlowest optical loss Total loss of the 10minus6 level are available today Superpol-ished surfaces which show roughnesses with less than 1 A root mean square(integrated over spatial scales from approximately 1 micron to 100 microns)and thus very low scattering are necessary to achieve these low numbersMinimizing the total number of optical components is essential From thisperspective a monolithic squeezing resonator as shown in Fig 20 (a) is theoptimum choice The squeezed mode needs to be matched to the mode ofthe laser interferometer or to the mode of the balanced homodyne detectorVisibilities of up to 998 have been achieved [Eberle et al (2010)] whichcorresponds to a loss of about 04 Of great importance also is the quantumefficiency of the photo-diodes used for detecting the squeezed field (togetherwith the interferometric signal) Recently a quantum efficiency of photo-diodes in a squeezing experiment of (995plusmn 05) was measured [Vahlbruchet al (2016)] To minimize photon loss the photo-diodes had no protectionwindow an anti-reflection coating on the semi-conductor material and theremaining reflection was re-focussed with an external mirror

Also the dark-noise spectral density of the detection electronics reducesthe observable squeezing and needs to be as low as possible Similar to optical

51

noise it also provides a contribution to the observed variance The dark noiseof the detection electronics needs to be much lower than the detected photoncounting noise In [Vahlbruch et al (2016)] it was 28 dB below shot noisebut still reduced the observable squeeze factor from 153 dB to 150 dB

Excess noise emerges if the squeezed mode couples to a nonzero tem-perature bath or to a mode whose excitation is strongly fluctuating (Thecoupling process can always be understood as a beam splitter coupling andis physically described by overlapping electric fields Coupling to a zero tem-perature bath leads to Eq (16)) The captured excess noise variance thenneeds to be added to the initial squeezing variance which deteriorates theobserved squeezing stronger than just mixing in the vacuum mode Excessnoise is less likely to occur at MHz frequencies but can be significant ataudio-band sideband frequencies and below and is thus a serious issue ingravitational-wave detectors [Chua et al (2014)] The reason for that isthat acoustically or thermally excited motions of surfaces produce frequencyshifts of back-scattered light mainly at these low frequencies [Vahlbruch et al(2007)]

Phase noise corresponds to stochastic phase fluctuations between thesqueezed field and the local oscillator within the measuring time It cor-responds to mixing the squeezed mode with itself with a fluctuating squeezeangle [Suzuki et al (2006) Franzen et al (2006)] Phase noise in squeez-ing experiments typically is less of an issue than optical loss [Dwyer et al(2013) Oelker et al (2016) Vahlbruch et al (2016)] The setuprsquos phasenoise can be reduced by making the squeezing resonator more compact andthus mechanically more stable against acoustic and thermal fluctuations ofthe environment and by improving the quality of the servo loops for cavitylength and propagation length controls Operating a squeezed-light resonatorin vacuum might also be beneficial The ability to run a high performancesqueezed-light generator in vacuum was demonstrated in [Wade et al (2015)]

442 Squeezing in the gravitational-wave (GW) detection band

High squeeze factors have been first demonstrated at sideband frequen-cies of a few MHz and above where excess noise is generally negligible whenworking with visible or near-infra-red light Today we know that extendingthe squeezing spectrum towards the audio-band and even below is technicallynot always easy but straight forward once a high squeeze factor is achieved

52

at MHz frequencies In most squeezing experiments the main laser light

Figure 21 Photograph of the GEO 600 squeezed-light source ndash The breadboarddimensions are 135 cm times 113 cm The squeezing resonator is high-lighted by the white ar-row and is set up as a standing-wave hemilithic cavity containing a plano-convex PPKTPcrystal of about 10 mm length (see also Fig 20 b) It is pumped with continuous-wave532 nm light that is produced by frequency doubling of light (at angular frequency ω)from a commercial NdYAG laser Two more laser fields at about 1064 nm having fre-quency offsets of more than 10 MHz with respect to ω(2π) serve as optical control fieldsBoth fields are mode-matched and injected into the squeezing resonator together with thesecond-harmonic pump field

at the squeezing wavelength is accompanied by significant noise up to thelaser relaxation oscillation For this reason laser control fields at the opticalcarrier-frequency in the optical path of the squeezed mode need to be avoided[Bowen et al (2002) Schnabel et al (2004) McKenzie et al (2004)] and thesqueezing resonator length and the optical path stabilized by other means[McKenzie et al (2005) Vahlbruch et al (2006)] Furthermore and mostimportantly excess noise due to back-scattering is an issue Back-scattering(also called lsquoparasitic interferencesrsquo) is produced if DC light scatters out ofthe optical path hits a vibrating surface and re-scatters back into the opticalpath [Vahlbruch et al (2007)] Significant back-scattering can be produced

53

in interferometers for the detection of gravitational waves since light powersof several hundreds of kilowatts are used Even back-scattering from the mil-liwatt local oscillator of balanced homodyne detectors is an issue at acousticsideband frequencies and below The recipe for avoiding parasitic interfer-ences turns out to be threefold (i) avoiding scattering by using ultra-cleansuperpolished optics with close to perfect anti-reflex coatings (ii) avoidingback-scattering by carefully blocking all residual (faint) light fields and (iii)reduce the vibrationally and thermally excited motion of all mechanical andoptical parts that could potentially act as a re-scattering surface with re-spect to the optical path [Vahlbruch et al (2007) McKenzie et al (2007)]The insights described above led to the first demonstration of audio-bandsqueezing at frequencies down to 200 Hz [McKenzie et al (2004)] and laterto the first demonstration of squeezing over the full gravitational-wave de-tection band even from as low as 1 Hz [Vahlbruch et al (2007)] Whilea standing-wave squeezing resonator [Ou et al (1992)] can be built in avery compact way that is rather insensitive against mechanical vibrations[Chelkowski et al (2007)] a traveling-wave bow-tie squeezing resonator [Wuet al (1986)] is more tolerant against back-scattered light [Chua et al (2011)]The strongest squeezing in the audio-band of up to 116 dB was reported inRef [Stefszky et al (2012)]

443 The first squeezed-light source for GW detection

The first squeezed-light source for the continuous operation in GW de-tectors had been designed and completed between 2008 and 2010 [Vahlbruch(2008) Vahlbruch et al (2010)] Since then this source has been producingsqueezed vacuum states in a fully phase controlled way using co-propagatingfrequency-shifted bright control beams [Vahlbruch et al (2006)] as an inte-gral part of the GW detector GEO 600 The source is a turn-key device witha fully automated re-lock system [Vahlbruch et al (2010) Khalaidovski et al(2012)] Re-locking is required if the temperature of the environment changessignificantly which drives the actuators outside their dynamic ranges

Up to 9 dB of squeezing over the entire GW detection band was observedusing a balanced homodyne detector (BHD) located close to the squeezingresonator The squeeze factor has been limited by optical loss due to ab-sorption in the PPKTP crystal transmission of the back-surface and thenon-perfect AR-coating of the crystalsrsquos intra-cavity surface The adjustableair gap has been necessary to allow for an easy way to apply length control

54

-12

-8

-4

0

4

8

12

16

20

10 100 1k 10k

Rel

ativ

e no

ise

pow

er [

dB]

Frequency [Hz]

shot noise (a)

squeezed noise (b)

anti-squeezed noise (c)

~ 9dB

Figure 22 Broadband squeezing spectrum ndash Noise power spectra measured on theoutput of the GEO 600 squeezed-light source shown in Fig 21 with a balanced homodynedetector The traces correspond to the spectra of quadrature amplitude variances normal-ized to vacuum noise The resolution bandwidth used increases towards higher frequenciesto reduce the measurement time (a) Shot noise normalized to unity which serves as thereference level (0 dB) (b) Squeezed noise covering the complete detection band of ground-based GW detectors (c) Anti-squeezed noise Peaks at 50 Hz and 100 Hz are the electricmains frequency and its first harmonic The data was published in Ref [Vahlbruch et al(2010)]

Additional optical loss in the path to the balanced homodyne detector mainlyarose due to a Faraday isolator that eliminated parasitic interferences Fi-nally the mode missmatch to the BHD as well as its non-perfect quantumefficiency provided additional loss Inferring the squeeze factor without theBHD detection loss more than 10 dB of squeezing are provided by the sourceSince 2010 it has been used in basically all observational runs of the GEO 600GW detector see Section 6

444 Generation of two-mode (bi-partite) squeezing

lsquoTwo-mode squeezed lightrsquo or lsquobi-partite squeezed lightrsquo is light that allowsfor joint measurements at two locations A and B These joint quadraturemeasurements reveal correlations and anti-correlations with a remaining un-certainty smaller than the ground-state uncertainty which certifies the pres-

55

ence of entanglement cf Subsec 34 Bi-partite squeezed light has been gen-erated by type I and by type II parametric down-conversion In case of type Ithe squeezed fields from two squeezing resonators as described in Subsec 44are overlapped on a balanced beam splitter with a 90 phase shift The twooutput fields together represent the entangled mode [Furusawa et al (1998)Bowen et al (2003c) Eberle et al (2013)] In case of type II signal and idlerfields are non-degenerate regarding polarisation and a single cavity contain-ing an appropriate crystal and a polarising beam splitter are sufficient for theproduction of bi-partite squeezing Also in this case the measurements ofthe quadrature amplitudes of signal and idler fields show large uncertaintiestogether with bi-partite correlations and anti-correlations that are strongerthan the ground state uncertainty of individual subsystems [Ou et al (1992)Villar et al (2006) Jing et al (2006)]

To date the strongest entanglement of bi-partite squeezed light has beenproduced based on type I parametric down-conversion [Eberle et al (2013)]The requirements of producing strong entanglement are identical to those ofproducing strong squeezing outlined above The strength of bi-partite en-tanglement can be given in decibels in full analogy to the squeeze factorPractically the strength of bi-partite squeezing is always somewhat smallerthan that of single party squeezing since it requires an additional mode-matching that results in additional optical loss

45 Conclusions

The first observation of squeezed light was achieved in 1985 [Slusher et al(1985)] Shortly after cavity-enhanced optical parametric amplification forsqueezed-light generation was demonstrated [Wu et al (1986)] which todayenables the observation of up to 15 dB of squeezing [Vahlbruch et al (2016)]Quite generally the maximum squeezing level that is observed does not de-pend on the strength of the optical nonlinearity Squeezing cavities can easilybe operated at their oscillation threshold where they should provide infinitesqueezing if decoherence is zero The main limiting factor is optical lossincluding that of the photo-electric detection

Dedicated experimental research and development towards a squeezed-light source for applications in gravitational-wave detectors can be tracedback to 2002 [McKenzie et al (2002) Bowen et al (2002)] Since then a sur-prising amount of progress has been made culminating in the first squeezed-light source specifically built for the integration into a gravitational-wave

56

detector For the future squeeze factors above 15 dB will certainly be possi-ble by further reducing optical loss This statement is supported by the highdegree of matching between experimental data and a theoretical loss modelas presented in Fig 3 of [Vahlbruch et al (2016)]

5 Quantum noise in laser interferometers

51 Interferometric measurements

The purpose of a laser interferometer is the precise measurement of smallchanges of an optical path length with respect to a reference path Forthis the interferometer transfers the change of the phase difference betweentwo light fields into an amplitude quadrature change of the interferometerrsquosoutput light The latter can easily be detected by a single photo diode Ofgeneral interest are differential changes of the optical path length that aremuch smaller than the laser wavelength ie differential phase changes thatare much smaller than 2π In this case the differential phase change can bedescribed in very good approximation as a differential change of the phasequadrature instead

In order to transfer the phase quadrature signal with minimum loss ahigh interference contrast at the interferometerrsquos beam splitter is essentialAdditionally instrumental noise in terms of unwanted excitations of the out-putrsquos amplitude quadrature needs to be reduced to a minimum Noise arisesdue to power fluctuations of the input laser light back-scattered laser lightinside the interferometer thermally driven displacements of mirror surfacesand in many more ways The important measure of the sensitivity of aninterferometer obviously is its signal-to-noise-ratio The most useful measureis given in terms of the noise spectral density S(f=Ω2π) that is normalizedto the physical unit of the signal S(f) is in fact a lsquonoise-to-signal-ratiorsquoand can be seen as the signal-normalized variance of the photo diode out-put decomposed into spectral components versus sideband frequency f withthe resolution bandwidth of 1 Hz As an example S(100 Hz) = 10minus39 m2Hzmeans that the instrumental noise in the one hertz band around 100 Hz equalsa signal that would be produced if the mirror of one interferometer arm os-cillates with an amplitude of just

radic10 middot 10minus20 m in the very same band Such

small spectral densities are achieved by gravitational-wave detectors [Abbott(2016)]

57

52 Quantum measurement noise and shot noise

The most fundamental noise source in laser interferometers is due to thequantum noise of light which is in fact two-fold [Caves et al (1980)] Firstof all there is lsquoquantum measurement noisersquo which arises in the processof photo-electric detection For coherent states the quantum measurementnoise is the lsquophoton counting noisersquo of mutually independent photons andusually simply called lsquoshot-noisersquo Fig 2 (b i) shows a time series of suchnoise hiding the actual signal The frequency components of the shot noiseare well described by the quantum uncertainty of the output fieldrsquos amplitudequadratures XΩ∆Ω see Subsec 22 (Recall this quantity corresponds to the

differential phase quadrature YΩ∆Ω of the light beams in the interferometerarms) The photon counting noise has a white Fourier spectrum howeverthe lsquoshot noisersquo of an interferometer is usually normalized to the signal whosetransfer function is usually not white for instance due to the presence of armcavities or a signal-recycling cavity

All current and planned gravitational-wave detectors are Michelson-typelaser interferometers with operating points very close to a dark fringe Thelight power in the output port is just a couple of tens of mW which canbe handled by a single photo diode In this configuration the signal-to-shot-noise-ratio is actually maximized which can be shown in three steps [Bachorand Ralph (2004)] For the first step we use plane waves to describe theelectric field in the output port of a Michelson interferometer For perfectinterference contrast at the balanced beam splitter ie for perfect modematching and for defining φ = 0 as the dark port condition we get

Eout(t φ) =1

2E0 sin(ωt+ φ)minus 1

2E0 sin(ωt) (31)

where E0 is the amplitude of the total internal field whose two parts hasaccumulated a differential phase It directly follows for the squared fields

E2out(t φ) =

(sin

φ

2

)2

(E0 cos(ωt+ φ2))2 (32)

We now turn to a light beam with a localized transversal mode that can befocussed onto a photo-electric detector The photo diode has perfect quantumefficiency ie the rate of photo electrons is not only proportional to the rateof output field photons but also has a unity slope efficiency Since the optical

58

frequency is too high to be resolved we consider the averaged light power

P out(φ) =

(sin

φ

2

)2

P (33)

The next step is a formulation of the signal being the derivative of detectedphoton number versus phase Let n be the average value of the photonnumber per measuring time interval Eq (33) can then be rewritten as

nout(φ) =

(sin

φ

2

)2

n (34)

rArr dnout(φ) = n sinφ

2cos

φ

2dφ (35)

The final step is the calculation of the signal-to-shot-noise-ratio Shot noiserefers to coherent states which have a standard deviation of the photonnumber of σ(n) =

radicn

dnout(φ)

σ(nout)=n sinφ

2cosφ

2dφ

radicn sinφ

2

(36)

and find for a signal-to-noise ratio of unity for coherent states and for anon-zero but still small phase difference ∆φCoh 2π

1 =radicn cos

φ

2∆φCoh with φ 6= 0 (37)

In this equation the smallest measurable phase difference is given for φrarr 0

∆φCohmin =

1radicn (38)

This is the well-known shot-noise limit of high-precision phase sensing ∆φCohmin

is the smallest phase shift that can be measured with a signal-to-noise ra-tio of one when using n mutually independent photons per measuring time(those of a coherent state) when the loss of photons is assumed to be zeroThe typical purpose of a laser interferometer is the continuous sensing (moni-toring) of a continuously changing phase An illustrative example is the phasesignal produced by the black hole merger measured by Advanced LIGO onSept 14 in 2015 (Fig 1 in [Abbott (2016)]) The measuring interval should

59

be short to be able to resolve the time-evolution of the signal Generally themeasurement of an arbitrary signal that lasts for a finite time thus needs tobe understood as l subsequent measurement intervals using n photons eachIt can be shown that Eq (38) is also valid for interferometers operated athalf fringe ie when each output port contains the same light power In thiscase photo diodes need to be placed in both output ports and the actualsignal is provided by their difference voltage

Due to its importance the shot-noise limit deserves some remarksThe phase φ in Eq (31) is the phase difference of two mode-matched fieldsand might be accumulated by a single pass along the length L such as ina Mach-Zehnder interferometer or in a double pass such as in a (simple)Michelson interferometer or in four passes as realized in a Michelson inter-ferometer with folded arms [Grote (2005)] The shot-noise limit in Eq (38)and its scaling therefore holds independent of the number of passes Theclaim in Ref [Higgins et al (2007)] that the scaling according to Eq (38) canbe surpassed by multiple passes is not justifiedThe fact that Eq (38) is derived by approaching φ rarr 0 correctly describesthe actual operation point of gravitational-wave detectors which is close tobut not exactly at a dark port In practice a tiny offset from dark port ischosen at which the shot noise is well above the photo diodersquos electronic darknoiseEq (38) solely depends on the number of quanta but not on the lightrsquos wave-length λ Of course the shot-noise limit for the change of an optical pathlength ∆L does depend on the wavelength and ∆φCoh

min needs to be replacedby ∆φCoh

min = 2π∆LCohminλ

Finally an essential result of the shot-noise limit is that the ideal precisemeasurement should use lsquoas much quanta as possible per measuring inter-valrsquo which translates to lsquoas much light power in the interferometer armsas possiblersquo Eq (38) is indeed the one and only reason why gravitational-wave detectors use high power lasers power-recycling and arm resonatorsExtending the measuring time for a given light power can also improve thesensitivity but only if the signal repeats ie is periodic Let us assume thatone period of the signal is resolved by l intervals using n photons each Inthis case repeating the overall measurement k times improves Eq (38) by1radick The fundamental statement of Eq (38) however does not change

since the actual photon number n may then simply incorporate the factor k

60

For a given average photon number the shot-noise limit in Eq (38) canonly be surpassed by using photons that are quantum correlated ie by usingnonclassical states of light How is the shot-noise limit surpassed with thehelp of squeezed states A Michelson interferometer that is operated closeto a dark fringe acts like an almost perfect mirror for both input ports Allthe input light is back-reflected towards the laser source This also accountsfor the quantum uncertainty of the input light The quantum uncertaintythat impinges onto the photo diode thus (mainly) enters the interferometerthrough its (almost) dark port An interferometer that uses displaced co-herent states entering from one port can thus be improved by replacing theordinary vacuum entering the signal output port by a squeezed vacuum stateThis was the proposal by CM Caves in 1981 [Caves (1981)] which is labeledhere with lsquoCSVrsquo If the differential phase quadrature of the interferometer issqueezed Eq (38) then within the limit of large coherent state displacementα sinh2r improves to

∆φCSVmin asymp

eminusrradicn (39)

(The above expression is an approximation since the squeezing operationproduces a small number of photons that are not accounted for here) Ofcourse the mode of the squeezed vacuum needs to be precisely matchedto the mode of the interferometer The first experimental demonstrations ofsqueezed phase measurements used a Mach-Zehnder [Xiao et al (1987)] and apolarization interferometer [Grangier et al (1987)] Fig 2 shows how spatialdegeneracy between an externally generated squeezed mode and the signalmode in a Michelson interferometer is achieved using a polarizing beam split-ter and a Faraday rotator Again the limit in Eq (39) can only be achievedif optical loss is zero Optical loss not only reduces the signal but here alsoreduces the squeeze parameter see Eqs (15) and (16)

Let us consider an example The sensitivity of a laser interferometer thatuses coherent states with an excitation of 1023 photons per second can beimproved by a factor of

radic10 by either adding 09 middot 1024 photons per second

or by adding about just 2 photons per second and bandwidth in hertz thatbelong to the 10 dB squeezed vacuum confer Eq (30) Since the full sig-nal band of ground-based GW detectors covers sideband frequencies up to10 kHz just 2 middot 104 photons per second are necessary At a wavelength ofλ = 1064 nm these values correspond to a power increase by 168 kW and

61

37 fW respectively

The question arises whether a scaling of the sensitivity better than propradic1n is possible It was theoretically shown that in principle the scaling

can indeed considerably be improved yielding the so-called Heisenberg limitor Heisenberg scaling [Bondurant and Shapiro (1984) Yurke et al (1986)Braunstein (1992) Holland and Burnett (1993)]

∆φHLmin prop

1

n (40)

The Heisenberg scaling requires nonclassical states that have a certain num-ber of quanta similar to Fock states ie n = n The theoretically optimalstates describe a superposition of n (N) indistinguishable photons in oneinterferometer arm while having zero (0) photons in the second arm and viceversa and were named lsquoN00Nrsquo-states [Dowling (2008)] A specific propertyof these states is lsquosuper-resolutionrsquo The output ports of the interferometershow an n-times faster oscillation of the interference fringes when changingthe phase between the two interferometer arms Super-resolution correspondsto an n-times improved signal transfer function and was demonstrated forinstance in Refs [Rarity et al (1990) Kuzmich and Mandel (1998) Mitchellet al (2004) Afek et al (2010)] The presence of this nonclassical phe-nomenon however does not prove a sensitivity better than the semi-classicalbound according to Eq (38) Sensitivity is rather related to the signal-to-noise-ratio and needs to take into account all imperfections as well as theprobability of a successful detection of the sensing state [Thomas-Peter et al(2011)] All experiments so far used post-selection on particular measure-ment outcomes and neglected the typically large probability that nothingwas detected

Super-resolution was demonstrated with up to n = 5 [Afek et al (2010)]In addition to the fact that super-resolution does not prove a sensitivity bet-ter than the semi-classical bound photon numbers in state-of-the-art super-resolution experiments are extremely small compared to the photon numberof about 1023 (within a measuring interval of one second) of coherent statesused in Ref [Abbott (2016)] and of about 1022 using coherent states plussqueezed vacuum states used in Ref [Abadie (2011)]( which did prove a sen-sitivity better than the semi-classical bound)

Another interesting and related question is what the smallest phase is

62

that can be estimated in a single measurement again using a given numberof quanta Taking into account that no prior information about the phaseshift exists still a scaling proportional to 1n is possible In the limit of largen however an additional factor of π is required in the nominator of Eq (40)[Sanders and Milburn (1995) Berry and Wiseman (2000)] yielding

∆φHLmin asymp πn (41)

The state that can actually achieve this bound is different from the N00Nstate and was found in [Summy and Pegg (1990) Luis and Perina (1996)Berry and Wiseman (2000)] A N00N state is not the optimum state forphase estimation (via a single measurement) since it only provides one bit ofinformation A recent review on generell aspects on phase measurements isgiven by Ref [Demkowicz-Dobrzanski et al (2015)]

It is important to note that Eqs (38-40) do not consider photon lossExperiments that demonstrated super-resolution and aimed for achievingthe scaling in Eq (40) were conditioned on zero photon loss Let η gt 0 bethe average efficiency of detecting (all) photons Eq (38) then reads

∆φηCohmin =

radic1

η n (42)

Eq (39) turns into

∆φηCSVmin asymp

radicηeminus2r + 1minus η

η n(43)

and Eq (40) turns into [Demkowicz-Dobrzanski et al (2012 2013)]

∆φηmin =

radic1minus ηη n

for 0 lt η lt 1 (44)

For non-zero photon loss most interestingly the ultimate sensitivity of aphase measurement for a given photon number also shows a 1

radicn -scaling

The difference between the CSV strategy of using bright coherent states incombination with squeezed vacuum states which is bounded by Eq (43) andthe strategy of using the optimal nonclassical state which is bounded byEq (44) is marginal in practice [Demkowicz-Dobrzanski et al (2013)] Forgravitational-wave detectors and for any other laser interferometer using in-tense light there is no need for an alternative to the CSV strategy

63

We now turn back to the shot noise according to Eq (38) Generally noisecan be decomposed into its spectral contributions For a simple Michelsoninterferometer without arm resonators and without a signal-recycling cavitythe square-root of the single-sided shot-noise spectral density normalized tothe differential arm length change x in units of m

radicHz is given by [Saulson

(1994)] radicSMI

SNx =

radic~c2

2ωPprop 1radic

P (45)

where ω is the optical angular frequency of the quasi-monochromatic carrierlight and P the total light power in both arms including the built-ups fromcavities In combination with a squeezed vacuum whose relative phase gen-erates squeezing of the output lightrsquos amplitude quadrature the right handside reduces according to the factor eminusr Note that the single-sided spectraldensity is only defined for positive sideband frequencies and thus twice aslarge as the double-sided spectral density

The spectral density of the measurement of a GW induced strain is givenby the same expression but normalized to h = xL (If the gravitational waveis oriented in an optimal way with respect to the Michelson interferometerone arm is squeezed while the other is expanded by the same amount of ∆L =x2 and h then corresponds to the actual gravitational-wave amplitude)The square-root of the single-sided shot-noise spectral density normalized tostrain in units 1

radicHz is given by

radicSMI

SNh =

radic~c2

2L2ωP (46)

Equations (45) and (46) show that the smallest measurable signal (corre-sponding to unity signal-to-shot-noisendashratio) is inversely proportional to thesquare root of the laser power and has a white spectrum for sideband frequen-cies much smaller than the carrier frequency see horizontal line in Fig 23

All first- and second-generation GW detectors use power-recycling andadditional cavities to improve their sensitivities Fabry-Perot arm resonatorsdo not only increase the light power but additionally also increase the signalfor signal frequencies inside the resonator linewidth For lossless Fabry-Perot

64

arm resonators the spectral densities in Equations (45) and (46) need to bemultiplied by the following factor [Kimble et al (2001)]

HFP =

radicL2(γ2

FP + Ω2)

c2 (47)

where γFP = cTFP(4L) is the Fabry-Perot arm resonatorrsquos half bandwidthand TFP is the light power transmission of the input mirror The end mirrorsare assumed to have perfect reflectivity A similar expression can be derivedfor describing the improvement due to signal-recycling [Buonanno and Chen(2001)]

In summary shorter laser wavelengths higher light powers and squeezingof the amplitude quadrature of the interferometer output reduce shot noise ina broadband way ie for all signal frequencies Fabry-Perot arm resonatorsas well as signal-recycling provide improvements mainly for frequencies insidethe resonator linewidths

53 Quantum back-action and quantum radiation pressure noise

In laser interferometers quantum back-action noise results from the un-certainty of the lightrsquos radiation pressure force on the interferometer mirrorsand is also called lsquo(quantum) radiation pressure noisersquo (RPN) Its origin isthe quantum uncertainty of the differential amplitude quadrature XΩ∆Ω ofthe fields in the interferometer arms It results in an uncertain momentumtransfer to the mirrors and thus in an position uncertainty of the mirrors atfuture times with respect to their differential mode of motion [Caves et al(1980)] The physical mechanism of radiation pressure corresponds to anintensity dependent phase shift [Pace et al (1993)]The higher the light power in the arms of a laser interferometer the loweris its shot-noise spectral density see Eq (45) Unfortunately the spectraldensity of quantum back-action noise increases with light power The single-sided force noise spectral density reads

radicSRPNF =

radic8~ωPc2

(48)

Whereas the force noise of the quantum radiation pressure has a white spec-trum the RPN does not since the mirrorrsquos reaction to external periodicforces depends on frequency The link between the Fourier component of an

65

external force F (Ω) and the Fourier component of the displacement x(Ω) isgiven be the mechanical susceptibility HM It reads for an harmonic oscillatorwith mass M

HM(Ω) =1

M | minus Ω2 + Ω2M + iΩΩMQ|

(49)

where ΩM is the oscillatorrsquos resonance frequency and Q its quality factorThe square root of the single-sided spectral density of the RPN normalized

to the displacement of an harmonic oscillator with mass M is then given by

radicSRPNx = HM(Ω)

radic8~ωPc2

(50)

In GW detectors the test mass mirrors are suspended as pendula with highmechanical Q-factors and their centre of mass motion corresponds to that ofa harmonic oscillator The resonance frequencies of the pendula are lowerthan the detection band of interest The mechanical susceptibility is thereforeoften approximated for the so-called free-mass regime as H fm

M (Ω) = (mΩ2)minus1The square root of the single-sided spectral density of the RPN normalizedto differential displacement of two mirrors with each of mass M in a simpleMichelson interferometer is given by [Saulson (1994)]radic

SfmMIRPNx =

radic2~ωPc2m2Ω4

propradicP (Ω ΩM) (51)

where m = M2 is the mirrorsrsquo reduced mass In case of a simple Michel-son interferometer that is enhanced with arm cavities the spectral density inEq (51) needs to be multiplied with the expression given in Eq (47) In com-bination with a squeezed vacuum whose relative phase generates squeezing ofthe output lightrsquos phase quadrature the right hand side reduces according tothe factor eminusr Note if the radiation pressure noise is squeezed the shot noisemust be anti-squeezed or vice versa The radiation pressure noise calibratedto strain of space time is given by the right side of Eq (51) divided by theinterferometer arm length LIn summary heavier masses longer laser wavelengths lower light powersand squeezing of the amplitude quadrature in the interferometer arms re-duce radiation pressure noise

66

1 10 100 1000Frequency [Hz]

Radiation pressure noise

Shot noise

SQL

10-21

10-18

10-15

10-12

radicSx

[mradic

Hz]

ndashndash

ndashndash

Figure 23 Displacement-normalized quantum noise spectral densities ndash Consid-ered is a simple Michelson interferometer with neither arm cavities nor signal recyclingThe two end mirrors (m = 100 g) of the interferometer arms are suspended as pendulahaving a resonance frequency of ΩM2π = 1 Hz and a Q-factor of 107 The interferometeruses quasi-monochromatic light (in coherent states) with a total power of 4 kW Opticalloss and the offset from a dark output fringe is assumed to be negligible Wavelengthλ = 1550 nm The standard quantum limit (SQL) corresponds to the lowest noise achiev-able at a given sideband frequency when varying the light power without using quantumcorrelations

54 Interferometer total quantum noise and the standard quantum limit

Both shot noise and radiation pressure noise contribute to the total quan-tum noise of a given interferometer If they are not quantum correlatedwhich is the case for a conventional Michelson interferometer when detectingthe output lightrsquos amplitude quadrature their variances add up (The resultis not shown in Fig 23) It can easily be deduced from the previous sectionsthat changing the laser power will shift the two quantum noise contributionsHowever the total quantum noise never goes below the standard quantumlimit (SQL) [Braginsky and Manukin (1967)]

Let us consider Fig 23 for sideband frequencies much greater than thependulum resonance Here the test mass mirrors react as free masses whenexerted to external forces The SQL in this free-mass regime is calculated

67

by minimizing the sum of the squares of Eqs (45) and (51) [Saulson (1994)]

Sfmtotx =

~c2

2ω

[1

P+

4ω2

c4m2Ω4P

] (52)

Its derivative reads

dSfmtotx

dP=minus1

P 2+

4ω2

c4m2Ω4 (53)

Setting the above equation to zero provides the optimum laser power versussideband frequency in order to achieve the lowest total quantum noise

P fmopt =

c2mΩ2

2ω (54)

Inserting the optimal light power into Eq (57) provides the square root ofthe single-sided noise spectral density of the free-mass SQL in m

radicHzradic

SfmSQLx =

radic2~mΩ2

(55)

Again m is the reduced mass and dividing by the interferometer arm lengthL yields normalization to the GW-induced strain h Eq (55) shows that theSQL falls off with sideband frequency The corresponding equation for aMichelson interferometer that uses arm cavities readsradic

SfmFPSQLx =

radic~

mΩ2

(1

HFP

+HFP

) (56)

with HFP according to Eq (47)Using the expression for the SQL the square root of the total quantum

noise spectral density of a Michelson interferometer in the free-mass approx-imation can be written asradic

SfmFPtotx =

radicSfmFP

SQLx

2

[1

k+ k

] (57)

with the radiation pressure coupling parameter

k(Ω) =2ωP

mc2Ω2 (58)

68

For a fixed light power and fixed reduced mass of the mirrors the quantumnoise limited interferometer reaches the SQL when k = 1 which is realizedat the angular sideband frequency ΩSQL =

radic2ωP(mc2)

Note that neither squeezing the phase quadrature nor squeezing the am-plitude quadrature of the interferometer light leads to sub-SQL performance[Caves (1981)] also confer Ref [Schnabel (2005)] Fig 3 (left) As we willsee in the next sections the standard quantum limit can be surpassed if shotand radiation pressure noise are correlated Then the total quantum noise isnot given by the sum of the variances ie the sum of the squares in Eqs (45)and (51)

55 Squeezed light for surpassing the standard quantum limit

A measurement with sensitivity better than the standard quantum limit(SQL) is also called a lsquoquantum non-demolition (QND)rsquo measurement [Bra-ginsky and Khalili (1995 1996) Kimble et al (2001)] Several QND tech-niques for laser interferometers were proposed in recent decades [Jaekel andReynaud (1990) Kimble et al (2001) Purdue and Chen (2002) Chen (2003)McClelland et al (2011) Danilishin and Khalili (2012) Graf et al (2014)]What they have all in common is they exploit quantum correlations betweenobservable uncertainties

Arguably the most extensive way of introducing quantum correlationsand surpassing the SQL is the injection of squeezed states of light [Jaekeland Reynaud (1990)] If the squeezed quadrature angle of the injected statesis neither 0 nor 90 the quantum uncertainties of the amplitude and phasequadrature amplitudes that describe the differential field in the two interfe-rometer arms become correlatedLet us consider a very simplified setup that just consists of a quasi-mono-chromatic light field that is back-reflected from a quasi-free mirror Thelight power and the mass be such that reflected light in a coherent stateresults in a measurement of the mirror position with a noise spectral den-sity at the SQL at sideband angular frequency ΩSQL At this frequencyquantum measurement noise and back-action noise are of the same sizeie the uncertainty in X produces an equally large additional uncertaintyin Y Upon reflection the quadrature amplitude variances change from∆2X = ∆2Y = 14 to 2∆2X = ∆2Y = 12 This result corresponds tothe situation in Fig 23 at the crossing frequency of shot noise and radiation

69

pressure noise The coupling of the uncertainty variances can be describedby the matrix K = (1 minusk 0 1) where k = 1 at the SQL If the modulationstate at ΩSQL is the ground state its variances are transferred according to

KT

(1 00 1

)K =

(1 0minus1 1

)(1 00 1

)(1 minus10 1

)=

(1 minus1minus1 2

) (59)

In accordance with Fig 23 the variance of YΩSQLis twice as large as the vac-

uum noise varianceNow let the quantum noise of the light field be 10 dB squeezed at 45

(Eq (19)) The projection of the quantum uncertainty onto the X-observableproduces the radiation pressure noise by being transferred with the couplingfactor k = 1 at the SQL into the Y -observable in fact with negative signsince a larger value of X produces a larger optical path length and thus aretardation of the phase Due to the squeezing at 45 the initial uncertaintyin Y cancels with the additional uncertainty that originates from the onein X The following calculation shows that the strength of the cancellationcorresponds to the initial squeezing strength Upon reflection the quantumuncertainties transform in the following way(

1 0minus1 1

)(505 495495 505

)(1 minus10 1

)=

(505 minus01minus01 02

) (60)

The state of light after reflection has a squeezed phase quadrature amplitudeThe improvement in comparison to Eq (59) is exactly 10 dB The quantumnoise improvement corresponds to the input squeeze factor and is also a mea-sure by what factor the SQL is surpassed Squeezed vacuum injection thusallows surpassing the SQL upon measuring the conventional Y -quadrature(which is realized by a single photo diode in the interferometerrsquos output port)as first realized by [Unruh (1983) Yuen (1983) Jaekel and Reynaud (1990)]

In the example above the input squeeze angle is optimized for a sin-gle sideband frequency Injecting a broadband squeezed vacuum field withfrequency-independent squeeze angle of 45 would result in a rather bad inter-ferometer quantum-noise performance at frequencies far smaller or largerthan ΩSQL Fig 24 shows the quantum-noise performance if the input fieldhas squeeze angles that are optimized for every k(Ω) as given in Eq (58)Injected squeezing can thus lead to a broadband sub-SQL performance ifthe quantum measurement noise (shot noise) and the quantum back-actionnoise (radiation pressure noise) are correlated in an optimal way Due to the

70

10 50 100 500 1000 5000Frequency [Hz]

Dis

plac

emen

t noi

se sp

ectra

l den

sity

[mradic

Hz]

10-20

10-19

10-18

Total quantum noise 0dB

Total quantum noise -10dB

SQL

Y

X

Y

X

Y

X

Figure 24 Surpassing the SQL with squeezed-light injection ndash At shot-noise lim-ited sideband frequencies squeezing of the Y -quadrature amplitude improves the noisespectral density of the interferometer At radiation-pressure-noise limited sideband fre-quencies squeezing of the X-quadrature amplitude improves the noise spectral density ofthe interferometer If both kinds of quantum noise contribute equally (at the SQL markedwith a dot) a squeeze angle of 45 results in surpassing the SQL by the full squeeze factorsee Eq (60) In the graph here the squeeze angle is optimized for all frequencies result-ing in a broadband quantum noise reduction [Jaekel and Reynaud (1990)] Measurementsensitivities beyond the SQL (shaded area) are in the so-called quantum non-demolition(QND) regime [Kimble et al (2001)] Dashed horizontal lines represent the (squeezed)shot noise Dashed straight lines with negative slope represent the (squeezed) radiationpressure noise The calculations use 10 dB of squeezing a conventional Michelson inter-ferometer with neither arm resonators nor signal recycling a light power at the beamsplitter of 1 MW at a wavelength of λ = 1550 nm and mirror masses of 1 kg

correlation shot noise and radiation pressure noise can be squeezed simulta-neously

71

Light with a frequency-dependent squeeze angle

The discovery that shot noise and radiation pressure noise can be squeezedsimultaneously and thus a broadband reduction of quantum noise beyond theSQL be achieved required the insight that the spectral analysis of light definesa spectrum of many lsquosideband modulation modesrsquo that all can be in differ-ent quantum states An ordinary squeezing resonator which is on resonancefor light at twice the pump wavelength produces a spectrum of modulationmodes that all have the same squeeze angle A frequency-dependent squeezeangle can be introduced by reflecting such a field from a detuned single-ended filter cavity which was suggested by Kimble and coworkers [Kimbleet al (2001)] They showed that the optimal frequency dependence thatleads to the broadband improvement shown in Fig 24 can be realized byusing altogether two filter cavities as shown in Fig 25 Motivated by this re-sult research and development on filter cavities for optimizing the frequencydependence of broadband squeezed fields has been very active in recent years[Corbitt et al (2004) Chelkowski et al (2005) Dwyer et al (2013) Kweeet al (2014) Straniero et al (2015) Oelker et al (2016)]

Photo diode Squeezed vacuum

Faraday Rotator

Coherent light

YΩΔΩ

Figure 25 Frequency dependent squeezing injection ndash A broadband squeezed fieldwith a frequency-dependent squeeze angle that is optimal for gravitational-wave detectorsis produced by reflecting off an ordinary broadband squeezed field from two detuned opticalfilters [Kimble et al (2001)]

A light field with a frequency-dependent squeeze angle was first demon-strated in Ref [Chelkowski et al (2005)] see Figs 26 and 27 The experi-

72

Detuned filter cavityLockingphoto diode

Isolator

EOM R asymp 1

SHG

EOM

DBSOPA

LaserIsolator

Mode cleaner

LO

Homodyne detector

Figure 26 Generation of a frequency-dependent squeezing ndash A frequency-dependent orientation of the squeeze ellipse was first demonstrated in Ref [Chelkowskiet al (2005)] Initially a conventional spectrum of squeezed vacuum states of light was gen-erated in a squeezing resonator (lsquoOPArsquo) The squeezed vacuum was transmitted throughan optical isolator to a detuned filter cavity After reflection the squeezed vacuum stateswere absorbed in a balanced homodyne detector (BHD) The phase of the BHDrsquos localoscillator (LO) was changed for quantum state tomography of the squeezed states in diffe-rent regions of the spectrum The result showed a frequency-dependent orientation of thesqueeze ellipse see Fig 27 SHG second harmonic generation EOM electro-optical mod-ulator for applying phase modulation sidebands for cavity length control DBS dichroicbeam splitter R mirror reflectivity λ4 quarter wave plate for turning linear polarizedlight into circular polarised light and vice versa

ment consisted of a standing-wave squeezing resonator which produced ans-polarized broadband amplitude quadrature squeezed field accompanied bya dim continuous-wave DC control field with a wavelength of λ = 2πcω =1064 nm The squeeze bandwidth covered sideband frequencies up to aboutΩ(2π) = 30 MHz which corresponded to the linewidth of the squeezing res-onator The optical cavity for producing the frequency dependence of thesqueeze angle was a standing-wave cavity composed of a plane incouplingmirror of reflectivity r1 =

radic097 and a concave end mirror of reflectivity

r2 =radic

09995 The cavity length was L = 50 cm resulting in a linewidthof 147 MHz The squeezed field first passed a Faraday isolator to preventinterference effects between the filter cavity and the squeezing resonator A

73

Phase quadrature

min

max

-2 -1 0 1 2

141 MHz

-2

-1

0

1

2

Am

plit

ud

e q

uad

ratu

re

Figure 27 Frequency-dependent squeezing ndash Picture top right Reconstructed con-tour plot of the Wigner function of the sideband modulation at Ω(2π) = 141 MHz afterreflection from a 1515 MHz detuned filter cavity The state shows quantum correlationsbetween phase and amplitude quadratures ie squeezing at an angle of here about 40The white circle visualizes the standard deviation of the vacuum state uncertainty Thewhite ellipse represents the standard deviation of the squeezed uncertainty Small picturesMeasurement results on the same continuous-wave laser beam at various sideband frequen-cies around 15 MHz For each tomographic picture noise histograms of 100 equidistantquadrature angles were measured In each case the laser beam was phase locked to a ref-erence beam and the quadrature angle stably controlled and stepwise rotated The phasereference was given by a phase modulation at 198 MHz [Chelkowski et al (2005)] Thepicture was first published in Ref [Schnabel (2005)] (copyright ccopy2007 by Imperial CollegePress)

λ4-waveplate turned the s-polarized field into a circularly polarized beamwhich was then mode matched into the detuned cavity The retro-reflectedfield was analyzed by a balanced homodyne detector (BHD) for quantumstate tomography The filter cavity was electro-optically controlled to be de-tuned by 1515 MHz with respect to the DC control field The cavity length

74

control was achieved by the Pound-Drever-Hall (PDH) locking techniqueutilizing a circularly polarized laser beam that carried 15 MHz phase mod-ulation sidebands and was coupled into the filter cavity from the back Theoutput voltage of the BHD was characterized by a spectrum analyser as wellas used to perform quantum state tomography In the latter case the BHDoutput voltage was mixed down with an electronic local oscillator at differentradio-frequencies around 15 MHz and low-pass filtered to set the resolutionbandwidth to ∆Ω(2π) = 100 kHz The final electric signal corresponds to atime series of quadrature amplitude measurements XθΩi∆Ω Quantum statetomography is a method to reconstruct the phase space quasi-probability dis-tribution (Wigner function) of quadrature amplitudes from sets of measuredXθΩ∆Ω distributions when varying the angle θ [U Leonhardt (1997)] Forevery sideband frequency Ωi 100000 quadrature values were measured di-vided up on 100 equidistant quadrature angles Each quadrature angle wasstably controlled with a precision of plusmn1 Fig 27 shows the reconstructedWigner functions which were all measured on the same laser beam but atdifferent sideband frequencies For these measurements the detuned filtercavity was locked to the lower sideband at minus1515 MHz The result clearlyshows the frequency-dependent orientation of the squeeze ellipse In a morerecent experiment a frequency-dependent squeeze angle was also realized inthe kHz regime [Oelker et al (2016)]

56 Optomechanically induced (ponderomotive) squeezing

The radiation pressure of light when acting on a movable mirror re-sults in an intensity dependent phase shift [Pace et al (1993)] The cou-pling produces a so-called lsquoponderomotive effectrsquo [Braginsky and Manukin(1967)] which is of third order optical nonlinearity and which transforms abright coherent state inside an interferometer into a squeezed state of light[Vyatchanin and Matsko (1993)] This type of squeezed-light generationis usually called lsquoponderomotive squeezingrsquo or lsquooptomechanical squeezingrsquoConsequently even if no squeezed field is injected into the interferometercorrelations between the quadrature amplitudes are generated that allow forsurpassing the SQLPonderomotive squeezing as produced by the interferometer itself can onlybe exploited for evading back-action (radiation pressure noise) It can notbe used to squeeze the interferometer shot noise This is why ponderomotivesqueezing is fundamentally less extensive than injecting externally producedsqueezed states of light [Corbitt et al (2006)] suggested an external pon-

75

deromotive squeezing source for gravitational-wave detectors In this casedue to its external generation also the interferometerrsquos shot noise can besqueezed Recently ponderomotive squeezing was observed for the first time[Brooks et al (2012) Purdy et al (2013)] The achieved squeeze factors aremuch smaller than those produced by optical-parametric down-conversion[Vahlbruch et al (2016)]

Let us have a look again at Eq (59) Rotating the covariance matrix onthe right by arctan(minus

radic54minus 12) asymp minus58 indeed reveals squeezing

(cos 58 minussin 58

sin 58 cos 58

)(1 minus1minus1 2

)(cos 58 sin 58

minussin 58 cos 58

)asymp(

262 00 038

) (61)

The vacuum-noise normalized variance of 038 corresponds to about 42 dBof ponderomotive squeezing This is the general value that is produced atthe angular sideband frequency ΩSQL At higher frequencies the squeezefactor gets smaller at lower frequencies higher The squeezing strength of42 dB can be observed if the photo diode in the interferometer output portis replaced by a balanced homodyne detector using a local oscillator phase ofabout minus58 It can be shown however that the optimal signal-to-quantum-noise-ratio at the SQL is achieved for a local oscillator phase of exactly 45At this angle back-action is fully evaded

Full evasion of radiation pressure noise at all frequencies requires an op-timized frequency dependence of the relative local oscillator phase This canbe achieved by reflecting off the interferometer output field from two detunedfilter cavities [Kimble et al (2001)] The scheme was called lsquovariational out-putrsquo In the case of zero optical loss this scheme can fully evade radiationpressure noise just leaving the shot noise as the only quantum noise contri-bution

The variational-output scheme can be used to enhance the frequency-dependent squeezed input scheme The right site of Eq (60) shows thatthe output statersquos squeezing is not optimally detected in the Y -quadratureRather than with a single photo diode the detection should be done witha balanced homodyne detector with optimized phase of its local oscillatorIn this case the output lightrsquos quantum noise is solely given by squeezedshot noise The total quantum noise in Fig 24 would then be given by thelowest (dashed) horizontal line This combined scheme was called lsquosqueezed

76

variationalrsquo [Kimble et al (2001)] It can be realized by reflecting off theinterferometer output light from in total two optical filter cavities placed infront of the balanced homodyne detector

57 Conclusions

The highest quantum-noise-limited sensitivities of high-precision laserinterferometers are achieved by employing a large number of quanta tomaximize the signal strength in combination with strongly squeezed statesto minimize the quantum noise From this perspective it is clear that thequantum-noise-limited sensitivity of future gravitational-wave detectors willbe further improved ndash by increasing the light power and the squeeze factorTo be able to do so the optical loss in these devices needs to be reduced

In principle the optical loss in laser interferometers can be made smallbut never zero Recent theoretical research has shown that for any non-zero loss the sensitivity scales proportional to 1

radicn at best where n is the

average photon number per measurement This scaling is efficiently achievedby combining strongly displaced coherent states with squeezed vacuum statesof light

If a repeated measurement is not only limited by quantum measurementnoise but also by quantum back-action noise squeezed states of light can beused to simultaneously reduce both ie in the case of an interferometer shotnoise and radiation pressure noise

6 The first application of squeezed light in an operating gravita-tional-wave detector

Squeezed states of light have been successfully used to improve the sensi-tivity of the gravitational-wave detector GEO 600 from 2010 up to the pointwhen this Review was written [Abadie (2011) Grote et al (2013)] Afterdecades of proof-of-principle experiments [Xiao et al (1987) Grangier et al(1987) McKenzie et al (2002 2004) Vahlbruch et al (2005 2006 20072008) Goda et al (2008)] the implementation of a squeezed-light source inGEO 600 has resulted in the first sensitivity improvement beyond shot noiseof a measurement device that targets new observations in nature The im-plementation of squeezed states in GEO 600 was not done to provide anotherproof-of-principle demonstration but was realized because it offered a rela-tively cheap way of further improving the measurement sensitivity Of course

77

the sensitivity of GEO 600 can also be further increased by purely classicalmeans however the implementation of arm resonators to enable higher lightpowers without increasing the thermal load on the beam splitter or eventhe realization of longer interferometer arms are much more expensive Inthis respect the sensitivity improvement of GEO 600 with squeezed light canarguably be regarded as the first lsquotruersquo application that developed out of thefield of lsquononclassical (quantum) metrologyrsquo (Note that the term rsquoquantummetrologyrsquo is currently defined in different ways [Giovannetti et al (2006)Gobel and Siegner (2015)] and the term rsquononclassicalrsquo referring to a non-classical P-function gives a distinct description)

61 Gravitational waves

Einsteinrsquos General Theory of Relativity [Einstein (1916)] or simply lsquoGen-eral Relativityrsquo (GR) predicts that accelerating mass distributions producegravitational radiation analogous to electromagnetic radiation from accel-erating charges Experimental evidence of their existence is given by theobservation of the slow spiraling together of two neutron stars caused bythe loss of orbital energy to gravitational waves The inspiral rate exactlymatches the predictions of Einsteinrsquos theory [Weisberg and Taylor (2005)]Recently Advanced LIGO observed gravitational waves for the first time[Abbott (2016)] thereby giving the go-ahead for gravitational-wave astron-omy The gravitational-wave source was the final inspiraling and the mergerof two black holes 13 billion light years away from earth

Gravitational-waves evolve in the far field of the source propagate withthe speed of light and are measurable on earth with laser interferometersFig 28 displays a gravitational wave propagating along a certain directionGravitational waves are dynamical changes of space-time They are transver-sal and quadrupolar in nature and have two polarization states

A variety of known astrophysical and cosmological sources are predictedto emit gravitational radiation that should reach the Earth with a measur-able strength [Sathyaprakash and Schutz (2009)] The first gravitational waveevent detected was produced by two black holes of 36 and 29 solar massesDuring the final 02 seconds of their inspiraling they produced a peak gravi-tational strain in our solar system of 10minus21 covering frequencies up to 250 Hz[Abbott (2016)] Other predicted sources are mergers of neutron stars supernovae and background signals from the Big Bang According to GR GWsfrom complex astrophysical sources carry a plethora of information that willhave a major impact on gravitational physics astrophysics and cosmology

78

L +Δ L

Binary system

fBS

Figure 28 Space-time oscillation ndash Gravitational waves are dynamical deformations ofspace-time that form in the plane perpendicular to the direction of wave propagation Asa result distances between free-falling test masses in a transverse plane will change witha strain h = ∆LL For black hole or neutron star binary systems with orbital frequencyfBS distances will oscillate at frequency fGW = 2fBS The wavelength of this oscillation isgiven by λGW = cfGW where c is the speed of light The wave of orthogonal polarizationwith respect to the one shown is rotated by 45 around the propagation axis

62 Interferometric detection of gravitational waves

Current gravitational wave detectors are kilometre-scale laser interfero-meters [Dooley et al (2016) Aasi (2015) Acernese (2015) Aso et al (2013)]Continuous-wave laser light is split into two beams traveling in orthogonaldirections Both beams are reflected back towards the central beam splitterwhere they interfere Gravitational waves change the optical path lengthdifference and thus the light power directed towards the photo-diode thatis positioned in the signal output port of the beam splitter A gravitationalwave at frequency fGW = ΩGW(2π) reveals itself as a light-power modula-tion at the same frequency The spectral decomposition of the output signalis described by a spectrum of the quadrature amplitude YΩ∆Ω introduced inSec 3 It corresponds to the amplitude quadrature amplitude of the outputlight and relates to the differential phase quadrature of the interferometerarms

79

Photo diode

Squeezed vacuum

Faraday Rotator

Signal

Quantum noise

Coherent light XΩΔΩ

YΩΔΩ

(a) (b)

YΩΔΩFigure 29 Squeezed-light-enhanced interferometric measurement ndash (a) Michelsoninterferometer with arm cavities power recycling and signal recycling (see main text forexplanation) The interferometer is operated close to a dark fringe such that the quantumnoise entering from the dark port is back-reflected The squeezed field is mode-matched tothe signal output field (b) Phase space diagram of the gravitational-wave signal outputat sideband frequency Ω(2π) The quantum noise is squeezed below the ground stateuncertainty and thus the signal to quantum noise ratio improved

The first key ingredient of an interferometric gravitational-wave detectorare suspended heavy mirrors that can be regarded as quasi-free in the di-rection of laser light propagation thereby acting as test masses that probespacetime Being on ground current detectors are located in rather noisy en-vironments that allow the realization of undisturbed quasi-free mirrors onlyabove a sideband frequency of the order of 10 Hz Since sufficiently strongGW signals are expected up to a frequency of 10 kHz todayrsquos gravitationalwave detectors target at signals in the acoustic band from 10 Hz to 10 kHzThe quasi-free motion of the test mass mirrors in this frequency regime isachieved by suspending the mirrors as sophisticated multiple-stage pendulain vacuum chambers [Aasi (2015)] Far above the pendularsquos resonant frequen-cies which are typically around 1 Hz the centre of masses of the mirrors areisolated from vibrations of the ground and they react on frequency compo-nents of small external forces approximately as free masses The mirrors andtheir suspensions are built from materials having exquisitely high mechani-cal quality factors This helps to concentrate the thermal energy that causes

80

displacements of the mirror surface into well-defined vibrational frequencymodes At these particular very sharp frequencies no gravitational wavescan be detected

The second key ingredient of an interferometric gravitational-wave de-tector is laser light with a power of up to hundreds of kilowatts or evenmegawatts The light is quasi-monochromatic and needs to show very lowamplitude (quadrature) noise and phase (quadrature) noise at sideband fre-quencies within the detection band Low amplitude noise is necessary toprovide a shot noise limited output field It is also necessary to avoid classi-cal radiation pressure noise which becomes an issue if the light power or themirror masses in the two arms are not identical Low phase noise is requiredif the storage time of the light in the two arms is not identical This mightaccidentally occur due to different linewidths of the arm cavities or mightbe part of the interferometer design to allow for the length control schemeproposed by Schnupp [Heinzel et al (1998)] To maximize the light power in-side the interferometerrsquos cavities it should be produced in an almost perfecttransversal spatial distribution of a Gaussian TEM00 mode

Light sources of gravitational-wave detectors are ultra-stable NdYAGmaster-slave systems that provide up to 200 W of light at 1064 nm [Winkel-mann et al (2011) Kwee et al (2012)] The high power in the interferometerarms is achieved by cavity built-ups in the so-called power-recycling cavityand in the arm cavities Power recycling uses a partially reflective mirror thatis located between the light source and the interferometer beam splitter Itssurface is matched to the lightrsquos wave front and forms an optical cavity to-gether with the rest of the interferometer Since gravitational-wave detectorsare operated close to a dark fringe large power built-ups can be achievedThe highest power built-up is achieved for a mirror transmission equal to the(given) interferometer round trip loss In this case an impedance-matchedcavity is achieved The power-recycling cavity as well as the arm cavitiesare stabilized on resonance for the input light The difference between theirfunctionality is that the power-recycling cavity does not limit the detectionbandwidth of the interferometer GEO 600 as well as Advanced LIGO em-ploy a third type of cavity the so-called signal-recycling cavity Similarlyto power recycling a partially reflecting mirror that is placed between theoutput port of the beam splitter and the photodiode is used to resonantly en-hance the GW signal [Meers (1988)] The signal-recycling cavity resonantlyenhances the signal modulation fields within its linewidth without furtherenhancing the carrier light power In combination with low linewidth Fabry-

81

Perot arm resonators it can also be used to extract the signal by reducing theeffective finesse of the arm resonators for the signal sidebands This schemeis called resonant sideband extraction [Heinzel et al (1996)] The signal-recycling cavity has also been tested in a detuned setting in which just theupper or lower sideband is extracted or resonantly enhanced respectively[Heinzel et al (2002)] Current gravitational-wave detectors however usecarrier-tuned signal recycling

All these techniques are lsquoclassicalrsquo approaches for maximizing the signal-to-shot-noise ratio At frequencies above a few hundred Hertz howevershot-noise is still the limiting noise source in gravitational-wave detectorsFuture gravitational-wave detectors will therefore use even higher light pow-ers but further increasing the light power becomes more and more challeng-ing Optical materials with less light absorption need to be found to coun-teract an increasing thermal load inside the mirrors Mirror masses need tobe further increased to counteract the increasing radiation pressure noiseNonclassical approaches are superior and become more and more attractivethe farther classical approaches are pushed to the extremes Nonclassicalapproaches allow for simultaneously increasing the signal-to-shot-noise ratioand the signal-to-radiation-pressure-noise ratio without changing light poweror mirror masses see Fig 24 They also allow for a complete evasion of radia-tion pressure noise [Braginsky and Khalili (1995 1996) Kimble et al (2001)]see Subsec 56

63 Squeezed-light enhancement of the gravitational-wave detector GEO 600

In 2010 GEO 600 was equipped with the squeezed-light source shownin Fig 18 The location of the squeezed-light source close to the outputport is shown in Fig 30 It was known that GEO 600 was shot-noise limitedat sideband frequencies above about 700 Hz In this frequency regime thereplacement of the ordinary vacuum states that entered the interferometerfrom the output port by a spectrum of squeezed vacuum states was expectedto reduce the noise spectral density into the nonclassical regime It was notprecisely clear what squeezing factors could be expected since the opticalloss upon mode-matching an external field into the output port propagationalong the arms and the final photo-electric detection was not determined

Critical components were the quantum efficiency of the photo-diode aswell as the optical loss of Faraday rotator for overlapping the squeezed fieldwith the output mode Also the transversal output mode of GEO 600 was an

82

issue since it contained significant contributions from higher order modesto which a good mode matching of the squeezed mode was not possible Theimplementation of the squeezed-light source thus was accompanied with anew custom made InGaAs photo-diode with 3 mm diameter The goal was aquantum efficiency of greater 99 [Vahlbruch et al (2016)] Also the Fara-day rotator was custom-made and optimized for lowest optical loss which

Figure 30 GEO 600 ndash View into the central building of the British-German GW detectorlocated close to Hannover Germany The vacuum chambers contain the suspended beamsplitter power- and signal recycling mirrors additional input and output optics as well asmirrors to realize a double pass of the laser light through the 600 m long interferometerarms By courtesy of the AEI

83

involved a rather precise rotation of the polarisation of 45 plusmn 05 over anaperture of more than 15 mm Finally a ring cavity (output mode cleaner)was placed in front of the photo-diode which acted as a passive filter forhigher transversal modes Since GEO 600 was not limited by radiation pres-sure noise and since it used a carrier-tuned signal-recycling cavity a frequencyindependent orientation of the squeezing angle was optimum After several

100 200 300 500 700 1000 2000 3000 500010-22

10-21

10-20

Obs

erva

tory

noi

se c

alib

rate

d to

str

ain

[1radic

Hz]

10-19

10-18

10-17

Noise calibrated to test m

ass displacement [m

radicHz]

Sideband frequency f

Figure 31 Nonclassical reduction of the GEO 600 instrumental noise ndash Shownare the square roots of the noise spectral densities without squeezed vacuum states (top)and with squeezed vacuum states (bottom) [Abadie (2011)] Their calibrations [Affeldtet al (2014)] to gravitational-wave strain and differential arm length change are shownon the left and right y-axes respectively Note that both traces increase towards higherfrequencies due to the frequency-dependent signal enhancement of the signal-recyclingcavity The injection of squeezed vacuum states leads to a broadband noise reductionof up to 35 dB at shot-noise limited frequencies The spectral features are for instancecaused by excited violin modes of the mirror suspensions (600ndash700 Hz and harmonics)Data by courtesy of the AEI

months the combination of the squeezed-light source and the gravitational-wave detector succeeded The main laser of the squeezed-light source wasphase locked to the main laser of GEO 600 and a stable mode matching be-

84

tween the squeezed field and the interferometer output field was achieved[Abadie (2011)] The effect on GEO 600rsquos (strain normalized) noise spectraldensity is shown in Fig 31 At frequencies above about 700 Hz the squareroot of noise spectral density was reduced by up to one third This value cor-responds to a quantum noise variance of 045 (minus35 dB) with the shot noisevariance normalized to unity For isotropically distributed gravitational-wavesources this factor produces a detection rate increase by a factor of 153 asymp 34After its integration into GEO 600 the squeezed-light source was used in allscientific runs seeking for gravitational waves for instance in the observa-tional run S6eVSR4 that was undertaken from June 3rd to September 5th

in 2011 [Grote et al (2013)]Towards the end of 2011 right before the start of the detector upgrade

to Advanced LIGO a nonclassical sensitivity improvement was also demon-strated in one of the LIGO detectors [Aasi (2013)] In this experiment anonclassical sensitivity improvement corresponding to up to 215 dB abovefrequencies of about 150 Hz was achieved The successful test is a strongmotivation for a squeezed-light upgrade of Advanced LIGO Note that thedesign of Advanced LIGO was completed in 1999 and squeezed-light sourceswere not mature at those times

In the past years the squeezing enhanced GEO 600 detector was not onlyused for observations but also was the control of the injected squeezed modefurther improved Stabilizing the overlap between squeezed mode and brightmode of the interferometer to close to perfect is necessary to reduce theeffective optical loss and to maximize the measurable squeezing factor Re-cently superior methods for stabilizing the longitudinal phase of squeezedvacuum mode were found [Dooley et al (2015)] and the first automatic align-ment system for stabilizing and optimizing the transversal mode overlap wasdemonstrated [Schreiber et al (2016)]

64 Are squeezed states the optimal nonclassical resource ingravitational-wave detectors

For a given number of photons Eq (41) quotes the ultimately smallestphase change that can be measured with a signal-to-noise-ratio of one Thescaling with number of photons per measuring time of this Heisenberg limitseems appealing compared to the scaling achievable with coherent states orsqueezed states according to Eqs (38) and (39) respectively The Heisenberglimit however is only valid for precisely zero photon loss Since the non-classical states required to achieve Eqs (40) and (41) show an exponentially

85

increasing sensitiveness to loss when increasing the photon number the ac-tual scaling can not be deduced from Eq (40) Proposals to use Fock statesand the so-called N00N states for optimizing interferometer sensitivities [Hol-land and Burnett (1993) Dowling (1998) Mitchell et al (2004) Afek et al(2010)] are thus only applicable when the experiment is conditioned on zerophoton loss As discussed in recent publications the correct expression forthe fundamental sensitivity limit needs to consider not only the total photonnumber inside the interferometer but also the total photon loss [Dorner et al(2009) Ko lodynski and Demkowicz-Dobrzanski (2010) Knysh et al (2011)Escher et al (2011) Demkowicz-Dobrzanski et al (2012)]

Based on these earlier works Ref [Demkowicz-Dobrzanski et al (2013)]proved that the nonclassical sensitivity enhancement of GEO 600 reportedin Ref [Abadie (2011)] has been exceedingly close to fundamental quantuminterferometry bound under given energy constraints and photon loss lev-els More than that it was generally proven that the approach of com-bining displaced coherent states and squeezed vacuum states is optimal forgravitational-wave detectors

In Ref [Abadie (2011)] the gravitational-wave detector GEO 600 used aneffective number of photons per second of approximately n = 2 middot 1022 whichcorresponded to a total optical power inside the interferometer arms of P asymp37 kW at a wavelength of 1064 nm The total optical loss was 1minus η asymp 038The injected squeezing factor was eminus2r asymp 01 For these numbers the ratioof Eqs (43) and (44) is calculated to

∆φCSVmin

∆φgenmin

asymp

radicηeminus2r + 1minus η

1minus ηasymp 108 (62)

which is a good approximation within the limit of large coherent state dis-placements α sinh2r The quantum noise of GEO 600 including thesqueezed-light source was just 8 above the fundamental quantum inter-ferometry bound An increased squeezing strength of 16 dB (eminus2r asymp 0025)which is in reach would bring the approach based on coherent states andsqueezed vacuum states to within just 2 above the fundamental bound

Future GW detectors will have significantly reduced optical loss values(1 minus η) lsquoLossrsquo includes scattering and absorption at mirrors non-perfectfringe contrasts and the non-perfect quantum efficiency of the photo detec-tor Optical loss reduction is important for at least four reasons First itleads to an increased signal second it leads to a reduced quantum noise when

86

employing squeezed states third less absorption reduces the thermal load onthe test mass mirrors and fourth less scattering reduces the probability ofback-scattered light which produces disturbance signals [Billing et al (1979)Vahlbruch et al (2007) Punturo et al (2014)] The higher the finesse valuesof the arm and signal-recycling cavities are the more significant is opticalloss at mirror test masses the beam splitter and the signal-recycling mirrorThe finesse value of the power-recycling cavity and the loss of mirrors andlenses that guide the output field to the photo-diode are less critical Suitablephoto detectors of 995 quantum efficiency are available today [Vahlbruchet al (2016)] but achieving a total optical loss of 10 is still challengingThe reason for that is that first of all a measurement device aiming for bestabsolute sensitivity should use as much quanta (photons) as possible Highfinesse values for the enhancement cavities are thus essential but results inan unavoidable scaling-up of the effect of mirror losses A realistic exampleof future gravitational wave detectors thus considers η = 09 with a squeezingfactor of 20 dB (eminus2r = 001) In this case the quantum noise will be about4 above the ultimate fundamental bound for a given photon number

From Eq (62) it can be concluded that there is no need for any moresophisticated nonclassical states than squeezed states In particular non-classical states with a defined photon number such as N00N states are notrequired Within the approximation quoted this result is independent ofthe photon number This result is also independent of the physical systemused for interferometric phase estimation and can also be made for quantum-enhanced atomic clock calibration in the presence of dephasing Here the-oretical results indicate that the precision of Ramsey interferometry withspin-squeezed states is close to the optimal one in the asymptotic regime ofa large number of atoms [Huelga et al (1997) Ulam-Orgikh and Kitagawa(2001) Escher et al (2011)] as already stated in Ref [Demkowicz-Dobrzanskiet al (2013)] More sophisticated nonclassical states with fixed number ofn quanta might still be useful for the exceptional case when the absorptionof one quantum already results in zero measurement sensitivity anyway Anexample is an ensemble measurement where the absorption of a single photondemolishes the source of the phase change to be characterized A typicallyused approach of conditioning the measurement result on n clicks of n singlephoton counters conditions on precisely zero loss and is thus able to use theadvantage of Eq (40) over Eq (39)

87

65 Conclusions

Squeezed states of light will contribute to realizing gravitational-wave ob-servatories with much higher sensitivities than existing or planned ones Tobenefit from squeezed states in a most efficient way optical loss in terms ofabsorption and scattering must be minimized In particular the optical lossof mirror coatings and mirror substrates need to be minimized The rele-vant mirrors include the test masses the balanced beam splitter the signalrecyclingextraction mirror and all optical components between the latterand the photo diode Excellent spatial mode matching between the brightinterferometer field and the squeezed vacuum field is also of great impor-tance Achieving this requires further improvement of the surface figures ofall reflective optical components of the interferometer as well as improvedhomogeneity of all optical components that the light passes through

The quantum noise reduction achieved in a gravitational-wave detectoris of course always smaller than the highest squeeze factor provided by thesqueezed-light source As an example let us consider the observation of 15 dBof nonclassical noise suppression directly at the source If the squeezed fieldsenses an additional loss of 5 when propagating through the interferometerwhich is a very challenging number from todayrsquos point of view the remainingsqueezing level is about 11 dB see Eq (16)

7 The application of 2-mode-squeezed light in laser interferome-ters

71 Quantum Dense Metrology

At first glance the application of bi-partite (two-mode) squeezed statesto a device whose goal is measuring a single observable seems meaninglessSqueezing the uncertainty of that observable should be the optimum one cando This is indeed true when concerning just quantum noise but recentlyit was discovered that in the presence of classical disturbances bi-partitesqueezing can improve such measuring devices [Steinlechner et al (2013)]The concept was named quantum dense metrology (QDM) The potentialimprovement of a gravitational-wave detector with bi-partite squeezed statesis shown in Fig 32 (a) A description is given in the caption The pre-condition for a potential improvement can be best understood within a phasespace diagram Fig 32 (b) contains two different kinds of lsquosignalsrsquo The firstis the actual signal which always shows up as a phase space displacement

88

along the Y axis The second is a disturbance signal that can produce adisplacement in arbitrary direction in phase space A prominent example

Faraday Rotator

Coherent light

Photo detectors Balanced homodyne detectors

Squeezed vacuum

Squeezed vacuum

Quantum noise in

Y (A) and X (B)

XΩΔΩ(B)

YΩΔΩ(A)

(a) (b)

Disturbance projected

onto Y

and X

XΩΔΩ(B)

YΩΔΩ(A)

Example of (unknown) disturbance

Signal

Figure 32 Bi-partite-squeezed-light-enhanced measurement ndash (a) Setup for theapplication of bi-partite (two-mode) squeezed light in a laser interferometer on the basisof QDM Two squeezed vacuum fields are overlapped on a balanced beam splitter withprogrammable squeeze angles for instance with a relative angle of 90 which produces abi-partite state as shown in Fig 12 The beam splitter outputs are entangled for any rela-tive angle greater than zero One part is matched to the interferometer mode The secondpart is kept outside as a reference beam The interference of the interferometer outputand the reference beam is arranged with such a phase difference that it reproduces the twosqueezed inputs on the photo detectors The two squeezed beams are photo-electricallydetected measuring the respective squeezed quadrature (using balanced homodyne detec-tors) Both beams carry half of all interferometer induced modulations which includesignals as well as disturbances A single readout as shown in Fig 21 cannot distinguishbetween the two kinds The double readout shown here provides additional informationand allows for recognition of the disturbance [Steinlechner et al (2013)] as well as in prin-ciple a modeling of the disturbance and with a correct model an improvement of thenoise spectral density of the interferometer [Ast et al (2016)] (b) Phase space diagramdescribing phase quadrature readout A as well as amplitude quadrature readout B Bothshow squeezed quantum noise The amplitude quadrature readout does not contain anygravitational-wave signal ie any feature in this channel must be due to disturbancesThis information can be used to improve the interferometer

89

for such a disturbance is parasitic interference due to back-scattered laserlight [Vahlbruch et al (2007)] Back-scattering is a limiting noise at lowsignal frequencies of gravitational-wave detectors [Billing et al (1979) Vinetet al (1997) Hild (2007) Ottaway et al (2012) Punturo et al (2014)] Notethat all noise that couples in via unwanted motions of the test mass mirrorsso-called lsquodisplacement noisersquo always produces a phase space displacementalong the Y axis and cannot be tackled with QDM

Fig 33 shows measurement results obtained in Ref [Steinlechner et al(2013)] In a table-top experiment one part of a bi-partite squeezed state ofa continuous-wave mode at 1064 nm was mode-matched into the output portof a Michelson laser interferometer operated at its dark fringe in full analogyto Fig 32(a) A lsquosignalrsquo was produced by driving the piezo behind one of theend mirrors at a frequency of 555 MHz The lsquodisturbancersquo was introducedby re-injecting a small amount of light that leaked through the second endmirror with an additional piezo-mounted mirror The piezo was driven ata frequency of 517 MHz to produce a phase modulation An additionalDC voltage defined an arbitrary and unknown optical path length of thelight before being re-injected and as such the phase space orientation of thedisturbance signal This mechanism of a parasitic interference is realizednaturally in any interferometric device due to back-scattering of quanta frommoving surfaces in the environment

The interferometer output consisted of the signal as well as the distur-bance with a quantum uncertainty given by one subsystem of the bi-partiteentanglement It was overlapped with the second subsystem of the entan-gled state on a balanced beam splitter and the two outputs were analysedwith balanced homodyne detectors The phases of the bi-partite entangle-ment and the BHD local oscillators were controlled to resemble Fig 32(b)ie both BHDs measured a squeezed uncertainty regardless of the phase ofthe (generally unknown) disturbance

The beam splitter that combines interferometer output and the entangledreference beam unavoidably splits the signal as well as the disturbance intotwo paths For a balanced beam splitter this generally reduces the signaland disturbance power by 3 dB for both quadrature measurements Fig 33shows however that both BHDs performed about 6 dB below shot noisewhich demonstrates the usefulness of the scheme The squeeze factor can inprinciple be infinite which thus qualifies the lsquo3 dB penaltyrsquo In the abovefigure the additional information from the second BHD output was used torecognize the parasitic interference in the first BHD output providing a lsquovetorsquo

90

50 51 52 53 54 55 56 57

50 51 52 53 54 55 56 57-70

-75

-80

-85

-90

-70

-75

-80

-85

-90

Y (A) (Ω2π)Ω∆Ω

Frequency ( f )

Noi

se p

ower

(dB

m)

Noi

se p

ower

(dB

m)

X (B) (Ω2π)Ω∆Ω

~ ndash6 dB

~ ndash6 dB

Shot noise reference

Shot noise reference

Signal Projected disturbance

Projected disturbance

Vet

o

No signal

Figure 33 Bi-partite-squeezed-light-enhanced measurement ndash The result wasachieved in a table-top setup [Steinlechner et al (2013)] In the two panels the lower

(blue) traces show the squeezed quadrature noise-power spectra ∆2Y(A)Ω∆Ω(Ω2π) (top) and

∆2X(B)Ω∆Ω(Ω2π) (bottom) as simultaneously measured with balanced homodyne detectors

lsquoArsquo and lsquoBrsquo respectively The conventional Y -measurement (top) cannot distinguish be-tween signal and disturbances The additional X-measurement (bottom) does not detectany phase quadrature signal thus any feature in this measurement is a parasitic signal dueto a disturbance The respective projection onto the Y -measurement can thus be lsquovetoedrsquoIn a more sophisticated approach the X-data might be used to model and then to elim-inate the disturbance as well as its projections on both quadrature measurements Theresult is a reduced spectral density of the actual phase quadrature measurement [Ast et al(2016)] Traces shown here are slightly sloped due to the decreasing transfer functions ofthe balanced homodyne detectors The resolution bandwidth was ∆Ω(2π) = 10 kHz thevideo bandwidth was 100 Hz All traces were averaged three times

signal to trigger its removal from the data streamThe question arose whether the additional information can be used to

reduce the actual noise spectral density of the first measurement ie to re-cover signals that were buried by parasitic interferences Very recently it wasshown that this is indeed possible Ref [Ast et al (2016)] reports a table-top proof-of-principle experiment in which the additional information in the

91

QDM approach could be used for improving the sensitivity of an interfero-meter The measurement sensitivity was improved from above-shot-noise tosub-shot-noise (sub-Poissonian) performance This result was possible notbecause the way the parasitic interference arose was known but because theadditional information provided by QDM allowed for fitting a model of theexcess noise to the readout data

Quantum dense metrology (QDM) as shown in Fig 32 improves a mea-surement by simultaneously reading out two conjugate observables Bothreadout observables show a squeezed quantum noise and act as estimatorsof independent physical quantities This situation was recently described aslsquoquantum-mechanics freersquo [Tsang and Caves (2012)] QDM is based on anEinstein-Podolsky-Rosen (EPR) entangled [Einstein et al (1935)] bi-partitesystem as described in Subsec 34 EPR entanglement was previously con-sidered for the quantum-informational task of dense coding which doublesthe capacity of quantum communication channels [Bennett et al (1992)Braunstein and Kimble (2000)] The application of EPR entanglement inmetrology was first proposed by DrsquoAriano et al [DrsquoAriano et al (2001)]

72 Conclusions

A single beam that carries an optimized spectrum of squeezed vacuumstates and that is injected into the interferometerrsquos dark port provides themost efficient and practically optimal approach to reduce the quantum noisein laser interferometers by means of nonclassical states (see Section 5) Theconclusion of the section here is that two entangled beams provide a superiorapproach if the interferometerrsquos sensitivity is limited by classical noise thatis not exclusively restricted to the actual observable which is the phasequadrature amplitude Y Parasitic interferences due to laser light that isbackscattered from vibrating surfaces are an example Current gravitational-wave detectors use light fluxes of about 1024 photons per second [Abbott(2016)] Just a single photon per second and hertz which leaves the mainlight beam and is backscattered from a vibrating surface and in this way getsfrequency shifted into the detection band produces a significant disturbancesignal The lsquoquantum-dense metrologyrsquo approach might provide a powerfultechnique to tackle this problem

Very recently it turned out that QDM is not the only technique that mayexploit EPR entanglement to improve phase measurements Ref [Ma et al(2017)] proposes to use EPR entanglement to simultaneously suppress shotnoise and radiation pressure noise in a gravitational-wave detector without

92

the need for an additional filter cavity (confer subsection 55) In this caseEPR entanglement is exploited that is carried by one broadband squeezedbeam and that is present between quadrature amplitudes defined with re-spect to different optical frequencies ω and ωprime as investigated in Ref [Hageet al (2010)] Such lsquofrequency multiplexedrsquo EPR entanglement might resultin considerably lower costs of building a gravitational-wave detector with abroadband simultaneous squeezing of shot noise and radiation pressure noiseAlso this proposal does not lead to a fundamentally lower quantum noise butrather improves on classical aspects of an interferometer

8 Summary and Outlook

In many cases experiments that involve interference of quantum statescan be described in a semi-classical way This description uses the classicalwave picture for the interference part of the experiment and subsequently theclassical particle picture when the states transfer their energy to a detectoror more generally to a thermal bath This semi-classical description is notpossible when using the specific class of lsquononclassicalrsquo states Squeezed statesof light are a prominent example of these Squeezed states and other nonclas-sical states allow for observations that made Einstein Podolsky and Rosenformulate their critical and seminal paper on quantum theory [Einstein et al(1935)]

In the review here it is argued that after many successful proof-of-principle experiments with nonclassical states in the past decades the routineuse of squeezed-light in observational runs of the gravitational-wave detectorGEO 600 goes beyond proof-of-principle and is a true application of nonclassi-cal light Since 2010 the squeezed-light source has improved the measurementsensitivity of GEO 600 in basically every observational run [Abadie (2011)Grote et al (2013) Dooley et al (2016)] At quantum noise limited frequen-cies ie above a few hundreds of hertz the sensitivity has been improvedcorresponding to a squeezing strength in the noise spectral density of up to37 dB which corresponds to an increase of the average gravitational-wavedetection rate by a factor of 043minus32 = 36 This success is a strong motiva-tion to also equip the Advanced LIGO Virgo and Kagra gravitational-wavedetectors with squeezed light Similar improvement factors even down tolower signal frequencies are expected [LSC (2013)] The achievable improve-ment factors are mainly limited by the optical loss on the squeezed states

93

and much higher factors are achievable in principleUp to now squeezed states have not been used to reduce the radiation-

pressure noise in gravitational-wave detectors The reason is that so far othernoise sources are larger than radiation pressure noise and such an effect can-not be observed It is expected however that future gravitational-wavedetectors will eventually be partly limited by radiation pressure noise Fromthis point on squeezed light will be used to simultaneously reduce shot noiseand radiation pressure noiseSqueezed states are the optimum nonclassical states for gravitational-wavedetectors or more generally for all laser interferometers operating with largeaverage photon numbers per measuring interval [Demkowicz-Dobrzanski et al(2013)] In addition to using higher light powers and heavier test mass mir-rors higher squeeze factors will thus contribute to mitigate the lightrsquos quan-tum noise in laser interferometers

Two-mode (bi-partite) squeezed light has not been used in gravitational-wave detectors so far They are not capable of further reducing the quantumnoise in laser interferometers but they can be used to mitigate classicalnoise that originates from fluctuating phase space displacements A well-known such noise source is back-scattered light Proof-of-principle experi-ments were performed recently [Steinlechner et al (2013) Ast et al (2016)]This new technique could turn out to be valuable in next generations ofgravitational-wave detectors in particular in those targeting high sensitivi-ties at low sub-audio signal frequencies and using high light powers Suchan implementation in gravitational-wave detectors does not require any newtechnology Compared to a squeezed-light enhanced interferometer just asecond squeezed-light source is required

It is certainly remarkable that those quantum states that made EinsteinPodolsky and Rosen falsely think quantum theory incomplete are now ex-ploited as new technologies in measurement devices targeting new observa-tions in nature

Acknowledgements

RS thanks M Ast J Bauchrowitz C Baune S Chelkowski J DiGugliel-mo A Franzen B Hage J Harms A Khalaidovski L Kleybolte NLastzka M Mehmet S Steinlechner and H Vahlbruch for their contri-butions many fruitful discussions and their support with the figures and JFiurasek for many valuable comments on the manuscript Thanks are also

94

due to Y Chen F Khalili and H Miao for fruitful discussions within thequantum noise working group of the LIGO Scientific Collaboration (LSC)Special thanks are due to H Vahlbruch and H Grote together with theGEO 600 team for their pioneering work on the squeezed-light implementa-tion in GEO 600 RS is supported by the Deutsche Forschungsgemeinschaft(Grant No SCHN 757-6) and by the European Research Council (ERC)project lsquoMassQrsquo (Grant No 339897)

References

References

Aasi J et al Jul 2013 Enhanced sensitivity of the LIGO gravitationalwave detector by using squeezed states of light Nature Photonics 7 (8)613ndash619URL httpwwwnaturecomdoifinder101038nphoton2013177

Aasi J et al Apr 2015 Advanced LIGO Classical and Quantum Gravity32 (7) 074001URL httparxivorgabs14114547httpstacks

ioporg0264-938132i=7a=074001key=crossref

20895763c84bce3f8929251031b2475c

Abadie J et al Sep 2011 A gravitational wave observatory operatingbeyond the quantum shot-noise limit Nature Physics 7 (12) 962ndash965URL httparxivorgabs11092295httpwwwnaturecom

doifinder101038nphys2083

Abbott B P et al Feb 2016 Observation of gravitational waves from abinary black hole merger Phys Rev Lett 116 061102URL httplinkapsorgdoi101103PhysRevLett116061102

Acernese F et al 2015 Advanced virgo a second-generation interferome-tric gravitational wave detector Classical and Quantum Gravity 32 (2)024001URL httpstacksioporg0264-938132i=2a=024001

95

1 Introduction

Laser interferometers are used to monitor small changes in refractiveindices rotations or surface displacements such as mechanical vibrationsThey transfer a differential phase change between two light beams into achanging power of the output light which is photo-electrically detected forexample by a photo diode The light is produced in a lasing process thatusually aims for a coherent (Glauber) state In practice laser light is oftenin a mixture of coherent states producing excess noise in the interferometricmeasurement But even if the laser light is in a (pure) coherent state its de-tection is associated with noise usually called lsquoshot-noisersquo This arises fromthe quantisation of the electro-magnetic field which for a coherent stateresults in Poissonian counting statistics of mutually independent photons

1000099009800 10100 10200

Photon number n

0000

0002

0004

0006

0008

0010

0012

Pro

babi

lity Squeezing

Poisson distribution

Figure 1 Poissonian and squeezed photon statistics ndash The upper boundary of eacharea represents the probability distribution of detected photon number n when perform-ing a large number of measurements on an ensemble of identical states having an averagephoton number of n = |α|2 = 10000 where α is the coherent field excitation or lsquodis-placementrsquo The broader curve shows the lsquoPoissonianrsquo distribution which describes thecounting statistic of mutually independent particles ie those of the coherent state Dueto the large value of α the distribution is almost Gaussian with a standard deviation ofplusmnradicn The narrow curve corresponds to the equally displaced 10 dB squeezed state which

obviously has a lsquosub-Poissonianrsquo photon statistic Note that squeezed states with smallor even without any coherent excitation (squeezed vacuum states) exhibit quite differentphoton statistics ndash see Fig 13 for example

4

If the coherent state is highly excited and thus the average number ofphotons n per detection interval is large the Poissonian distribution canbe approximated by a Gaussian distribution with a standard deviation ofplusmnradicn During the past decades squeezed states of light have attracted a lot

of attention because they can exhibit less quantum noise than a coherent stateof the same coherent excitation ie they can show sub-Poissonian countingstatistic see Fig 1

θ

Shot noise Squeezed noise

(a) (b)

Brightlaser input

Squeezedvacuum input

Faradayrotator

Photo diode

Michelsoninterferometer

Signal5050

(i)Ph

oto

curr

ent [

rel

units

]

Time [ms]

(ii)

0 5 10

Figure 2 Squeezed-light enhanced Michelson interferometer ndash (a) In addition tothe conventional operation of a Michelson laser interferometer with bright coherent lighta broadband squeezed-vacuum field is injected into the signal output port and overlappedwith the bright interferometer mode The interferometer is operated close to a dark fringesuch that most of the bright coherent light as well as most of the squeezed vacuum areback-reflected from the Michelson interferometer respectively Due to interference withthe broadband squeezed vacuum the interferometerrsquos output light on the photo diodeshows reduced variance in the photon number statistic as shown in Fig 1 Overlappingthe two light fields is possible with theoretically zero loss by the combination of a Faradayrotator and a polarizing beam splitter (PBS) A signal is produced by modulating therelative arm length (b) Simulated data for photo diode measurements Without squeezing(i) the signal of the laser interferometer is not visible With squeezing (ii) the shot noiseis reduced and here a sinusoidal signal visible

Squeezed states belong to the class of lsquonon-classicalrsquo states which areconsidered to be at the heart of quantum mechanics These states are de-fined as those that cannot be described as a mixture of coherent states Inthis case their Glauber-Sudarshan P -functions [Sudarshan (1963) Glauber(1963)] do not correspond to (classical) probability density functions ie theyare not positive-valued functions As a lsquoclassicalrsquo example the P -function ofa coherent state corresponds to a δ-function

5

But the question remains what property of coherent states justifies thename lsquoclassicalrsquo even though coherent states are quantum states and showquantum uncertainties My answer to this question is the following Allexperiments which only involve coherent states and mixtures of them allowfor a description that uses a combination of classical pictures As we will seebelow this description swaps between two different classical pictures and isthus not truly classical but semi-classical (A more precise description of thenature of coherent states uses the term lsquosemi-classicalrsquo)

Let us consider a laser interferometer that uses light in a coherent stateFirstly the light beam is split in two halves by a beam splitter The twobeams travel along different paths and are subsequently overlapped on abeam splitter where they interfere exactly as classical waves would do Theelectric fields superimpose thereby producing the phenomenon of interfer-ence Up to this point there is no reason to argue light might be composedof particlesSecondly the new (still coherent) beams that result from the interference areabsorbed for instance by a photo-electric detector In the case of coherentstates the detection process can be perfectly described in the classical parti-cle picture in which the particles appear independently from each other in atruly random fashion yielding the aforementioned Poisson statistic Duringthe detection process no wave feature of the light is present Let us havea closer look A truly random (lsquospontaneousrsquo) event is an event that hasnot been triggered by anything in the past This allows us to make a clearcut between the first part of the experiment described by the classical wavepicture and the second part of the experiment described by the classicalparticle picture Both lsquoworldsrsquo are disconnected The subsequent applicationof two classical pictures is not truly classical but lsquosemi-classicalrsquo It is in-deed the observation that the photons occur individually with truly randomstatistics that allows this semi-classical description In the case of a mixtureof coherent states the photon statistics are super-Poissonian which can beunderstood as a mixture of different Poissonian distributions In the caseof a slowly changing coherent state the mean value n depends on time Inall these cases the semi-classical description is appropriate Let me pointout that in this very reasonable description photons do not exist before theyare detected eg absorbed Further note that the famous double-slit exper-iment with coherent states also allows for the same semi-classical description

For squeezed states [Yuen (1976) Walls (1983)] the situation is different

6

As before the interference can be fully described by the classical wave pic-ture The result of the detection process however is different from that ofmutually independent random events It is also different from any super-Poissonian statistics that could be produced by mixing an arbitrary num-ber of different andor time-dependent Poissonian distributions Insteadthe squeezed probability distribution in Fig 1 suggests that the probabil-ity of detecting a photon decreases with the more photons that are alreadydetected in the same time interval over which a single measurement is in-tegrated From this observation one must conclude that the photons donot individually appear in a random fashion upon detection There mustbe lsquoquantumrsquo correlations between the photons These correlations mustexisted before detection since there is no interaction between the photonsduring their detection Pre-existing correlations between detected photonsseem to imply that the photons themselves existed before detection ie attimes when interference occurred In a semi-classical description howeverphotons are classical particles and cannot interfere for instance on a beamsplitter At this point the semi-classical picture breaks down Squeezedstates are therefor lsquononclassicalrsquoThe failure of the semi-classical model described above generally certifiesnonclassicality

Squeezed states are usually not characterized by counting their pho-tons but by measuring canonical continuous-variable phase-space observ-ables Measurements are performed as usual on an ensemble of identicalstates and quasi-probability density functions are calculated from the dataThe Glauber-Sudarshan P -function is the quasi-probability density distribu-tion over coherent states If the P -function of a state is entirely positivethe state is a coherent state or a (classical) mixture of coherent states Thestate is considered as semi-classical If the P -function is not a positive-valuedfunction the state cannot be expressed as a (classical) mixture of coherentstates and is thus nonclassical [Gerry and Knight (2005) Vogel and Welsch(2006)] A non-positive-valued P -function is the sufficient and necessary con-dition for the failure of the semi-classical model The Wigner function is thequasi-probability phase-space representation over the canonical continuous-variable phase-space observables themselves [Gerry and Knight (2005)] TheWigner functions of squeezed states are entirely positive Although subject todiscussion this fact does not mean that squeezed states are less nonclassicalthan Fock states or cat states which not only have a nonclassical P -function

7

but also a partially negative Wigner function (A cat state is a quantum su-perposition of two macroscopically distinct states [Monroe (2002)] referringto Schrodingerrsquos-cat gedanken experiment [Schrodinger (1935)]) In practicesqueezed states can even be regarded as superior nonclassical states becausethey represent the only nonclassical state that has been produced in a steadystate fashionIn almost all experiments so far the generation of Fock states and cat statesinvolves a probabilistic event such as the detection of a photon in anotherbeam path to herald these states In fact squeezed states provide the non-classical resource for the probabilistic preparation of Fock states as well as catstates But only the squeezed states themselves show a nonclassical effect in astationary way Limited only by the time duration and the frequency span ofthe mode that is in a squeezed state the squeezing effect can be continuouslyobserved independently of the time when the measurement is performed andalso independently of the measurement integration time This fact is of greatimportance for applications of squeezed states in measurement devices sincea squeezed-light-enhanced measurement remains unconditional and the ef-fective measurement time is not reduced

In past decades squeezed states of light were used in many proof-of-principle experiments to research their potential for improving the sensitivityof laser interferometers [Grangier et al (1987) Xiao et al (1987) McKen-zie et al (2002) Vahlbruch et al (2005) Goda et al (2008) Taylor et al(2013)] or the performance of imaging beyond the shot-noise limit [Lugiatoet al (2002) Treps et al (2003)] both accompanied by a huge number oftheoretical works Potential applications in secure optical communication(quantum key distribution) were also proposed and proof-of-principle ex-periments demonstrated [Ralph (1999) Furrer et al (2012) Gehring et al(2015)] This review restricts itself to the improvement of laser interfero-meters since only here has the application of squeezed light gone beyondproof-of-principle The gravitational-wave detector (GWD) GEO 600 hasoperated with squeezed light now for more than seven years starting in 2010[Abadie (2011) Grote et al (2013)] GEO 600 is a 600 m long Michelsonlaser interferometer built for the detection of gravitational waves Thesewaves are audio-band and sub-audio-band changes of space-time curvatureoriginating from cosmic events such as the merger of neutron stars or blackholes as detected recently [Abbott (2016)] In GWDs such as GEO 600 [Doo-ley et al (2016)] Advanced LIGO [Aasi (2015)] Advanced Virgo [Acernese

8

(2015)] and KAGRA [Aso et al (2013)] conventional laser technology hasbeen pushed to extremes over the past decades Noise spectral densities nor-malized to space-time strain of less than 10minus23 Hzminus12 have been measured[Abbott (2016)] Progress will continue and based on the successful appli-cation in GEO 600 squeezed light is now widely accepted to provide a newadditional technology to contribute to the new field of gravitational-waveastronomy It was also successfully tested in one of the LIGO detectors in2013 [LSC (2013)] and is an integral part of the European design study forthe 10 km Einstein-Telescope [Punturo et al (2010)]

GEO 600 has already taken several years of lsquosqueezedrsquo observational datawhich has increased its sensitivity at signal frequencies above 500 Hz Withthe implementation of a squeezed light source in GEO 600 the applicationof nonclassical states in metrology has been pushed beyond merely proof-of-principle

lsquoTwo-mode squeezed statesrsquo show a squeezed uncertainty in at least onejoint continuous variable of two subsystems lsquoArsquo and lsquoBrsquo Examples of jointvariables are differences and sums of phase-space observables of A and BTwo-mode squeezed states not only belong to the class of nonclassical statesbut due to their bi-partite character also to the class of lsquoinseparablersquo orlsquoentangledrsquo states They are the ideal states to demonstrate the Einstein-Podolsky-Rosen paradox [Einstein et al (1935)] as first achieved in [Ouet al (1992)] Apart from fundamental research on quantum mechanicsrecent proof-of-principle experiments demonstrated their usefulness in inter-ferometric measurements that go beyond the application of simple squeezedstates [Steinlechner et al (2013) Ast et al (2016)] This experiment is thefinal topic of this review

2 Observations on light fields in squeezed states

Generally there are two different kinds of observables that can be subjectof a measurement performed on a quantum system The first kind is associ-ated with the systemrsquos wave property In optics it corresponds to the electricfield strength at a given phase angle ϑ The according (dimensionless) opera-tors are called the quadrature amplitudes Xϑ and have a continuous spectrumof eigenvalues Quadrature amplitudes are measured in very good approx-imation with a balanced homodyne detector using the interference with a

9

bright local oscillator beam see Fig 3 (a) In practice any measurement ofXϑ integrates over some sideband (Fourier) spectrum within the angular fre-quencies Ω plusmn ∆Ω2 The sideband information always needs to be quotedA straight forward but rather untypical way is by adding subscripts whichleads to Xϑ

Ω∆Ω The classical analogue of the quadrature amplitude operator

XϑΩ∆Ω is the modulation depth of the optical field at modulation phase angle

ϑ and at angular modulation frequency Ω measured over the band ∆Ω lt 2ΩThe uncertainties of the statersquos quadrature amplitudes at different phases ϑare limited by a Heisenberg uncertainty relation see section 3 The secondkind of measurement is associated with the systemrsquos particle property andis given by the photon number operator n associated with a measuring timeinterval ∆t Its precise measurement requires a photon counter ideally withsingle photon resolution The measurement result obviously has a discretespectrum Continuous as well as discrete observables are usually subject toquantum uncertainties and thus quantum noiseUsually the measurementrsquos integration time and frequency band actuallydefine the physical system that is characterized In quantum optics experi-ments the interrogated physical system is called a lsquomodersquo

21 Definition of a lsquosingle modersquo

Let us define a light field or generally any quantum system to be asingle mode if it corresponds to the lsquosmallest entity of a waversquo In this caseits spectral and temporal distributions as well as waist size and divergenceare at their Fourier limits and all other properties such as optical axis waistposition and polarization are well defined For instance a linearly polarizedlongitudinal resonance of an optical standing-wave cavity defines such a singlemode if the cavity finesse is high and transversal modes are non-degenerateThe complete photo-electrical detection of a cavity mode however is notstraight forward Most quantum optical experiments are instead performedon propagating light In this case single modes are defined by spatial filtersand by temporal-spectral measurement windows both being at the Fourierlimit Examples for single modes are a laser pulse and a spectraltemporalcutout from a continuous observation of a quasi-monochromatic continuous-wave light beam in the spatial TEM00 mode both at the Fourier limits

In classical physics the only remaining free parameter of a given singlemode is its excitation energy In quantum physics the situation is differentFor a given energy a single mode can be in many different quantum states

10

which differ in their quantum statistics Examples are coherent states num-ber (Fock) states and squeezed states

22 Observations on squeezed states using a single PIN photo-diode

An ideal PIN photo-diode absorbs the full energy of a light mode andproduces one photo electron for every absorbed photon energy It uses theinternal photo-electric effect inside a semiconductor such as silicon or In-GaAs In contrast to avalanche photo-diodes PIN photo-diodes operatewith unity gain lsquoPINrsquo stands for lsquopositiversquo lsquointrinsicrsquo and lsquonegativersquo and isdescribing the doping of the semiconductor layers A PIN photo-diode is op-timally suited for the continuous monitoring of a rather bright light field ofup to several tens of milliwatts An example is the photo-diode in the outputport of a gravitational-wave detector as shown in Fig 2 (a) The prominentwavelength of 1064 nm which is emitted by NdYAG lasers has an opticalfrequency of ν = ω(2π) = 282 middot 1014 Hz The period of the field oscillationis a few femtoseconds and cannot be directly resolved with photo-electric de-tectors However variations of the electric field around the averaged opticalfield oscillation on longer time-scales can be resolved Applying an electronicbandpass filter at the sideband angular frequency Ω plusmn ∆Ω2 to the photovoltage provides information about the lsquodepth of the lightrsquos amplitude mod-ulationrsquo which is also called the lsquoamplitude of the amplitude quadraturersquo Itcan also slowly vary in time and reads

Xϑ=0

Ω∆Ω(t) equiv XΩ∆Ω(t) equiv X (1)

The subscript is usually skipped as it is done with the time dependenceas indicated on the right Applying the electronic bandpass filter in factdefines the mode of the light being detected The structure of the defini-tion in Eq (1) forms the basis of interferometric signals and quantum noisealso in the semi-classical case of coherent states Lets take an example Inthe recent observation of gravitational waves [Fig 1 bottom row in Abbott(2016)] the time-frequency representation of the gravitational-wave signalcorresponded to the amplitude quadrature amplitude XΩ∆Ω(t) of the inter-ferometer output light Note that a larger value of ∆Ω allows for changes ofthe quadrature amplitude on shorter time scales

If the light fieldrsquos lsquomodulation modersquo does not contain any quanta simplybecause there are no photons that have a frequency difference of plusmnΩ with

11

respect to the carrier it is in its ground state In this case lsquovacuum noisersquois observed which originates from the ground state uncertainty Since thevacuum noise only becomes measurable as a beat with a bright light field itcan also be seen as the carrierrsquos band-path filtered shot noise A modulationmode in a displaced vacuum state (a coherent state) corresponds to nonzerocoherent modulationThe measured level of the vacuum noise generally depends on the power ofthe bright carrier light and on the electronic amplification In any case itprovides the reference for certifying lsquosqueezingrsquo Observations using a singlePIN photo-diode require an independent measurement to quantify vacuumnoise A necessary condition is that attenuating the total fieldrsquos light powerresults in the same attenuation of the measured XΩ∆Ω values If they showa stronger attenuation a coherent modulation or thermal noise might bepresent If they show a weaker attenuation the photo-diode and its electron-ics might be saturated

Fig 2 (b) illustrates how a broadband squeezed field improves the mea-surement of an amplitude modulation in time domain based on a PIN photo-diode Shown is a simulated time sequence of XΩ∆Ω-data sampled from thephotoelectric voltage In this simulation all sideband frequencies from zero(DC) to the cutoff frequency of the detector electronics (Ωcut) are included(Ω = ∆Ω2 = Ωcut2) No additional band pass filter is applied making it amaximally broadband detection Although the data in Fig 2 (bi) contains aclassical amplitude modulation of the detected light this signal is not visibledue to random noise here representing shot noise Fig 2 (bii) shows thesame situation but with shot noise that is squeezed over the full detectionband The quantum uncertainty of the modulation depth is squeezed andthe classical signal becomes visible

It needs to be noted that a single PIN photo-diode can only measure theamplitude of the amplitude quadrature XΩ∆Ω(t) but not the non-commutingobservable the lsquoamplitude of the phase quadraturersquo

Xϑ=90

Ω∆Ω (t) equiv YΩ∆Ω(t) equiv Y (2)

For values that are small compared to the field strength of the bright fieldthe quantity Y approximately describes the bright fieldrsquos lsquophase modulationdepthrsquo

12

23 Observations on squeezed states using a balanced homodyne detector

-72

-69

-66

-63(i)

(ii)

(iii)

(a)

LO

(Squeezed)signal input

Phaseshifter

PD1

PD2

5050

01 0200 03

-75

-60

-57Balanced homodyne detector

(b)

Time [s]

Noi

se p

ower

[dB

m]

Figure 3 Balanced homodyne detection (BHD) ndash (a) Setup The quadrature atchoosable angle ϑ of the signal field is measured by overlapping the latter with a localoscillator (LO) field of the same mode parameters on a balanced beam splitter and record-ing the difference voltage from two PIN photo-diodes as shown In order to meet theBHD approximation the LO needs to be much more intense than the signal field A closeto perfect mode overlap between LO and signal input field is crucial For a non-perfectoverlap the detector measures the input state with unwanted contributions of the vacuumstate (b) Noise power measurements (i) on an electronically amplified and band-passfiltered quadrature amplitude of the vacuum field (Xvac

Ω∆Ω) (signal input blocked) (ii) on

a squeezed quadrature (XsqzΩ∆Ω) of a squeezed vacuum state (ϑ = 0) and (iii) on re-

spective quadratures of the same squeezed state where the phase angle ϑ was continuouslyshifted by changing the optical path length of the LO The measurement data shows about5 dB of squeezing and was first published in [Chelkowski et al (2007)] Ω2π = 5 MHz∆Ω2π = 100 kHz

In contrast to a single PIN photo diode a balanced homodyne detector(BHD) is suitable to measure the quantum statistic of all types of modu-lations ie for all angles ϑ Such a detector consists of two identical PINphoto-diodes a balanced beam splitter and an external homodyne local os-cillator field that is much brighter than the signal beam and that has anadjustable phase The signal beam corresponds to the squeezed field whichin many experiments is in a squeezed vacuum field having an optical powerthat usually corresponds to just a few photons per mode The two beams areoverlapped on the balanced beam splitter with close to perfect mode match-ing and the two interference outputs are focussed onto the photo diodes seeFig 3 (left) The electric output signal of the BHD is the difference of the

13

photo diode voltages The LO takes over the role of the carrier light fieldbut with the possibility to choose the phase shift ϑ This way eigenvaluesof X Y or Xϑ can be measured where the latter is given by the followinglinear combination of the first two

Xϑ(t) = cos(ϑ) X(t) + sin(ϑ) Y(t) (3)

If the modulation depths of signal and local oscillator beams are weak com-pared to their coherent amplitudes |α| and |αLO| the output voltage of aBHD corresponds to eigenvalues of the following operator

V (t) prop 2cos(ϑ) |αLO||α|+ |αLO| Xϑ(t) + |α| XϑLO(t) (4)

The lsquohomodyne approximationrsquo further involves |αLO| |α| such that theterm on the right can be neglected even if the local oscillator shows someclassical quadrature excitation The output voltage of a BHD is usually spec-trally analysed or at least spectrally filtered which removes the DC part infull analogy to a single photo diode (see previous subsection) Sampling thefiltered voltage provides eigenvalues proportional to the generalized quadra-ture amplitude in Eq (3)

V BHDΩ∆Ω(t) prop |αLO| Xϑ

Ω∆Ω(t) (5)

Fig 3 (a) shows the setup of a balanced homodyne detector for the char-acterization of squeezed states Setting ϑ = 0 eigenvalues of the ampli-tude modulation depths can be sampled from the photo voltage accordingto Eq (5) Setting ϑ = 90 eigenvalues of the phase modulation depths aremeasured The datarsquos expectation values 〈Xϑ〉 provide the coherent displace-ment of the squeezed state The datarsquos variances

∆2Xϑ equiv 〈(Xϑ)2〉 minus 〈Xϑ〉2 (6)

provide the statersquos (quantum) noise A pure squeezed state as well as asqueezed state that experienced photon loss have Gaussian quantum statisticsand are thus fully described by the expectation values and variances (first andsecond moments) of two orthogonal quadratures but only if one quadraturereflects the lowest quadrature variance

14

In most experiments with squeezed light the photo electric voltage ac-cording to Eq (5) is not sampled with a data aquisition system but the signalis directly fed into a spectrum analyser measuring the noise power of the volt-age If the expectation value 〈Xϑ〉 is zero the noise power is proportionalto the variance ∆2Xϑ in Eq (6) The reference for quantifying the squeezefactor is measured by blocking the (squeezed) signal field in Fig 3 (a) Themeasured vacuum noise level corresponds to the LOrsquos (electronically ampli-fied) shot noise level

Traces (ii) and (iii) in Fig 3 (b) show measured noise powers of the mod-ulation mode (Ω2π = 5 MHz ∆Ω2π = 100 kHz) being in a squeezed vac-uum state (i) is proportional to the variance of the ground state uncertainty∆2Xvac

Ω∆Ω (ii) is proportional to the quantum noise variance of the squeezed

quadrature amplitude ∆2XsqzΩ∆Ω (iii) is proportional to the quantum noise

variance of the quadrature amplitude with scanned phase ∆2XΩ∆Ω(ϑ(t))

To fully characterize a quantum state ie to do quantum state tomogra-phy [Vogel and Risken (1989)] a BHD is a prerequisite But also interfero-metric measurements with balanced homodyne detectors instead of singlePIN photo-diodes have several advantages A correctly implemented BHDreadily provides the vacuum noise level when the signal beam is blockedWith a BHD the optimum operating point of the interferometer is preciselyat a dark fringe If a perfect dark fringe can practically be achieved ampli-tude noise of the laser does not couple into the signal port If the interfero-meter has balanced arm length also frequency noise of the laser then does notcouple into the signal port Some quantum non-demolition schemes with theprospect of evading quantum radiation pressure noise require the detection ofa non-canonical quadrature angle [Jaekel and Reynaud (1990) Kimble et al(2001)] Here the adjustable phase of a BHD provides a straight forwardapproach The experimental exploration of BHDs for gravitational-wave de-tectors only has started recently [Steinlechner et al (2015)]

A light field can be analysed with respect to many different modulationfrequencies Ω The result constitutes a spectrum [Breitenbach et al (1998)]where in principle every modulation mode can be in a different quantumstate Fig 4 shows spectra of squeezed states from 5 MHz to 100 MHz with∆Ω2π = 1 MHz The lower curve shows the spectrum of the most strongly

15

-14-12-10-8-6- 4-20 2 4 6 8

10 12 14 16 18

6 8 10 20 40 60 80 1005

Squeezed noise

Anti-squeezed noise

Vacuum noise

Frequency [MHz]

Nor

mal

ized

noi

se v

aria

nce

[dB

]

Figure 4 Spectrum of quadrature amplitude variances ndash Shown are the quantumnoise properties of a large number of modulation modes having a resolution bandwidth of∆Ω(2π) = 1 MHz For all traces the balanced homodyne detector output was analysedwith a spectrum analyser Squeezing of XΩ∆Ω (bottom trace) and anti-squeezing of

YΩ∆Ω (top trace) versus f = Ω(2π) are shown relative to the vacuum noise variance Thespectrum below 5 MHz is not shown since it contained less squeezing due to laser relaxationoscillation in the carrier field as well as disturbances from back-scattered light [Vahlbruchet al (2007)] Disturbances at frequencies above 70 MHz originated from relatively largedetector dark noise which was subtracted from all traces shown here The thin linerepresents a theoretical model that takes into account for the linewidth of the squeezingcavity The data was first presented in in Ref [Mehmet et al (2010)]

squeezed variances in this case the variances of XΩ∆Ω The upper spectrum

shows the variance in the orthogonal quadrature amplitude (YΩ∆Ω) Allvariances are normalized to those of the corresponding vacuum state Thesqueeze factor reduces towards higher frequencies due to the linewidth ofthe squeezing cavity The anti-squeezing is always higher than the absolutevalue of the squeezing due to Heisenbergrsquos uncertainty relation and due to thepresence of optical loss The curves do not represent pure squeezed states butmixed squeezed states with a significant contribution from vacuum statesdue to optical loss Pure squeezed states can only be produced by making

16

the influence of all decoherence processes negligibleThe choice of the resolution bandwidth (RBW ∆Ω) during data taking

and processing defines the spectral-temporal modulation modes includingtheir number within the detected spectrum For any setting of the RBWthe quantum mechanical properties of the quadrature amplitudes XΩ∆Ω and

YΩ∆Ω [Caves (1985)] fully correspond to those introduced for quadratures instandard text books and which are reviewed in Sec 3

24 Observations on two-mode squeezed states using balanced homodyne de-tectors

Two-mode squeezed states are composed of two subsystems lsquoArsquo and lsquoBrsquoand are bi-partite entangled states with a Gaussian quantum statistic Toavoid conflicts with different usage of the term lsquomodersquo they can synony-mously be named lsquobipartite Gaussian entangled statesrsquo or lsquobipartite squeezedstatesrsquo which will be mainly used in this Review In the same way multi-partite Gaussian entangled states correspond to multi-partite squeezed states

The measurement observables that prove or disprove the bi-partite squeez-ing property are XA

Ω∆Ω minus XBΩ∆Ω and Y A

Ω∆Ω + Y BΩ∆Ω where the minus and

plus signs may be swapped Bi-partite squeezed states are precisely thosestates that were discussed by Einstein Podolsky and Rosen (EPR) in theirseminal paper [Einstein et al (1935)] Fig 5 shows a measurement resulton bi-partite squeezed light [Eberle et al (2013)] The variances of bothjoined observables are squeezed as shown in the two lower traces They wererecorded consecutively by adding or subtracting the outputs of two balancedhomodyne detectors But by interfering the subsystems on a beam splitterone could even measure both joined observables simultaneously This pos-sibility is correctly described in quantum theory since their commutator iszero

The so-called EPR paradox arises as follows If we either measure XAΩ∆Ω

and XBΩ∆Ω or Y A

Ω∆Ω and Y BΩ∆Ω it is obvious from the data in Fig 5 that

we can always predict the measurement result at subsystem lsquoBrsquo when know-ing the result at subsystem lsquoArsquo This seems to suggest that both quantitiesat lsquoBrsquo are precisely defined simultaneously before the measurement on lsquoArsquowhich contradicts the rigorous (and correct) interpretation of their non-zerocommutator that they are not precisely defined simultaneously

To solve this paradox EPR conjectured that the wavefunction as definedby quantum theory does not provide the full information This led to a

17

discussion of whether hidden variables existed that needed to be included ina complete theory of quantum mechanics (see also Bell [Bell (1966)]) Theexperimentally observed violation of Bellrsquos inequality [Bell (1964) Aspectet al (1981) Giustina et al (2013) Hensen et al (2015)] however ruled outthe existence of (local) hidden variables

Based on that the EPR paradox needs to be solved in a different wayContrary to what EPR assumed it is in fact possible to predict the valueof an arbitrary observable of a physical system A with certainty via a mea-surement on system B although this observable was not defined before themeasurement Without any interaction a measurement on subsystem lsquoArsquo notonly creates lsquorealityrsquo of eg XA

Ω∆Ω simultaneously lsquorealityrsquo is also created

regarding the observable XBΩ∆Ω describing subsystem lsquoBrsquo Here the term

lsquorealityrsquo has the meaning as defined by EPR [Einstein et al (1935)] Simi-larly the detection of one photon of a two photon entangled number statenot only produces the reality of this photon but also that of a second oneA discussion of Einstein-Podolsky-Rosen entanglement can also be found in[Schnabel (2015)] Note that the EPR paradox can also be described aslsquoquantum steeringrsquo [Schrodinger (1935) Cavalcanti et al (2009) Handchenet al (2012)] It should also be mentioned that two-mode squeezing beingdetected with BHDs and not with photon counters cannot be used to violatea Bell inequality The latter topic is outside the scope of this Review

Bi-partite squeezed states were first characterized with balanced homo-dyne detectors by the group of J Kimble in 1992 [Ou et al (1992)] Gener-ally the EPR paradox becomes more pronounced the stronger the bi-partitesqueezing is A measure of the strength of EPR entanglement was introducedby M Reid [Reid and Walls (1985)] According to this measure the resultin Fig 5 can be quantified to ε2 = 00309 where the critical value is one Itcorresponds to the strongest Gaussian EPR entangled state generated so far

For a long time it looked like that two-mode squeezed states are notuseful for laser interferometers The reason for that belief was that a laserinterferometer as any other measurement device too is built to measureone observable It seems to be ideal already if the quantum noise in thissingle observable is squeezed The increased quantum noise in the orthog-onal observable is not harmful in this case and squeezing in two differentobservables useless Only recently realistic scenarios were discussed in whichtwo-mode squeezing in fact does improve the performance of a laser inter-

18

-12

-10

-8

-6

-4

-2

0

0 1 2 3 4 5 6 7 8 9 10

100

dB

109

dB

Δ (XAvac + X B

vac ) = Δ (Y

Δ (XA + X B)

Avacminus Y B

vac )2 2

2

Δ (YAminus Y B)2

Nor

mal

ized

noi

se v

aria

nce

[dB

]

Times [s]

Figure 5 Two-mode squeezing measurement ndash For this measurement the outputs oftwo balanced homodyne detectors are added or subtracted and the variances (noise pow-ers) of the results recorded The upper trace was measured with modes lsquoArsquo and lsquoBrsquo beingin their ground states This measurement served as a reference level Strong two-modesqueezing was observed as shown by the lower two traces The sideband frequency wasΩ(2π) = 8 MHz and the resolution bandwidth was ∆Ω(2π) = 200 kHz The measure-ment results were first published in Ref [Eberle et al (2013)]

ferometer [Steinlechner et al (2013)] The proof-of-principle experiment isreviewed in Sec 7

25 Observations using photon counters

Alternatively to field quadratures an optical mode in a squeezed statecan also be characterized at least partly by detecting its photon numberdistribution For a pure squeezed vacuum state such a measurement wouldreveal the existence of solely even photon numbers including a large prob-ability for zero photons The average photon numbers of squeezed vacuumstates with feasible squeeze factors are very small of the order of one persecond and bandwidth in hertz see Fig 13 (a) ndash (c) A distribution with closeto zero probability of odd photon numbers however has not been measuredso far The reason is the lack of ideal photon counters First of all theefficiency of these detectors ie their probability of converting one photoninto one click and no photon into no click must be almost perfect lsquoLostrsquophotons as well as dark counts wash out the oddeven oscillations Further-more most detectors available can only distinguish between zero and one

19

photon This problem can be solved by distributing the squeezed mode ontoa large number of single photon detectors using an array of beam splitterssuch that all paths have a low probability of carrying more than one pho-ton Photon number measurements on squeezed vacuum states neverthelessplay an extremely important role in quantum optics When the squeezingstrength is very low the probability of detecting more than 2 photons can beneglected and the detection of a photon heralds the existence of a second one

0

300

600

900

1200

-40 -20 0 20 400

5

10

15

20

25

Two-

fold

coi

ncid

ence

s [1

(4s

)](a) (b)

Non-degeneratetwo-mode squeezing

5050

APDA1

APDA2

APDHerald

Photon coincidencedetection

Delay [ns]

Thr

ee-f

old

coin

cide

nces

[1

(4s)

]1500

Figure 6 Coincidence clicks from non-degenerate photon pairs ndash The first suchexperiment was reported in Ref [Hong and Mandel (1986)] (a) shows a setup with threeavalanche photo-diodes (APDs) for proving the successful heralding of a single photonnumber state (b) Histograms of the two-fold coincidence detections at APDHerald andAPDA1 (red) and at APDHerald and APDA2 (yellow) with theoretical models (solid lines)If the two-mode squeezing just carried one photon in each spatial subsystem the three-foldcoincidence detection should be zero Indeed the according histogram (grey points righty axis) shows only a few events These are produced by false (dark) counts of the APDsThe delay for the three-fold coincidences is defined as the time between counts at lsquoA1rsquo andlsquoA2rsquo given that the trigger APDHerald detected a photon (within a 100 ns time window)The data was taken on photons that were up-converted from 1550 nm to 532 nm and itwas first published in Ref [Baune et al (2014)]

If a mode of light is always excited by either zero or two photons lsquocondi-tionalrsquo or lsquoheraldedrsquo one-photon Fock states can be realized (Measurementson an ensemble of the n-photon Fock state would always produce the mea-surement result n ie Fock states have a zero photon number uncertaintyThey are also called lsquonumber statesrsquo) The above concept of producing aone-photon Fock state obviously requires the deterministic and balanced dis-tribution of the down-converted signal and idler fields into two different paths

20

In order to achieve this the signal and idler fields need to be non-degenerateUsually a mode in a squeezed state is composed of degenerate signal andidler fields and this degeneracy thus needs to be removed Possible waysare producing the down-converted fields at well separated wavelengths [Vil-lar et al (2005) Su et al (2006) Li et al (2010) Samblowski et al (2011)]separating the upper and lower sidebands belonging to an ordinary squeezedmode by frequency filters [Schori et al (2002) Hage et al (2010)] and us-ing spatial filters [Hong et al (1987)] A frequently used approach is usingtype II parametric down-conversion where the photons within a pair are al-ways orthogonally polarized [Ou et al (1992) Kiess et al (1993) Kwiat et al(1995)]The list of experiments with conditional or heralded photon number states islong They showed for instance nonclassical g(2)-functions [Hong et al (1987)]and violations of Bell inequalities [Weihs et al (1998)] Fig (6) shows a re-sult from a more recent experiment in which a bipartite-squeezed state withsubsystems at 1550 nm and 810 nm was produced the subsystem at 1550 nmsubsequently up-converted to 532 nm and the lsquoquantum non-Gaussianityrsquo ofheralded up-converted single photons demonstrated [Baune et al (2014)]Squeezed states are also the resource for the conditional generation of super-positions of coherent states [Ourjoumtsev et al (2006) Neergaard-Nielsenet al (2006)] and so-called N00N-states [Afek et al (2010)]

The generation of nonclassical states mentioned in the paragraph aboveis not stationary but relies on a probabilistic trigger event The produc-tion of squeezed states themselves usually happens in a stationary fashionThis distinction has an important consequence for applications of nonclassi-cal states in measurement devices Only (stationary) squeezed states allowfor a continuous improvement of a measurement Avoiding any loss of mea-suring time is generally of high relevance for the detection of short-livedsignals with unknown arrival time as well as for the detection of long-livedquasi-monochromatic signals since the signal-to-noise-ratio (SN) improveswith measuring time

26 Conclusions

The detection of squeezed light produces measurement results that canbe considered as remarkable Let us focus on experiments where a mode in abright coherent state is overlapped with a mode in a squeezed vacuum state

21

as shown in Figs (1) and (3) In both setups the squeezed vacuum field caneasily be blocked which allows us to compare the measurement results ona bright coherent state with and without the interference with the squeezedvacuum state Without squeezing the photo-electric detectors measure alarge number of photon events with a large quantization noise (shot noise)The large noise reflects the fact that all photon events were independent fromeach other as shown in Fig 2 (bi) With squeezing the photo-electric de-tectors again measure a large number of photon events with an expectationvalue that is even slightly higher but nevertheless the quantization noise ofall detected photons is significantly reduced Fig 2 (bii)Based on the discussion of EPR entanglement in Subsec 24 the photo-electric detection of the output light of a squeezing-enhanced laser inter-ferometer (with αlowastα1) produces the reality of photons This way we cankeep the lsquowave picturersquo in which no photons exist when light travels alongthe interferometer arms and when it interferes at the beam splitter Whenthe energy of the beam is elevating electrons to the conductance band ofthe photo-diodersquos semi-conductor n photon events simultaneously appearwithin the measuring interval with probability P (n) What conclusion hasto be drawn if the probabilities resemble a sub-poissonian statistic ndash Theoccurrence of photon events is still truly random but in this case not forindividual photons The occurrence of photons is correlated in such a waythat the probability of detecting an additional photon in the same time in-terval reduces the larger the number of already detected photons is Whatfollows from the discussion of EPR entanglement for a photon counting ex-periment with pure squeezed vacuum and ideal photon counters Here theprobabilistic detection of one photon entails the detection of a second onewith certainty With some smaller probability a third photon is detectedwhich entails the detection of a fourth photon with certainty and so on

If a photon of a mode that was not interrogated by the environment beforeis absorbed its reality is created in this very moment If the photon belongsto a squeezed state this process instantaneously influences the probability ofother photons becoming reality

Of course a more general statement can be made based on the insightthat interaction with the environment creates the reality of any kind ofquanta including electrons atoms and molecules

22

3 Theoretical description of squeezed states

31 The quadrature amplitude operators

Consider a single mode of light at optical frequency ω Its Hamiltonoperator reads

Hω = ~ω(n+

1

2

)= ~ω

(adaggerωaω +

1

2

)= ~ω

(X2ω + Y 2

ω

) (7)

where n is the photon number operator and aω and adaggerω are the annihilationand creation operators which obey the commutation rule

[aω a

daggerω

]= 1 The

operator aω has a complex-valued dimensionless eigenvalue spectrum andcorresponds to the complex amplitude αω in classical optics Xω and Yω arethe hermitian amplitude and phase quadrature operators The eigenvaluesof the quadrature operators are also dimensionless and proportional to theelectric fields at the oscillationrsquos antinode and at the oscillationrsquos node Inthe above equation they are defined such that their variances are ∆2Xω =∆2Yω = 14 if the oscillator is in its ground state ie if 〈n〉 = 0

Although Eq (7) simply describes the energy of an harmonic oscillator itis the essence of quantum theory since it mathematically describes the wave-particle dualism Whereas the eigenvalues of n have a discrete spectrum theeigenvalues of Xω and Yω have a continuous spectrum In classical opticsthe phase quadrature is zero In quantum optics its expectation value is alsozero but its uncertainty contributes to the overall energy

Eq (7) describes a cavity mode as well as a section that is cut from apropagating quasi-monochromatic light beam The latter example is of highrelevance in actual experiments By setting the sectionrsquos time window ie themeasuring time interval the time-frequency (lsquomodulationrsquo) mode is defined

The quadrature operators introduced in Eq (7) and displayed in Fig 7 donot correspond to lsquoXrsquo and lsquoY rsquo that are of relevance in laser interferometry andin optical communication and which were already discussed in Subsec 22and 23 The optical frequency of visible and near-infrared light is far too highto be transferred to an oscillation of photoelectric voltage Quite general alaser interferometer targets signals at audio or radio band frequencies Ωi ωSuch a measurement is achieved as stated before by decomposing the photo-electric voltage from the photo diode at the interferometer output into asingle-sided spectrum (positive frequencies only) of intervals of Ωplusmn∆Ω2

23

(a)

(b)

(c)

(d)

2π 4π

2π 4π

2π 4π

2π 4π

Figure 7 Phase spaces and electric field oscillations of monochromatic lightndash Top Left Monochromatic light in a coherent state is represented by a phasor (whitearrow) including its quantum uncertainty (white dashed circle and fuzzy area) located inthe phase-space spanned by the quadratures Xω and Yω When the phase space rotateswith optical frequency ω2π the projection of the quantum phasor onto a fixed (vertical)axis corresponds to the electric field E(t) as shown on the right side (a) Weakly displacedcoherent state (b) Corresponding amplitude squeezed state The electric field uncertaintyaround the zero average field region is anti-squeezed (c) Vacuum state at the same opticalfrequency (d) Corresponding squeezed vacuum state The meaning of the uncertaintycould be carved out by supplementing them with monochromatic waves all having theoptical frequency ω2π Changing amplitudes then display amplitude quadrature noiseChanging shifts along the time axis model the electric field uncertainty at the expectedzero crossing They are not implemented in the graphics here however since any of thosewaves does not exist due to Heisenbergrsquos uncertainty relation

24

The signals as well as the quantum uncertainties carried by a beam oflight are thus described by a spectrum of pairs of non-commuting quadratureoperators Mathematically every such operator is defined by an integral overthe Fourier components within the bandwidth The spectral weighting of theFourier components is called the lsquowindow functionrsquo By going to sidebandintervals a spectrum of a new type of optical mode is defined which describesthe modulation of the electric field in the respective frequency interval Ω plusmn∆Ω2 In this Review we call it a lsquomodulation modersquo

The quadrature operators that are defined around a modulation frequencyΩ with a bandwidth of ∆Ω are the quadrature amplitude operators thatare relevant in laser interferometry Whenever they are not related to aspecific band we use the short form XΩ∆Ω(t) equiv X and YΩ∆Ω(t) equiv Y cfEqs (1) and (2) These operators can slowly vary with time where thetime dependence is limited by ∆Ω (The time dependence is not due toquantum uncertainty which usually is time independent but for instancedue to the time dependence of the signal eg a passing gravitational wave)Let us consider now a pair of quadrature operators for a particular sidebandΩ plusmn ∆Ω2 The Hamilton operator of the corresponding modulation modeis found by switching to the frame rotating at optical frequency ω Thetransition is done by applying the unitary transformation U = exp(iωadaggerat)generating a new Hamiltonian H = U daggerHωU minus i~UpartU daggerpartt The Hamiltonianof the modulation mode reads

H = ~Ω

(nΩ +

1

2

)= ~Ω

(adaggera+

1

2

)= ~Ω

(X2 + Y 2

) (8)

where nΩ is the (occupation) number operator for the modulation modeand a and adagger its annihilation and creation operators The commutation rule[a adagger

]= 1 is unchanged X and Y are the amplitude and phase quadra-

ture amplitude operators respectively They correspond to the depth of theamplitude modulation and for weak excitations to the depth of the phasemodulation respectively They are the conventional hermitian field operatorsin experimental quantum optics Note that modulation modes at angularfrequency Ω can be described by a superposition of three optical frequenciesa carrier at ω an upper sideband at ω + Ω and a lower sideband at ω minus ΩThe quantum mechanical description of modulation states in connection tooptical carrier and upper and lower sidebands is known as the lsquoTwo-PhotonFormalismrsquo [Caves and Schumaker (1985) Schumaker and Caves (1985)]

The quadrature amplitude operators in Eq (8) are again defined such

25

that the variances of the uncertainty of a modulation field in its ground stateor in a coherent state are

∆2Xvac = ∆2Yvac = 14 (9)

Generally quadrature operators X and Y as defined in Eqs (7) and (8) arethe real and imaginary parts of the annihilation operator

a = X + iY hArr adagger = X minus iY (10)

hArr X =1

2

(a+ adagger

) Y =

1

2i

(aminus adagger

) (11)

They satisfy the commutation relation[X Y

]=i

2 (12)

and their variances are limited by a Heisenberg uncertainty relation of thefollowing form

∆2X∆2Y ge 1

16 (13)

A quantum state is called a ldquosqueezed staterdquo [Bachor and Ralph (2004)]if ∆2Xϑ lt 14 for an arbitrary field quadrature Xϑ = X cosϑ + Y sinϑ seeEq (3) The angle of the lowest variance below 14 is called the squeeze angleθ The largest factor by which the variance is below 14 is called the squeezefactor often given on a decibel (dB) scale using the following transformation

minus 10 middot log10

(∆2Xθ

∆2Xvac

) (14)

The squeeze factor can also be described by the squeeze parameter r

eminus2r =∆2Xθ

∆2Xvac

(15)

When a squeezed state experiences optical loss it remains squeezed butthe squeeze factor is reduced Also the statersquos purity is reduced ie theproduct of the quadrature uncertainties increases above the minimum valueOptical loss corresponds to mixing the state with the vacuum state Let∆2Xϑ be the variance of a quadrature amplitude ∆2Xvac the variance of the

26

(quadrature angle independent) ground state uncertainty and (1 minus η2) therelative energy loss Then the resulting quadrature variance reads

∆2Xprime

ϑ = η2∆2Xϑ + (1minus η2)∆2Xvac (16)

To maximize the benefit from squeezed states in applications stronglysqueezed states need to be generated and optical loss minimized Opticalloss occurs due to absorption and scattering in the optical components inthe path of the squeezed beam including the squeezing resonator itself anddue to non-perfect matching to the interferometer mode non-perfect inter-ference contrast of the interferometer and non-perfect quantum efficiency ofthe photo diodes The sum of all losses including those outside the inter-ferometer need to be less then 10 to allow a nonclassical quantum noisesuppression of a factor of 10 in power ie 10 dB

32 Phase space representations of squeezed states

The Wigner function ndash The properties of squeezed states are nicely dis-played by the Wigner function W (X Y ) [Wigner (1932)] An example interms of a squeezed vacuum state is shown in Fig 8 It is a quasi-probabilitydistribution which contains the statersquos full information including its quan-tum statistic There are two ways how a Wigner function provides a sufficientcriterion for nonclassicality First by containing negative values second byfeatures that have a smaller (squeezed) width compared with the Wignerfunction of the ground state Integrating the Wigner function over Y pro-vides the probability density of measurement results ie of the eigenvaluesof the observable X and vice versa

infinintminusinfin

W (X Y )dY = p(X)

infinintminusinfin

W (X Y )dX = p(Y ) (17)

where p(X) and p(Y ) are the observed probability distributions also exem-plarily shown in Fig 8

The ground state coherent states as well as (quadrature) squeezed stateshave quadrature eigenvalue probability densities that are Gaussian TheirWigner functions are also Gaussian and thus entirely positive Wigner func-tions of other nonclassical states for instance Fock states exhibit negativevalues For this reason the Wigner function is called a quasi -probabilityfunction

27

X Y XY

Figure 8 Wigner function and its projections ndash Displayed is the full information ofa squeezed vacuum state The quasi probability density of the Wigner function (centre)is plotted along a third dimension and is color-coded Also shown are the actual mea-surement results from which the Wigner function is reconstructed They are representedby the squeezed and anti-squeezed Gaussian projections onto the X and Y axes Theirprobability densities p(X) and p(Y ) are given on the vertical axes The Gaussian measure-ment statistic of the first clearly shows squeezing compared to the ground state statistic(dashed) The squeeze factor is 116 dB and the squeeze parameter r = 1335 [Mehmetet al (2010)]

Fig 9 shows the Wigner functions for (a) a coherent state (b) the ground(vacuum) state (c) a displaced squeezed state and (d) a squeezed vacuumstate All Wigner functions describe a modulation of the carrier light atsideband frequency Ω integrated over the frequency interval ∆Ω The car-rier light is not part of these Wigner functions The displacement in (a)represents a classical amplitude modulation (b) corresponds to the absenceof any photons with a frequency offset of plusmnΩ from the local oscillator field(c) and (d) represent states whose amplitude modulation depth is more pre-cisely defined than that of the ground state Fig 10 shows Wigner functionspectrum for a broadband squeezed vacuum field Every Wigner functiondescribes the modulation field at some modulation frequency Ωi integrated

28

X

Y

X

(a) (b)

X X

(c) (d)

Y

Y Y

Figure 9 Simplified representation of Wigner functions ndash The darker the shadedareas the larger is the phase-space quasi-probability Shown are four different (time-independent) states of a modulation mode at frequency Ω for a specific resolution bandwidth ∆Ω Panel (a) represents a coherent state the displacement (α) corresponds to aclassical amplitude modulation Panel (b) represents the ground (vacuum) state (c) adisplaced squeezed state and (d) a squeezed vacuum state both with squeeze angle θ = 0The latter is in analogy to Fig 8 Again the light field that carries the modulation is notpart of the pictures

over the resolution bandwidth (RBW) of ∆Ω

The Glauber-Sudarshan P -function ndash The P -function [Glauber (1963)Sudarshan (1963)] is calculated by de-convoluting the Wigner function fromthe ground state uncertainty [Gerry and Knight (2005)] For displaced vac-uum states (coherent states) the P -function corresponds to a displaced δ-function The mathematical expression of the P -function of a squeezed statecontains infinitely high orders of derivatives of the δ-function [Vogel andWelsch (2006)] Such a function contains negativities but cannot be dis-played It is possible however to define a phase-space quasi probabilityfunction for squeezed states that can be displayed and that does show neg-

29

0

Ω1

Ω2

Ω

Single-sided spectrum

Y

X

Figure 10 Hint of a Wigner function spectrum ndash A single-sided spectrum (positivefrequencies only) with respect to the carrier field can be used to visualize a broadbandsqueezed field Shown are two examples displaying a squeezed vacuum state at Ω1 anda displaced amplitude squeezed state at Ω2 The individual Wigner functions cover theresolution bandwidth ∆Ω gt 0 (not shown) In general the squeezing strength as well asthe squeeze angle and the displacement are a function of sideband frequency

ativities as a sufficient and necessary condition for certifying the squeezingeffect This lsquononclassicality functionrsquo is calculated by de-convoluting theWigner function from an uncertainty distribution that is steeper than theGaussian distribution A pronounced negativity of a squeezed vacuum stateof up to 69 standard deviations was found [Kiesel et al (2011)]

The double-sided phasor picture ndash This phasor picture links quantumstates of modulations with the quantum states of the contributing opticalfields [Bachor and Ralph (2004)] and is mathematically described by thetwo-photon-formalism [Caves and Schumaker (1985) Schumaker and Caves(1985)] Generally a weak amplitude or phase modulation at frequency Ω ofa carrier field at optical frequency ω can be understood as the carrierrsquos beatwith two optical frequencies at ωplusmnΩ The double-sided phasor picture is ableto display a spectrum of different and independent modulation frequenciesin the rotating frame of the carrier field The carrier light field is time-independent but the upper and lower sidebands are not They rotate withplusmnΩi(2π) respectively around the frequency axis

30

ω0

Yω

Xω

Upper sideband

Lower sideband

ω0+Ω1

ω0+Ω2

ω0-Ω2

ω0-Ω1

ω

Upper sideband

Lower sideband

Yω

Xω

ω0+Ω1

ω0+Ω2

ω0-Ω2

ω0-Ω1

ω

ω0

Figure 11 Double-sided phasor pictures ndash Phase spaces at optical frequency ω0 plusmnΩirotate around the frequency axis with sideband frequency plusmnΩi Its sign determines thedirection of rotation with respect to the local oscillator in the rotating frame at ω0 Apair of phase spaces need to be superposed to provide a description of a modulation fieldat |Ωi| Top Amplitude quadrature squeezed field with a coherent displacement at |Ω2|The displacement corresponds to a classical amplitude modulation The uncertaintiesof all optical frequencies are circular and larger than that of the ground state (dashed)Quantum correlations are indicated by additional symbols More details are given in themain text Bottom Corresponding spectrum of (displaced) vacuum states which do nothave any quantum correlations

31

Fig 11 shows such a double-sided phase space picture where the carrierrsquosmodulation at Ω1 is in a squeezed vacuum state and where the modulationat Ω2 is in a displaced squeezed state The picture shows how a classicalamplitude modulation as well as the quantum statistic of a modulation fieldis decomposed into contributions from upper and lower sidebands For asqueezed modulation field the upper and lower sidebands show no squeezedbut circular thermally excited quantum uncertainties The uncertainties of apair of sidebands however show correlations as well as anti-correlations InFig 11 these (anti-) correlations are marked with times and + for the modulationfrequency Ω1 and with N and bull for the modulation frequency Ω2

33 Covariance matrix representation of (single-party) squeezed states

Since squeezed states have a Gaussian quantum statistic four numbersare sufficient for their full description These numbers are the second mo-ment of the quadrature amplitude showing the strongest squeezing and thesecond moment of its orthogonal quadrature amplitude as well as their firstmoments describing the displacement These four numbers are sufficientto calculate the Wigner function shown in Fig 8 In general the quadra-ture of strongest squeezing is not perfectly aligned with one of the axesof the measurementrsquos coordinate system The so-called covariance matrix(VXXVXY VY XVY Y ) [Simon et al (1994)] accounts for phase space rotationsand enables the calculation of how these states evolve within an interfero-metric arrangement Their components are normalized to the vacuum noisevariance ∆2Xvac = ∆2Yvac and read

VXY =

langXY + Y X

rangminus 2

langXranglang

Yrang

2∆2Xvac

(18)

The following examples represent the ground state a pure 10 dB am-plitude quadrature squeezed state and a pure 10 dB squeezed state with asqueeze angle of 45

Vvac =

(1 00 1

) V0

01 =

(01 00 10

) V45

01 =

(505 495495 505

) (19)

with V4501 =RT

45V001R45 where Rα = (cosα minussinα sinα cosα) is the rota-

tion matrix

32

34 Phase space representation of two-mode (bi-partite) squeezed states

XA

YA

XB

YB

Figure 12 Bi-partite squeezed vacuum state ndash Shown is a Wigner-function-basedphase space representation in close analogy to the single party version in Fig 9 (d) Thepicture describes a single modulation at frequency Ω with bandwidth ∆Ω Measurementsat party A and B reveal local Wigner functions that correspond to thermal states sincethe uncertainties (indicated by the color and by the large circles) are symmetric and largerthan that of the ground state (indicated by the small dashed circles) The uncertaintieshowever show correlations and anti-correlations here indicated by + and times respectivelyThe strength of these (anti-)correlations are indicated by ellipses Bi-partite squeezingie entanglement is present if the short axes of the ellipses are shorter than the diameterof the ground state uncertainty The picture in fact represents Einstein-Podolsky-Rosenentanglement [Einstein et al (1935)] From a measurement of XA or YA the correspondingmeasurement result at party B can always be inferred with an uncertainty that is smallerthan the ground state uncertainty

A bi-partite state enables a measurement on subsystem A and simulta-neous a measurement on subsystem B For a large number of simultaneousensemble measurements of the same quadrature amplitude Xϑ the followingtwo joint quadrature variance can be calculated

∆2(XAϑ plusmn XB

ϑ ) (20)

A state that is symmetrically shared between two parties (A and B) is calleda two-mode squeezed state if the variances of joint quadrature measurementsfulfill the following inequality [Duan et al (2000)] ie

∆2(XA minus XB)

∆2(XAvac minus XB

vac)+

∆2(Y A + Y B)

∆2(Y Avac + Y B

vac)lt 2 (21)

33

with ∆2(XAvac plusmn XB

vac) = ∆2(Y Avac plusmn Y B

vac) = 2∆2Xvac A lsquotwo-mode squeezedstatersquo reveals entanglement in the second moments of the measurement statis-tics It is thus a lsquobi-partite Gaussian entangled statersquo

Fig 12 displays a (pure) bi-partite squeezed vacuum state distributedbetween A and B The state shows full symmetry regarding its subsystemsat the two sites The large circles and the colored area represent Wignerfunctions of the subsystems Measurements of the quadrature amplitudesXA YA XB and YB show identical variances and the correlations and anti-correlations have identical strength since ∆2(XA minus XB) = ∆2(YA + YB) lt12 for our normalization of quadrature amplitudes having a ground statevariance of 14

Generally a symmetric bi-partite squeezed state fulfills another quantita-tive (Gaussian) entanglement criterion if less than 50 of the vacuum state issymmetrically mixed into the initially pure state Bi-partite squeezed statesare always entangled but in this case they are even Einstein-Podolsky-Rosen(EPR) entangled [Reid (1989)] allowing the demonstration of the quantumsteering effect [Einstein et al (1935) Schrodinger (1935) Reid (1989) Cav-alcanti et al (2009)] The first such experiment was performed by Ou etal [Ou et al (1992)] using type II parametric down-conversion (PDC) Laterexperiments produced bi-partite squeezed vacuum states by overlapping twosqueezed vacuum states each produced with type I PDC on a balanced beamsplitter and used the entangled output for the demonstration of quantumteleportation [Furusawa et al (1998) Bowen et al (2003ca)] The criterionin Eq (21) and the EPR criterion from [Reid (1989)] was experimentallycompared in Ref [Bowen et al (2003b)] The steering effect in asymmet-ric bi-partite squeezed states were recently experimentally characterized inRef [Handchen et al (2012)]

Fig 12 shows features similar to those in the top part of Fig 11 Thisis not a coincidence and shows that a bi-partite squeezed state can also begenerated by spatially splitting the upper and lower sideband of a (single-party) squeezed state This was first experimentally demonstrated by thegroup of E Polzik [Schori et al (2002)] and later used for EPR multiplex-ing of a single longitudinal mode of a squeezing resonator [Hage et al (2010)]

35 Covariance matrix representation of bi-partite squeezed states

Also the full information of bi-partite states including the entanglementcan be cast by the covariance matrix [Simon et al (1994)] which can be used

34

to calculate the propagation of these states in laser interferometers Againall variances are normalized to the vacuum noise variance in full analogy toEq (18) The generic bi-partite covariance matrix has dimension 4times4 andreads

Vbp =

VXAXA

VXAYA VXAXBVXAYB

VYAXAVYAYA VYAXB

VYAYBVXBXA

VXBYA VXBXBVXBYB

VYBXAVYBYA VYBXB

VYBYB

with

VXAYB =

langXAYB + YBXA

rangminus 2

langXA

ranglangYB

rang2∆2Xvac

ϑ

(22)

Due to the symmetry in Eq (22) the 4times4 covariance matrix is fully spec-ified by just ten independent coefficients If the phase spaces at A and Bare aligned along the strongest correlations and anti-correlations the ma-trix components referring to different quadrature amplitudes eg VXAYA arezero Such entangled states can be produced by overlapping two squeezedfields with a squeeze angle difference of 90 on a balanced beam splitter

A symmetric bi-partite squeezed vacuum state which is also called anlsquoS-classrsquo [DiGuglielmo et al (2007)] bi-partite squeezed vacuum state shows(anti-)correlations in two joint quadratures as defined in Eq (21) For a puresuch state of 10 dB squeezing the covariance matrix reads

Vbp10|10 =

505 0 495 0

0 505 0 minus495495 0 505 0

0 minus495 0 505

The following covariance matrix describes a so-called lsquoV-classrsquo 10 dB bi-partite squeezed vacuum state Here only one joint quadrature shows 10 dBsqueezing whereas the orthogonal joint quadrature shows vacuum noise Thestate is obtained by overlapping one 10 dB squeezed state with a vacuumstate on a balanced beam splitter

35

Vbp0|10 =

055 0 045 0

0 55 0 minus45045 0 055 0

0 minus45 0 55

The first measurement of all elements of such a covariance matrix was achievedin [DiGuglielmo et al (2007)]

36 Photon numbers of squeezed states

In contrast to the ground state squeezed vacuum states do have photonexcitations As said earlier quantum theory links the wave and the particlepictures Indeed the squeeze factor of a modulation mode is directly con-nected to a certain photon number excitation Squeezed states of light areproduced via spontaneous photon pair generation eg by parametric down-conversion The following operator S is called the lsquosqueeze operatorrsquo [Gerryand Knight (2005)] It creates and annihilates photon pairs

|r θ〉 = S(r θ) |0〉 (23)

where |r θ〉 is a squeezed vacuum state with squeeze parameter r and squeezeangle θ and |0〉 is the vacuum state The definition of the squeeze operatoris

S(r θ) = exp

[1

2

(reminusiθa2 minus reiθadagger 2

)] (24)

The following shows that this definition indeed results in a state with squeezedquadrature amplitude variances Lets set θ = 0

〈0| Sdagger(r 0) X S(r 0) |0〉 =1

2〈0| Sdagger(r 0)

(a+ adagger

)S(r 0) |0〉 (25)

〈0| Sdagger(r 0) Y S(r 0) |0〉 =i

2〈0| Sdagger(r 0)

(aminus adagger

)S(r 0) |0〉 (26)

Using the Baker-Hausdorff formula we get

Sdagger(r θ) a S(r θ) = a cosh r minus adaggereiθsinh r (27)

Sdagger(r θ) adagger S(r θ) = adaggercosh r minus aeminusiθsinh r (28)

36

Since 〈0| X |0〉 = 〈0| Y |0〉 = 0 also Eqs (25) and (26) are zero To finallycalculate the variances we need

〈0| Sdagger(r 0) X2 S(r 0) |0〉 =1

4〈0| Sdagger(r 0)

(a2 + adaggera+ aadagger + adagger2

)S(r 0) |0〉

〈0| Sdagger(r 0) Y 2 S(r 0) |0〉 = minus1

4〈0| Sdagger(r 0)

(a2 minus adaggeraminus aadagger + adagger2

)S(r 0) |0〉

Given that SSdagger is the identity and using again Eqs (27) and (28) we get theexpected variances

∆2X =1

4

[cosh2 r minus 2cosh r sinh r + sinh2 r

]=

1

4eminus2r

∆2Y =1

4

[cosh2 r + 2cosh r sinh r + sinh2 r

]=

1

4e2r

Since the squeeze operator can only create and annihilate photon pairsa squeezed vacuum state without photon loss must correspond to an evennumber of photons But not only photon loss also a coherent displacementleads to flattening out the odd-even oscillations The probability of detectingN photons in a pure displaced squeezed state are derived for instance in[Gerry and Knight (2005)] and read

P (N) = |〈N |α r θ〉|2 =(05 tanh r)N

N cosh rexp

[minus|α|2 minus 1

2(αlowast2eiθ + α2eminusiθ)tanh r

]times∣∣∣HN

[(αcosh r + αlowasteiθsinh r)

radiceiθsinh(2r)

]∣∣∣2 (29)

where HN is the N th Hermite polynomialFig 13 shows the photon number distributions for 5 different pure squeezed

states according to Eq (29) Panels (a) to (c) show squeezed vacuum stateswith 43 dB 86 dB and 172 dB of squeezing Panel (d) shows the moregeneral case of a squeezed state with a coherent displacement α 6= 0 Dueto θ = 0 the state is amplitude quadrature squeezed Panel (e) refers to thecorresponding phase quadrature squeezed state For comparison panel (f)shows the photon number distribution of the coherent state with the samedisplacement

The panels in Fig 13 represent the diagonal elements of the statersquos den-sity matrix in number basis Only the latter also contains the coherences

37

002

004

006

000 000

002

004

006

008

403020100 Photon number n

403020100Photon number n

Pro

babi

lity

403020100000

005

010

015

100 2 4 6 8100 2 4 6 800

04

06

02

100 2 4 6 800

03

02

01Pro

babi

lity

Pro

babi

lity

08

00

04

06

02

08

(a) (b)

(c) (d)

(e) (f)

Figure 13 Photon number distributions ndash All panels represent pure states (a) 43 dBsqueezed vacuum state (r = 05 α = 0) (b) 86 dB squeezed vacuum state (r = 1 α = 0)(c) 172 dB squeezed vacuum state (r = 2 α = 0) (d) displaced 86 dB squeezed state(r = 1 θ = 0 α = 4) (e) displaced 86 dB squeezed state (r = 1 θ = π2 α = 4) (f)Coherent state (r = 0 α = 4) The average photon numbers are about 027 14 131174 174 and exactly 16 see Eq (30)

between photon numbers [Gerry and Knight (2005)] Figures as shown heregenerally do not give full descriptions of the states

38

A squeezed vacuum state (r 6= 0) always has a non-zero photon numberand can not be the ground state The average photon number of a puresqueezed vacuum state can be calculated using Eq (8) With the maximallysqueezed quadrature variance ∆2Xsqz the average photon number is given by

n = 〈n〉|α=0θr〉 = ∆2Xsqz +(∆2Xsqz)

minus1

16minus 1

2=eminus2r

4+e2r

4minus 1

2 (30)

with the vacuum noise variance normalized to one quarter A coherent dis-placement further adds |α|2 photons on average

4 Squeezed-light generation

41 Overview

Squeezed light was first produced in 1985 by Slusher et al using four-wave-mixing in sodium atoms in an optical cavity [Slusher et al (1985)]Shortly after squeezed light also was generated by four-wave-mixing in an op-tical fibre [Shelby et al (1986)] and by degenerate parametric down-conversion(PDC) in a 2nd-order nonlinear crystal placed in an optical cavity [Wu et al(1986)] The pumped cavity was operated below its oscillation thresholdie the parametric gain did not fully compensate the round trip losses whichis also called lsquocavity-enhanced optical-parametric amplification (OPA)rsquo

The early day experiments achieved squeeze factors of a few percent up toabout 3 dB Today squeeze factors of more than 10 dB are directly observedin several experiments [Vahlbruch et al (2008) Eberle et al (2010) Stefszkyet al (2012) Vahlbruch et al (2016)] All of them are based on cavity-enhanced OPA (below threshold) The parametrically amplified mode isdegenerate ie signal and idler modes are identical In particular the down-conversion process is of lsquotype Irsquo which means that the amplified mode has awell-defined polarization Squeezed states can also be generated above oscil-lation threshold In Refs [Villar et al (2006) Jing et al (2006)] bi-partitesqueezing was generated with above-threshold PDC Both experiments usedtype II PDC which provides orthogonally polarized signal and idler fieldsType II PDC below threshold was also used to generate squeezed and bi-partite squeezed fields [Grangier et al (1987) Ou et al (1992)] All these

39

experiments were performed in the continuous-wave regime which is also thefocus of this Review Squeezed states of modulations of trains of laser pulseshowever have been also generated since the 1980s using either PDC or theoptical Kerr effect [Slusher et al (1987) Bergman and Haus (1991) Our-joumtsev et al (2006) Dong et al (2008)] For an overview of the develop-ments in squeezed-light generation in the continuous-wave as well as pulsedregime see Ref [Bachor and Ralph (2004)] Squeezed-light generation inopto-mechanical setups [Aspelmeyer et al (2014)] which use the intensitydependent phase shift from radiation pressure was discussed in Refs [Paceet al (1993) Rehbein et al (2005) Corbitt et al (2006)] and recently ex-perimentally achieved by several groups [Brooks et al (2012) Safavi-Naeiniet al (2013) Purdy et al (2013)]

42 Degenerate type I optical-parametric amplification (OPA)

This section provides a graphical description of how degenerate type IOPAPDC turns a vacuum state into a squeezed vacuum state and a coher-ent state into a displaced squeezed state The process requires a bright pumpfield and a 2nd-order nonlinear crystal For simplicity we set all nonlinearitiesabove 2nd-order to zero

Let us consider a short segment of the second-order nonlinear crystalpumped with light of optical frequency 2ν All other modes that enter thecrystal shall not contain any photons ie are in their vacuum states Ofthese the only mode of interest is that at optical frequency ν which spatiallyoverlaps with the pump mode Fig 14 shows the total electric field of theoptical input E in and the 2nd-order nonlinear dielectric polarisation of thecrystal P The latter is proportional to the total electric field of the outputEout The pump field at 2ν periodically drives the vacuum field at ν betweenregions of low and high polarisation This process transforms the vacuumstate into a squeezed vacuum state in the output [Bauchrowitz et al (2013)]The output further contains the hardly depleted pump field and frequencydoubled parts of the pump field at 4ν It is again emphasized that Fig 14displays OPA in a small segment of the crystal In reality the nonlinear effectaccumulates over the crystal length or even over several passages since thecrystal is usually put into an optical resonator A noticeable effect is achievedif all infinitesimal contributions constructively interfere This is achieved incase of phase matching ie if the wave fronts of the modes at 2ν and νpropagate with the same speed and thus do not run out of phase Note that

40

t

t

Squeezing

Figure 14 Optical parametric amplification of a vacuum state ndash The upper leftcorner shows the crystal polarization P(E) = ε0

(χ(1)E + χ(2)E2

) ie the separation of

charge carriers by the electric component of an optical field E The graph illustrates howan input quantum field (from below) is projected into an output quantum field (towardsthe right) In the example shown here the input field is composed of a classical pumpfield E in

2ν at frequency 2ν and zero-point fluctuations E inν of a field at frequency ν cf

Fig 7(c) The superposition E in of these two fields is transferred into a time-dependentdielectric polarization that is the source of (and thus directly proportional to) the elec-tric component of the output field Eout The quantum uncertainty of the output fieldshows a phase-dependent (parametric) amplification at frequency 2ν Spectral decompo-sition of the output field Eout reveals coherent amplitudes at frequencies 2ν and 4ν and asqueezed vacuum state Eout

sqzν cf to Fig 7(d) The concept of this figure was published inRef [Bauchrowitz et al (2013)]

41

t

t Amplitudequadraturesqueezing

Figure 15 Optical parametric amplification of a coherent state ndash The pictureshows how a displaced vacuum state is transformed into a displaced squeezed state Thepumprsquos electric field is maximal when the input field at fundamental frequency ν shows azero crossing The phase relation described results in an output state that is amplitudequadrature squeezed If the pump field at the input in phase was shifted by half of itswavelength the squeezed output field were a phase quadrature squeezed The squeezinggeneration displayed here corresponds to the transition from Fig 7(a) to Fig 7(b) but withan additional reduction of the displacement The concept of this figure was published inRef [Bauchrowitz et al (2013)]

in actual squeezing experiments the 4ν component is usually suppressed byphase miss-matching

Fig 15 shows the same process but now for an input field at frequencyν in a coherent state In this case the relative phase between the two inputstates ∆ϕ = ϕ2ν minus 2ϕν is relevant In Fig 15 the relative phase is set suchthat the expectation value of the field at frequency ν is zero when the pump

42

(a) (b)

(c) (d)

X

Y

X

YΔϕ = ndash50deg

X

Y

Δϕ = 0deg

X

Y

Δϕ = 90deg

Figure 16 Phase-space illustration of degenerate OPA ndash The (displaced) dashedcircle in each diagram represents the uncertainty of the initial state at optical frequencyν The (displaced) shaded area represents the state after degenerate optical parametricamplification All quantum uncertainties shown correspond to pure states The boldgreen arrow describes the bright second-harmonic pump field whose uncertainty can beneglected The phase between the 2nd harmonic pump and the initial state (∆ϕ = ϕ2ν minus2ϕν) determines the result of the parametric amplification

field reaches its maximum (∆ϕ = 90) The output at the fundamentalfrequency is then an amplitude squeezed state with a deamplified coherentamplitude

Fig 16 summarizes the squeezing operation on the vacuum state as wellas on displaced vacuum states for different phase relations ∆ϕ between thetwo input fields

43 Cavity-enhanced OPA

Placing the nonlinear crystal inside a cavity can greatly enhance thedown-conversion efficiency but not only that A cavity introduces a thresholdfor the pump power above which the parametric gain is infinite just limited

43

by the finite pump power In this case the vacuum uncertainty of the inputfield at frequency ν is amplified to a bright laser field at frequency ν Thedevice is then called an optical-parametric oscillator (OPO) For the gener-ation of squeezed states however the pump power is usually kept (slightly)below threshold Due to nonzero optical loss there exists a pump powersmaller than the threshold above which the tiny improvement of squeezingis not noticeable anymore Getting the pump power closer to the thresholdcould even reduce the observed squeeze factor if a fluctuating squeeze an-gle projects anti-squeezing into the observed quadrature amplitude [Franzenet al (2006) Suzuki et al (2006) Dwyer et al (2013)] The cavity has an-other important purpose It confines the transverse spatial mode usuallyto TEM00 This mode confinement is crucial for any efficient application ofthe squeezed state in laser interferometry since it allows the suppression ofanti-squeezing from other transversal modes The squeezing process requiresa nonlinear material that should show negligible absorption at both opticalfrequencies involved in particular at the wavelength of the squeezed modeIn Refs [Vahlbruch et al (2008) Mehmet et al (2009)] 10 dB and 116 dBof squeezing were achieved using MgOLiNbO3 The highest squeeze factorstoday are produced in (quasi phase matched) periodically poled KTP [Eberleet al (2010) Mehmet et al (2011) Stefszky et al (2012) Vahlbruch et al(2016)]

The optical cavity that is built around the nonlinear crystal is vital forsqueezed-light generation and it deserves a detailed consideration Gener-ally the mode propagating away from a cavity is the result of interferenceat the cavity coupling mirror One contribution is given by the intra-cavityfield attenuated by the amplitude transmission coefficient t of the outcouplingmirror The second contribution is given by the outside field that is reflectedby the same mirror with amplitude reflectivity r =

radic1minus t2 and spatially

overlapped with the first Also the mode from a squeezing resonator is suchan interference product

The impedance matched resonator

Let us consider first an empty optically stable and loss-less Fabry-Perotresonator built from two identical mirrors each with amplitude reflectivityr = r1 = r2 lt 1 A propagating field be perfectly mode-matched to one ofthe cavity resonances In this setup the resonator shows zero reflection andthe resonator is said to be impedance matched (for all such input fields)

44

|α〉

r1α

r12

= +

|α〉 r22 = r1

2

minusr1α equiv t1 αcav = 1minus r12 minusα sdot r1

1minus r12

0

XY

Figure 17 Empty impedance-matched resonator ndash Mode-matched and resonant lightthat enters the cavity from the left here displayed by a quantum phasor for a coherentstate is fully transmitted including its quantum uncertainty The back-reflected statedestructively interferes with itself for all frequencies well within the cavity linewidth The180 phase shift of the transmitted cavity field amplitude minusr1α is a necessary conditionin order to fulfill energy conservation on cavity resonance Impedance matching is forinstance achieved for a resonant cavity with matched mirror reflectivities (r2

2 = r21) and

zero optical loss The complex amplitude of the field traveling towards left inside the cavityis readily deduced from the figure and reads αcav = minusαr1(1minus r2

1)minus12 (highlighted by thedashed circle) In the displayed setup zero field uncertainties are reflected however alsothe vacuum state that enters the cavity from the right is fully transmitted (not shown)

r1α

XY

r12 r2

2 =1

= + |α〉

minus 1+ r1( )α equiv t1 αcav = 1minus r12 minus 1+ r1( )α

1minus r12

|α〉

Figure 18 Empty maximally overcoupled resonator ndash Maximal overcoupling isachieved for a resonant cavity with a perfect end-mirror reflectivity (r2

2 = 1) and zerooptical loss For a given input-mirror reflectivity r2

1 the intracavity light power is maximalMode-matched and resonating light entering the resonator from the left is fully reflectedThe complex amplitude of the field traveling towards left inside the cavity is readilydeduced from energy conservation to αcav = minusα(1 + r1)(1 minus r2

1)minus12 (dashed circle) Inthis setup no uncertainty from the right couples to the left side of the cavity

45

Obviously the interference described in the previous paragraph is fullydestructive The same resonator also shows zero reflection of the input fieldrsquosquantum uncertainty since the interference happens between parts of thesame quantum state The mode propagating away from such a resonatorhowever is not in a nonclassical but in a vacuum state because the vacuumstate that enters the cavity through the opposite site is also fully transmittedThe interference at the coupling mirror of an impedance matched resonatoris displayed in Fig 17

r1α

r12

+

=

OPA PDC

Xcavg r1(1+r1) Xcav

Δ2Xg 0 Δ2Yg

XY

r22 =1

|α〉

Figure 19 Squeezing resonator ndash Shown is the interference at the zero-loss squeezingresonator operated at threshold The lower line represents the perfectly squeezed modepropagating away from the cavity towards the left The parametric gain medium inside thecavity deamplifies the X quadrature of the cavity mode (Xcav) by the factor r1(1 + r1)which is the ratio of the intra-cavity field amplitudes of the two previous figures The Xquadrature of the field that is back-reflected towards the left destructively interferes withitself similar to the situation of the impedance matched cavity in Fig 17 The parametricpower gain per resonator round-trip (G) needs to mimic the effect of an end mirror withreflectivity r2

2 = r21 For this reason the deamplification of Xcav corresponds to a round-

trip deamplification factor of r1 equivradic

1G The round-trip amplification factor for Ycav

then is 1r1 equivradicG which exactly compensates for the outcoupling and thus determines

the parametric oscillation threshold (threshold for bright lasing) The variances of thequantum uncertainties ∆2Ycav and ∆2Y are thus infinite In this setup no field uncertaintyfrom the right couples to the left of the cavity and a perfectly X-quadrature-squeezed fieldoutside the squeezing resonator is produced

46

The perfectly over-coupled single-ended resonator

We now increase the reflectivity of the far mirror lsquo2rsquo to being perfect (r2 = 1)This way the counter-propagating vacuum state can not enter the cavityAgain a propagating field be perfectly mode-matched through mirror lsquo1rsquo toone of the cavity resonances For frequencies well inside the cavity linewidththe situation is displayed in Fig 18 The setup protects the left side of thecavity against vacuum fluctuations entering through mirror lsquo2rsquo but of coursedoes not squeeze quantum noise The intra-cavity built-up factor is too highfor achieving destructive interference below the vacuum uncertainty on theleft side of the resonator

The impedanced-matched single-ended squeezing resonator

Building on the two previous concepts the straight forward approach now isto start from the perfectly over-coupled single-ended resonator and insert anattenuator into the cavity that does not couple the cavity mode to any bathbut still results in a roundtrip efficiency of precisely r1(lt 1) in amplitudeOptical loss is not appropriate since it increases the coupling of the cavitymode to a thermal bath neither would any phase-insensitive atenuator beappropriate It is easy to show that a phase-insensitive attenuator adds ad-ditional uncertainty since otherwise the commutation relation [a adagger] = 1 isviolated The amplification process that matches our requirement is OPA Toachieve infinite squeezing in X on cavity resonance a second-order nonlinearcrystal needs to be put into the cavity and pumped such that the intra-cavityamplitude quadrature is attenuated by the factor (1 + r1)r1 (on cavity reso-nance) with respect to the empty cavity This factor is readily deduced fromFigs 17 and 18 Due to the symmetry in parametric amplification the intra-cavity phase quadrature is then amplified by (1 + r1)r1 and the round-tripgain has a value of 1r1 in amplitude In this situation not only infinitesqueezing but also the (laser) threshold of the resonator is achieved sincethe round-trip gain of the intra-cavity phase quadrature equals its roundtriploss here fully given by the incoupling mirror

The physical descriptions in Figs 17 to 19 are fully consistent with ob-servations in squeezing experiments The consideration above in particularshows that the intra-cavity field shows a finite squeezing strength while theexternal field shows infinite squeezing The strongest intra-cavity squeezefactor possible is (1 + r1)2r2

1 In the high reflectivity limit this factor corre-

47

sponds to 6 dB Averaged over the full cavity mode the squeeze factor of thecavity mode is in this limit even limited to 3 dB [Walls and Milburn (2008)]Higher intra-cavity squeeze factors are possible for lower mirror reflectivities

44 The generation of squeezed light for laser interferometry

With the insights gained in the previous subsection we now turn to ac-tual experiments The application of squeezed states in laser interferometrycertainly requires large squeeze factors (idealy accompanied with the highestpossible purity) to maximize the impact in terms of sensitivity improvementIn cavity-enhanced OPA the highest parametric gain is achieved on cavityresonance ie at zero sideband frequency But this is not the main reasonwhy this Subsection focusses on the generation of squeezed states at lowsideband frequencies The application of squeezed states in a laser inter-ferometer requires that their sideband frequencies cover the devicersquos signalband Ground-based gravitational wave (GW) detectors have a detectionband from about 10 Hz to 10 kHz frequencies which can be considered aslsquolowrsquo compared to typical frequencies in quantum optics experiments

Squeezing at MHz sideband frequencies is easier to observe than at acous-tic frequencies because the latter are often polluted with excess noise fromlight beams that serve as control beams [Bowen et al (2002) McKenzie et al(2004)] and parasitic interferences from back-scattered light [Vahlbruch et al(2007)] Furthermore the observation of squeezing at low sideband frequen-cies requires a more stable setup since larger measuring times are necessaryThe observation of strong squeezing at MHz frequencies however alreadysets an upper limit to the optical loss of the setup At least the same squeezefactor can be observed at lower frequencies

There are two different main topologies for squeezing resonators TheFabry-Perot-type standing-wave resonator consists of a minimum number ofmirror surfaces and has the advantage of being compact and thus robustagainst mechanical vibrations Usually one or even two mirror coatings aredirectly placed on the spherical and polished surfaces of the nonlinear crys-tal itself [Wu et al (1986) Grangier et al (1987) Breitenbach et al (1998)Vahlbruch et al (2008) Eberle et al (2010) Vahlbruch et al (2016)] TheBowtie traveling-wave resonator has the advantage of providing a separa-tely accessible counter propagating mode for cavity length control [Ou et al(1992) Takeno et al (2007)] It shows no direct back-reflection of incoupledlight which helps reducing parasitic interferences [Stefszky et al (2012)]

48

(a)

(b)

(c)

(d)Squeezingresonator

To inter-ferometer

OPA

SHG

BHD

Mode cleaner

LO

Laser

DBS

DBS

Figure 20 Generation of squeezed light ndash (a) Example of a 2nd-order nonlinearcrystal for the squeezed-light generation at 1064 nm Shown is a bi-convex 65 mm long7MgOLiNbO3 crystal whose polished surfaces also carry the mirror coatings of the res-onator The crystal thus realizes a monolithic squeezing resonator as it was used forthe first demonstration of 10 dB squeezing [Vahlbruch et al (2008)] (b) Optical con-figuration of a half-monolithic (hemilithic) standing-wave squeezing resonator Here thecavity length can be adjusted by displacing the coupling mirror The crystal surface insidethe cavity is anti-reflection coated The photograph shows a 10 mm long PPKTP crystalsqueezing resonator as used for the GEO 600 squeezed-light source [Abadie (2011)] (c)Mechanically stable housing of a standing-wave squeezing resonator The crystalrsquos temper-ature is stabilized at its phase matching condition using Peltier elements (d) Schematicfor the squeezed-light generation After spatial filtering of continuous-wave laser lighttwo hemilithic standing-wave resonators are employed The first generates second har-monic pump light (SHG) The second (OPA) generates a squeezed vacuum field at theinitial wavelength The squeezed states are observed by a balanced homodyne detector(BHD) or alternatively sent and mode-matched to the optical mode of an interferometerbeforehand LO local oscillator DBS dichroic beam splitter

49

Fig 20 (a) and (b) show photographs of typical nonlinear crystals used forsqueezed-light generation at near infra-red wavelengths The crystals shownhere form a monolithic standing-wave squeezing resonator (a) or are partof a half-monolithic standing-wave squeezing cavity (c) shows a tempera-ture stabilized and mechanically stable housing of the squeezing resonator(d) shows a schematic of a full setup for the generation of squeezed vacuumstates of light for an application in a laser interferometer The only brightinput required for the squeezing resonator (OPA) is the second-harmonicpump field The resonator mode at fundamental frequency is thus initiallynot excited by photons ie it is in its ground state characterized by vacuumfluctuations due to the zero point energy see Fig 7 (c) [Gerry and Knight(2005)] The pump field spontaneously decays in the degenerate pair of sig-nal and idler fields The combined down-converted field leaving the resonatorexhibits quantum correlations which give rise to a squeezed photon countingnoise when overlapped with a bright coherent local oscillator beam Thedetection is done either in a balanced homodyne detector (BHD) or with asingle photo diode The squeeze factor increases the closer the pump powerof the squeezing resonator gets to the oscillation threshold and the lower theoptical loss on down-converted photon pairs is

441 High squeeze factors ndash minimizing decoherence

Squeezed states of light have significant impact on the sensitivity of laserinterferometers if large squeeze factors can be produced Squeezing of 3 dBimproves the signal-normalized quantum-noise spectral density by a factorof 2 This factor corresponds to doubling the (coherent state) light powercirculating inside the interferometer Squeezing of 10 dB corresponds to aten-fold power increase The experimentally demonstrated squeeze factorswere considerably improved in recent years [Takeno et al (2007) Vahlbruchet al (2008) Polzik (2008) Eberle et al (2010) Stefszky et al (2012)]culminating in a value of as large as 150 dB [Vahlbruch et al (2016)] Thisvalue corresponds to the same reduction of signal-normalized quantum noisethat is achieved by increasing the light power by a factor of 32 (At this pointit is already noted that squeezing the quantum noise can simultaneouslyreduce quantum measurement noise (shot noise) as well as quantum backaction noise (radiation pressure noise) This is not possible with scaling thelight power of coherent states see Subsec 55)

50

Ideally a parametric squeezed-light source can produce an infinite squeez-ing level see Fig 19 fundamentally just limited by the energy provided bythe pump field In practice the limit is set by decoherence mechanismsThe by far most important one is optical loss Optical loss occurs duringsqueezed-light generation its propagation through the interferometric setupincluding imperfect mode matchings and finally the photo-electric detec-tion Also detector dark noise [Schneider et al (1998)] phase noise [Takenoet al (2007)] and excess noise [Bowen et al (2002)] impair the observablesqueezing strength

Optical loss is usually understood as coupling the squeezed mode to a zerotemperature bath ie overlapping it with a vacuum mode For any amountof loss the resulting state is still squeezed But to be able to directly observesay 10 dB of squeezing the total loss on the state needs to be less than 10in this example cf Eq (16) To minimize optical loss the nonlinear crys-tal as well as lenses and beam splitters in the interferometric path need toshow very low absorption and scattering at the wavelength of the squeezedlight PPKTP shows absorption of about 10minus4cm and below at near-infraredwavelengths Low OH content fused silica is a suitable material for all otheroptics Absorptions of less than 10minus6cm were measured [Hild (2007)] Coat-ings on crystal surfaces and on all other optical components should also showlowest optical loss Total loss of the 10minus6 level are available today Superpol-ished surfaces which show roughnesses with less than 1 A root mean square(integrated over spatial scales from approximately 1 micron to 100 microns)and thus very low scattering are necessary to achieve these low numbersMinimizing the total number of optical components is essential From thisperspective a monolithic squeezing resonator as shown in Fig 20 (a) is theoptimum choice The squeezed mode needs to be matched to the mode ofthe laser interferometer or to the mode of the balanced homodyne detectorVisibilities of up to 998 have been achieved [Eberle et al (2010)] whichcorresponds to a loss of about 04 Of great importance also is the quantumefficiency of the photo-diodes used for detecting the squeezed field (togetherwith the interferometric signal) Recently a quantum efficiency of photo-diodes in a squeezing experiment of (995plusmn 05) was measured [Vahlbruchet al (2016)] To minimize photon loss the photo-diodes had no protectionwindow an anti-reflection coating on the semi-conductor material and theremaining reflection was re-focussed with an external mirror

Also the dark-noise spectral density of the detection electronics reducesthe observable squeezing and needs to be as low as possible Similar to optical

51

noise it also provides a contribution to the observed variance The dark noiseof the detection electronics needs to be much lower than the detected photoncounting noise In [Vahlbruch et al (2016)] it was 28 dB below shot noisebut still reduced the observable squeeze factor from 153 dB to 150 dB

Excess noise emerges if the squeezed mode couples to a nonzero tem-perature bath or to a mode whose excitation is strongly fluctuating (Thecoupling process can always be understood as a beam splitter coupling andis physically described by overlapping electric fields Coupling to a zero tem-perature bath leads to Eq (16)) The captured excess noise variance thenneeds to be added to the initial squeezing variance which deteriorates theobserved squeezing stronger than just mixing in the vacuum mode Excessnoise is less likely to occur at MHz frequencies but can be significant ataudio-band sideband frequencies and below and is thus a serious issue ingravitational-wave detectors [Chua et al (2014)] The reason for that isthat acoustically or thermally excited motions of surfaces produce frequencyshifts of back-scattered light mainly at these low frequencies [Vahlbruch et al(2007)]

Phase noise corresponds to stochastic phase fluctuations between thesqueezed field and the local oscillator within the measuring time It cor-responds to mixing the squeezed mode with itself with a fluctuating squeezeangle [Suzuki et al (2006) Franzen et al (2006)] Phase noise in squeez-ing experiments typically is less of an issue than optical loss [Dwyer et al(2013) Oelker et al (2016) Vahlbruch et al (2016)] The setuprsquos phasenoise can be reduced by making the squeezing resonator more compact andthus mechanically more stable against acoustic and thermal fluctuations ofthe environment and by improving the quality of the servo loops for cavitylength and propagation length controls Operating a squeezed-light resonatorin vacuum might also be beneficial The ability to run a high performancesqueezed-light generator in vacuum was demonstrated in [Wade et al (2015)]

442 Squeezing in the gravitational-wave (GW) detection band

High squeeze factors have been first demonstrated at sideband frequen-cies of a few MHz and above where excess noise is generally negligible whenworking with visible or near-infra-red light Today we know that extendingthe squeezing spectrum towards the audio-band and even below is technicallynot always easy but straight forward once a high squeeze factor is achieved

52

at MHz frequencies In most squeezing experiments the main laser light

Figure 21 Photograph of the GEO 600 squeezed-light source ndash The breadboarddimensions are 135 cm times 113 cm The squeezing resonator is high-lighted by the white ar-row and is set up as a standing-wave hemilithic cavity containing a plano-convex PPKTPcrystal of about 10 mm length (see also Fig 20 b) It is pumped with continuous-wave532 nm light that is produced by frequency doubling of light (at angular frequency ω)from a commercial NdYAG laser Two more laser fields at about 1064 nm having fre-quency offsets of more than 10 MHz with respect to ω(2π) serve as optical control fieldsBoth fields are mode-matched and injected into the squeezing resonator together with thesecond-harmonic pump field

at the squeezing wavelength is accompanied by significant noise up to thelaser relaxation oscillation For this reason laser control fields at the opticalcarrier-frequency in the optical path of the squeezed mode need to be avoided[Bowen et al (2002) Schnabel et al (2004) McKenzie et al (2004)] and thesqueezing resonator length and the optical path stabilized by other means[McKenzie et al (2005) Vahlbruch et al (2006)] Furthermore and mostimportantly excess noise due to back-scattering is an issue Back-scattering(also called lsquoparasitic interferencesrsquo) is produced if DC light scatters out ofthe optical path hits a vibrating surface and re-scatters back into the opticalpath [Vahlbruch et al (2007)] Significant back-scattering can be produced

53

in interferometers for the detection of gravitational waves since light powersof several hundreds of kilowatts are used Even back-scattering from the mil-liwatt local oscillator of balanced homodyne detectors is an issue at acousticsideband frequencies and below The recipe for avoiding parasitic interfer-ences turns out to be threefold (i) avoiding scattering by using ultra-cleansuperpolished optics with close to perfect anti-reflex coatings (ii) avoidingback-scattering by carefully blocking all residual (faint) light fields and (iii)reduce the vibrationally and thermally excited motion of all mechanical andoptical parts that could potentially act as a re-scattering surface with re-spect to the optical path [Vahlbruch et al (2007) McKenzie et al (2007)]The insights described above led to the first demonstration of audio-bandsqueezing at frequencies down to 200 Hz [McKenzie et al (2004)] and laterto the first demonstration of squeezing over the full gravitational-wave de-tection band even from as low as 1 Hz [Vahlbruch et al (2007)] Whilea standing-wave squeezing resonator [Ou et al (1992)] can be built in avery compact way that is rather insensitive against mechanical vibrations[Chelkowski et al (2007)] a traveling-wave bow-tie squeezing resonator [Wuet al (1986)] is more tolerant against back-scattered light [Chua et al (2011)]The strongest squeezing in the audio-band of up to 116 dB was reported inRef [Stefszky et al (2012)]

443 The first squeezed-light source for GW detection

The first squeezed-light source for the continuous operation in GW de-tectors had been designed and completed between 2008 and 2010 [Vahlbruch(2008) Vahlbruch et al (2010)] Since then this source has been producingsqueezed vacuum states in a fully phase controlled way using co-propagatingfrequency-shifted bright control beams [Vahlbruch et al (2006)] as an inte-gral part of the GW detector GEO 600 The source is a turn-key device witha fully automated re-lock system [Vahlbruch et al (2010) Khalaidovski et al(2012)] Re-locking is required if the temperature of the environment changessignificantly which drives the actuators outside their dynamic ranges

Up to 9 dB of squeezing over the entire GW detection band was observedusing a balanced homodyne detector (BHD) located close to the squeezingresonator The squeeze factor has been limited by optical loss due to ab-sorption in the PPKTP crystal transmission of the back-surface and thenon-perfect AR-coating of the crystalsrsquos intra-cavity surface The adjustableair gap has been necessary to allow for an easy way to apply length control

54

-12

-8

-4

0

4

8

12

16

20

10 100 1k 10k

Rel

ativ

e no

ise

pow

er [

dB]

Frequency [Hz]

shot noise (a)

squeezed noise (b)

anti-squeezed noise (c)

~ 9dB

Figure 22 Broadband squeezing spectrum ndash Noise power spectra measured on theoutput of the GEO 600 squeezed-light source shown in Fig 21 with a balanced homodynedetector The traces correspond to the spectra of quadrature amplitude variances normal-ized to vacuum noise The resolution bandwidth used increases towards higher frequenciesto reduce the measurement time (a) Shot noise normalized to unity which serves as thereference level (0 dB) (b) Squeezed noise covering the complete detection band of ground-based GW detectors (c) Anti-squeezed noise Peaks at 50 Hz and 100 Hz are the electricmains frequency and its first harmonic The data was published in Ref [Vahlbruch et al(2010)]

Additional optical loss in the path to the balanced homodyne detector mainlyarose due to a Faraday isolator that eliminated parasitic interferences Fi-nally the mode missmatch to the BHD as well as its non-perfect quantumefficiency provided additional loss Inferring the squeeze factor without theBHD detection loss more than 10 dB of squeezing are provided by the sourceSince 2010 it has been used in basically all observational runs of the GEO 600GW detector see Section 6

444 Generation of two-mode (bi-partite) squeezing

lsquoTwo-mode squeezed lightrsquo or lsquobi-partite squeezed lightrsquo is light that allowsfor joint measurements at two locations A and B These joint quadraturemeasurements reveal correlations and anti-correlations with a remaining un-certainty smaller than the ground-state uncertainty which certifies the pres-

55

ence of entanglement cf Subsec 34 Bi-partite squeezed light has been gen-erated by type I and by type II parametric down-conversion In case of type Ithe squeezed fields from two squeezing resonators as described in Subsec 44are overlapped on a balanced beam splitter with a 90 phase shift The twooutput fields together represent the entangled mode [Furusawa et al (1998)Bowen et al (2003c) Eberle et al (2013)] In case of type II signal and idlerfields are non-degenerate regarding polarisation and a single cavity contain-ing an appropriate crystal and a polarising beam splitter are sufficient for theproduction of bi-partite squeezing Also in this case the measurements ofthe quadrature amplitudes of signal and idler fields show large uncertaintiestogether with bi-partite correlations and anti-correlations that are strongerthan the ground state uncertainty of individual subsystems [Ou et al (1992)Villar et al (2006) Jing et al (2006)]

To date the strongest entanglement of bi-partite squeezed light has beenproduced based on type I parametric down-conversion [Eberle et al (2013)]The requirements of producing strong entanglement are identical to those ofproducing strong squeezing outlined above The strength of bi-partite en-tanglement can be given in decibels in full analogy to the squeeze factorPractically the strength of bi-partite squeezing is always somewhat smallerthan that of single party squeezing since it requires an additional mode-matching that results in additional optical loss

45 Conclusions

The first observation of squeezed light was achieved in 1985 [Slusher et al(1985)] Shortly after cavity-enhanced optical parametric amplification forsqueezed-light generation was demonstrated [Wu et al (1986)] which todayenables the observation of up to 15 dB of squeezing [Vahlbruch et al (2016)]Quite generally the maximum squeezing level that is observed does not de-pend on the strength of the optical nonlinearity Squeezing cavities can easilybe operated at their oscillation threshold where they should provide infinitesqueezing if decoherence is zero The main limiting factor is optical lossincluding that of the photo-electric detection

Dedicated experimental research and development towards a squeezed-light source for applications in gravitational-wave detectors can be tracedback to 2002 [McKenzie et al (2002) Bowen et al (2002)] Since then a sur-prising amount of progress has been made culminating in the first squeezed-light source specifically built for the integration into a gravitational-wave

56

detector For the future squeeze factors above 15 dB will certainly be possi-ble by further reducing optical loss This statement is supported by the highdegree of matching between experimental data and a theoretical loss modelas presented in Fig 3 of [Vahlbruch et al (2016)]

5 Quantum noise in laser interferometers

51 Interferometric measurements

The purpose of a laser interferometer is the precise measurement of smallchanges of an optical path length with respect to a reference path Forthis the interferometer transfers the change of the phase difference betweentwo light fields into an amplitude quadrature change of the interferometerrsquosoutput light The latter can easily be detected by a single photo diode Ofgeneral interest are differential changes of the optical path length that aremuch smaller than the laser wavelength ie differential phase changes thatare much smaller than 2π In this case the differential phase change can bedescribed in very good approximation as a differential change of the phasequadrature instead

In order to transfer the phase quadrature signal with minimum loss ahigh interference contrast at the interferometerrsquos beam splitter is essentialAdditionally instrumental noise in terms of unwanted excitations of the out-putrsquos amplitude quadrature needs to be reduced to a minimum Noise arisesdue to power fluctuations of the input laser light back-scattered laser lightinside the interferometer thermally driven displacements of mirror surfacesand in many more ways The important measure of the sensitivity of aninterferometer obviously is its signal-to-noise-ratio The most useful measureis given in terms of the noise spectral density S(f=Ω2π) that is normalizedto the physical unit of the signal S(f) is in fact a lsquonoise-to-signal-ratiorsquoand can be seen as the signal-normalized variance of the photo diode out-put decomposed into spectral components versus sideband frequency f withthe resolution bandwidth of 1 Hz As an example S(100 Hz) = 10minus39 m2Hzmeans that the instrumental noise in the one hertz band around 100 Hz equalsa signal that would be produced if the mirror of one interferometer arm os-cillates with an amplitude of just

radic10 middot 10minus20 m in the very same band Such

small spectral densities are achieved by gravitational-wave detectors [Abbott(2016)]

57

52 Quantum measurement noise and shot noise

The most fundamental noise source in laser interferometers is due to thequantum noise of light which is in fact two-fold [Caves et al (1980)] Firstof all there is lsquoquantum measurement noisersquo which arises in the processof photo-electric detection For coherent states the quantum measurementnoise is the lsquophoton counting noisersquo of mutually independent photons andusually simply called lsquoshot-noisersquo Fig 2 (b i) shows a time series of suchnoise hiding the actual signal The frequency components of the shot noiseare well described by the quantum uncertainty of the output fieldrsquos amplitudequadratures XΩ∆Ω see Subsec 22 (Recall this quantity corresponds to the

differential phase quadrature YΩ∆Ω of the light beams in the interferometerarms) The photon counting noise has a white Fourier spectrum howeverthe lsquoshot noisersquo of an interferometer is usually normalized to the signal whosetransfer function is usually not white for instance due to the presence of armcavities or a signal-recycling cavity

All current and planned gravitational-wave detectors are Michelson-typelaser interferometers with operating points very close to a dark fringe Thelight power in the output port is just a couple of tens of mW which canbe handled by a single photo diode In this configuration the signal-to-shot-noise-ratio is actually maximized which can be shown in three steps [Bachorand Ralph (2004)] For the first step we use plane waves to describe theelectric field in the output port of a Michelson interferometer For perfectinterference contrast at the balanced beam splitter ie for perfect modematching and for defining φ = 0 as the dark port condition we get

Eout(t φ) =1

2E0 sin(ωt+ φ)minus 1

2E0 sin(ωt) (31)

where E0 is the amplitude of the total internal field whose two parts hasaccumulated a differential phase It directly follows for the squared fields

E2out(t φ) =

(sin

φ

2

)2

(E0 cos(ωt+ φ2))2 (32)

We now turn to a light beam with a localized transversal mode that can befocussed onto a photo-electric detector The photo diode has perfect quantumefficiency ie the rate of photo electrons is not only proportional to the rateof output field photons but also has a unity slope efficiency Since the optical

58

frequency is too high to be resolved we consider the averaged light power

P out(φ) =

(sin

φ

2

)2

P (33)

The next step is a formulation of the signal being the derivative of detectedphoton number versus phase Let n be the average value of the photonnumber per measuring time interval Eq (33) can then be rewritten as

nout(φ) =

(sin

φ

2

)2

n (34)

rArr dnout(φ) = n sinφ

2cos

φ

2dφ (35)

The final step is the calculation of the signal-to-shot-noise-ratio Shot noiserefers to coherent states which have a standard deviation of the photonnumber of σ(n) =

radicn

dnout(φ)

σ(nout)=n sinφ

2cosφ

2dφ

radicn sinφ

2

(36)

and find for a signal-to-noise ratio of unity for coherent states and for anon-zero but still small phase difference ∆φCoh 2π

1 =radicn cos

φ

2∆φCoh with φ 6= 0 (37)

In this equation the smallest measurable phase difference is given for φrarr 0

∆φCohmin =

1radicn (38)

This is the well-known shot-noise limit of high-precision phase sensing ∆φCohmin

is the smallest phase shift that can be measured with a signal-to-noise ra-tio of one when using n mutually independent photons per measuring time(those of a coherent state) when the loss of photons is assumed to be zeroThe typical purpose of a laser interferometer is the continuous sensing (moni-toring) of a continuously changing phase An illustrative example is the phasesignal produced by the black hole merger measured by Advanced LIGO onSept 14 in 2015 (Fig 1 in [Abbott (2016)]) The measuring interval should

59

be short to be able to resolve the time-evolution of the signal Generally themeasurement of an arbitrary signal that lasts for a finite time thus needs tobe understood as l subsequent measurement intervals using n photons eachIt can be shown that Eq (38) is also valid for interferometers operated athalf fringe ie when each output port contains the same light power In thiscase photo diodes need to be placed in both output ports and the actualsignal is provided by their difference voltage

Due to its importance the shot-noise limit deserves some remarksThe phase φ in Eq (31) is the phase difference of two mode-matched fieldsand might be accumulated by a single pass along the length L such as ina Mach-Zehnder interferometer or in a double pass such as in a (simple)Michelson interferometer or in four passes as realized in a Michelson inter-ferometer with folded arms [Grote (2005)] The shot-noise limit in Eq (38)and its scaling therefore holds independent of the number of passes Theclaim in Ref [Higgins et al (2007)] that the scaling according to Eq (38) canbe surpassed by multiple passes is not justifiedThe fact that Eq (38) is derived by approaching φ rarr 0 correctly describesthe actual operation point of gravitational-wave detectors which is close tobut not exactly at a dark port In practice a tiny offset from dark port ischosen at which the shot noise is well above the photo diodersquos electronic darknoiseEq (38) solely depends on the number of quanta but not on the lightrsquos wave-length λ Of course the shot-noise limit for the change of an optical pathlength ∆L does depend on the wavelength and ∆φCoh

min needs to be replacedby ∆φCoh

min = 2π∆LCohminλ

Finally an essential result of the shot-noise limit is that the ideal precisemeasurement should use lsquoas much quanta as possible per measuring inter-valrsquo which translates to lsquoas much light power in the interferometer armsas possiblersquo Eq (38) is indeed the one and only reason why gravitational-wave detectors use high power lasers power-recycling and arm resonatorsExtending the measuring time for a given light power can also improve thesensitivity but only if the signal repeats ie is periodic Let us assume thatone period of the signal is resolved by l intervals using n photons each Inthis case repeating the overall measurement k times improves Eq (38) by1radick The fundamental statement of Eq (38) however does not change

since the actual photon number n may then simply incorporate the factor k

60

For a given average photon number the shot-noise limit in Eq (38) canonly be surpassed by using photons that are quantum correlated ie by usingnonclassical states of light How is the shot-noise limit surpassed with thehelp of squeezed states A Michelson interferometer that is operated closeto a dark fringe acts like an almost perfect mirror for both input ports Allthe input light is back-reflected towards the laser source This also accountsfor the quantum uncertainty of the input light The quantum uncertaintythat impinges onto the photo diode thus (mainly) enters the interferometerthrough its (almost) dark port An interferometer that uses displaced co-herent states entering from one port can thus be improved by replacing theordinary vacuum entering the signal output port by a squeezed vacuum stateThis was the proposal by CM Caves in 1981 [Caves (1981)] which is labeledhere with lsquoCSVrsquo If the differential phase quadrature of the interferometer issqueezed Eq (38) then within the limit of large coherent state displacementα sinh2r improves to

∆φCSVmin asymp

eminusrradicn (39)

(The above expression is an approximation since the squeezing operationproduces a small number of photons that are not accounted for here) Ofcourse the mode of the squeezed vacuum needs to be precisely matchedto the mode of the interferometer The first experimental demonstrations ofsqueezed phase measurements used a Mach-Zehnder [Xiao et al (1987)] and apolarization interferometer [Grangier et al (1987)] Fig 2 shows how spatialdegeneracy between an externally generated squeezed mode and the signalmode in a Michelson interferometer is achieved using a polarizing beam split-ter and a Faraday rotator Again the limit in Eq (39) can only be achievedif optical loss is zero Optical loss not only reduces the signal but here alsoreduces the squeeze parameter see Eqs (15) and (16)

Let us consider an example The sensitivity of a laser interferometer thatuses coherent states with an excitation of 1023 photons per second can beimproved by a factor of

radic10 by either adding 09 middot 1024 photons per second

or by adding about just 2 photons per second and bandwidth in hertz thatbelong to the 10 dB squeezed vacuum confer Eq (30) Since the full sig-nal band of ground-based GW detectors covers sideband frequencies up to10 kHz just 2 middot 104 photons per second are necessary At a wavelength ofλ = 1064 nm these values correspond to a power increase by 168 kW and

61

37 fW respectively

The question arises whether a scaling of the sensitivity better than propradic1n is possible It was theoretically shown that in principle the scaling

can indeed considerably be improved yielding the so-called Heisenberg limitor Heisenberg scaling [Bondurant and Shapiro (1984) Yurke et al (1986)Braunstein (1992) Holland and Burnett (1993)]

∆φHLmin prop

1

n (40)

The Heisenberg scaling requires nonclassical states that have a certain num-ber of quanta similar to Fock states ie n = n The theoretically optimalstates describe a superposition of n (N) indistinguishable photons in oneinterferometer arm while having zero (0) photons in the second arm and viceversa and were named lsquoN00Nrsquo-states [Dowling (2008)] A specific propertyof these states is lsquosuper-resolutionrsquo The output ports of the interferometershow an n-times faster oscillation of the interference fringes when changingthe phase between the two interferometer arms Super-resolution correspondsto an n-times improved signal transfer function and was demonstrated forinstance in Refs [Rarity et al (1990) Kuzmich and Mandel (1998) Mitchellet al (2004) Afek et al (2010)] The presence of this nonclassical phe-nomenon however does not prove a sensitivity better than the semi-classicalbound according to Eq (38) Sensitivity is rather related to the signal-to-noise-ratio and needs to take into account all imperfections as well as theprobability of a successful detection of the sensing state [Thomas-Peter et al(2011)] All experiments so far used post-selection on particular measure-ment outcomes and neglected the typically large probability that nothingwas detected

Super-resolution was demonstrated with up to n = 5 [Afek et al (2010)]In addition to the fact that super-resolution does not prove a sensitivity bet-ter than the semi-classical bound photon numbers in state-of-the-art super-resolution experiments are extremely small compared to the photon numberof about 1023 (within a measuring interval of one second) of coherent statesused in Ref [Abbott (2016)] and of about 1022 using coherent states plussqueezed vacuum states used in Ref [Abadie (2011)]( which did prove a sen-sitivity better than the semi-classical bound)

Another interesting and related question is what the smallest phase is

62

that can be estimated in a single measurement again using a given numberof quanta Taking into account that no prior information about the phaseshift exists still a scaling proportional to 1n is possible In the limit of largen however an additional factor of π is required in the nominator of Eq (40)[Sanders and Milburn (1995) Berry and Wiseman (2000)] yielding

∆φHLmin asymp πn (41)

The state that can actually achieve this bound is different from the N00Nstate and was found in [Summy and Pegg (1990) Luis and Perina (1996)Berry and Wiseman (2000)] A N00N state is not the optimum state forphase estimation (via a single measurement) since it only provides one bit ofinformation A recent review on generell aspects on phase measurements isgiven by Ref [Demkowicz-Dobrzanski et al (2015)]

It is important to note that Eqs (38-40) do not consider photon lossExperiments that demonstrated super-resolution and aimed for achievingthe scaling in Eq (40) were conditioned on zero photon loss Let η gt 0 bethe average efficiency of detecting (all) photons Eq (38) then reads

∆φηCohmin =

radic1

η n (42)

Eq (39) turns into

∆φηCSVmin asymp

radicηeminus2r + 1minus η

η n(43)

and Eq (40) turns into [Demkowicz-Dobrzanski et al (2012 2013)]

∆φηmin =

radic1minus ηη n

for 0 lt η lt 1 (44)

For non-zero photon loss most interestingly the ultimate sensitivity of aphase measurement for a given photon number also shows a 1

radicn -scaling

The difference between the CSV strategy of using bright coherent states incombination with squeezed vacuum states which is bounded by Eq (43) andthe strategy of using the optimal nonclassical state which is bounded byEq (44) is marginal in practice [Demkowicz-Dobrzanski et al (2013)] Forgravitational-wave detectors and for any other laser interferometer using in-tense light there is no need for an alternative to the CSV strategy

63

We now turn back to the shot noise according to Eq (38) Generally noisecan be decomposed into its spectral contributions For a simple Michelsoninterferometer without arm resonators and without a signal-recycling cavitythe square-root of the single-sided shot-noise spectral density normalized tothe differential arm length change x in units of m

radicHz is given by [Saulson

(1994)] radicSMI

SNx =

radic~c2

2ωPprop 1radic

P (45)

where ω is the optical angular frequency of the quasi-monochromatic carrierlight and P the total light power in both arms including the built-ups fromcavities In combination with a squeezed vacuum whose relative phase gen-erates squeezing of the output lightrsquos amplitude quadrature the right handside reduces according to the factor eminusr Note that the single-sided spectraldensity is only defined for positive sideband frequencies and thus twice aslarge as the double-sided spectral density

The spectral density of the measurement of a GW induced strain is givenby the same expression but normalized to h = xL (If the gravitational waveis oriented in an optimal way with respect to the Michelson interferometerone arm is squeezed while the other is expanded by the same amount of ∆L =x2 and h then corresponds to the actual gravitational-wave amplitude)The square-root of the single-sided shot-noise spectral density normalized tostrain in units 1

radicHz is given by

radicSMI

SNh =

radic~c2

2L2ωP (46)

Equations (45) and (46) show that the smallest measurable signal (corre-sponding to unity signal-to-shot-noisendashratio) is inversely proportional to thesquare root of the laser power and has a white spectrum for sideband frequen-cies much smaller than the carrier frequency see horizontal line in Fig 23

All first- and second-generation GW detectors use power-recycling andadditional cavities to improve their sensitivities Fabry-Perot arm resonatorsdo not only increase the light power but additionally also increase the signalfor signal frequencies inside the resonator linewidth For lossless Fabry-Perot

64

arm resonators the spectral densities in Equations (45) and (46) need to bemultiplied by the following factor [Kimble et al (2001)]

HFP =

radicL2(γ2

FP + Ω2)

c2 (47)

where γFP = cTFP(4L) is the Fabry-Perot arm resonatorrsquos half bandwidthand TFP is the light power transmission of the input mirror The end mirrorsare assumed to have perfect reflectivity A similar expression can be derivedfor describing the improvement due to signal-recycling [Buonanno and Chen(2001)]

In summary shorter laser wavelengths higher light powers and squeezingof the amplitude quadrature of the interferometer output reduce shot noise ina broadband way ie for all signal frequencies Fabry-Perot arm resonatorsas well as signal-recycling provide improvements mainly for frequencies insidethe resonator linewidths

53 Quantum back-action and quantum radiation pressure noise

In laser interferometers quantum back-action noise results from the un-certainty of the lightrsquos radiation pressure force on the interferometer mirrorsand is also called lsquo(quantum) radiation pressure noisersquo (RPN) Its origin isthe quantum uncertainty of the differential amplitude quadrature XΩ∆Ω ofthe fields in the interferometer arms It results in an uncertain momentumtransfer to the mirrors and thus in an position uncertainty of the mirrors atfuture times with respect to their differential mode of motion [Caves et al(1980)] The physical mechanism of radiation pressure corresponds to anintensity dependent phase shift [Pace et al (1993)]The higher the light power in the arms of a laser interferometer the loweris its shot-noise spectral density see Eq (45) Unfortunately the spectraldensity of quantum back-action noise increases with light power The single-sided force noise spectral density reads

radicSRPNF =

radic8~ωPc2

(48)

Whereas the force noise of the quantum radiation pressure has a white spec-trum the RPN does not since the mirrorrsquos reaction to external periodicforces depends on frequency The link between the Fourier component of an

65

external force F (Ω) and the Fourier component of the displacement x(Ω) isgiven be the mechanical susceptibility HM It reads for an harmonic oscillatorwith mass M

HM(Ω) =1

M | minus Ω2 + Ω2M + iΩΩMQ|

(49)

where ΩM is the oscillatorrsquos resonance frequency and Q its quality factorThe square root of the single-sided spectral density of the RPN normalized

to the displacement of an harmonic oscillator with mass M is then given by

radicSRPNx = HM(Ω)

radic8~ωPc2

(50)

In GW detectors the test mass mirrors are suspended as pendula with highmechanical Q-factors and their centre of mass motion corresponds to that ofa harmonic oscillator The resonance frequencies of the pendula are lowerthan the detection band of interest The mechanical susceptibility is thereforeoften approximated for the so-called free-mass regime as H fm

M (Ω) = (mΩ2)minus1The square root of the single-sided spectral density of the RPN normalizedto differential displacement of two mirrors with each of mass M in a simpleMichelson interferometer is given by [Saulson (1994)]radic

SfmMIRPNx =

radic2~ωPc2m2Ω4

propradicP (Ω ΩM) (51)

where m = M2 is the mirrorsrsquo reduced mass In case of a simple Michel-son interferometer that is enhanced with arm cavities the spectral density inEq (51) needs to be multiplied with the expression given in Eq (47) In com-bination with a squeezed vacuum whose relative phase generates squeezing ofthe output lightrsquos phase quadrature the right hand side reduces according tothe factor eminusr Note if the radiation pressure noise is squeezed the shot noisemust be anti-squeezed or vice versa The radiation pressure noise calibratedto strain of space time is given by the right side of Eq (51) divided by theinterferometer arm length LIn summary heavier masses longer laser wavelengths lower light powersand squeezing of the amplitude quadrature in the interferometer arms re-duce radiation pressure noise

66

1 10 100 1000Frequency [Hz]

Radiation pressure noise

Shot noise

SQL

10-21

10-18

10-15

10-12

radicSx

[mradic

Hz]

ndashndash

ndashndash

Figure 23 Displacement-normalized quantum noise spectral densities ndash Consid-ered is a simple Michelson interferometer with neither arm cavities nor signal recyclingThe two end mirrors (m = 100 g) of the interferometer arms are suspended as pendulahaving a resonance frequency of ΩM2π = 1 Hz and a Q-factor of 107 The interferometeruses quasi-monochromatic light (in coherent states) with a total power of 4 kW Opticalloss and the offset from a dark output fringe is assumed to be negligible Wavelengthλ = 1550 nm The standard quantum limit (SQL) corresponds to the lowest noise achiev-able at a given sideband frequency when varying the light power without using quantumcorrelations

54 Interferometer total quantum noise and the standard quantum limit

Both shot noise and radiation pressure noise contribute to the total quan-tum noise of a given interferometer If they are not quantum correlatedwhich is the case for a conventional Michelson interferometer when detectingthe output lightrsquos amplitude quadrature their variances add up (The resultis not shown in Fig 23) It can easily be deduced from the previous sectionsthat changing the laser power will shift the two quantum noise contributionsHowever the total quantum noise never goes below the standard quantumlimit (SQL) [Braginsky and Manukin (1967)]

Let us consider Fig 23 for sideband frequencies much greater than thependulum resonance Here the test mass mirrors react as free masses whenexerted to external forces The SQL in this free-mass regime is calculated

67

by minimizing the sum of the squares of Eqs (45) and (51) [Saulson (1994)]

Sfmtotx =

~c2

2ω

[1

P+

4ω2

c4m2Ω4P

] (52)

Its derivative reads

dSfmtotx

dP=minus1

P 2+

4ω2

c4m2Ω4 (53)

Setting the above equation to zero provides the optimum laser power versussideband frequency in order to achieve the lowest total quantum noise

P fmopt =

c2mΩ2

2ω (54)

Inserting the optimal light power into Eq (57) provides the square root ofthe single-sided noise spectral density of the free-mass SQL in m

radicHzradic

SfmSQLx =

radic2~mΩ2

(55)

Again m is the reduced mass and dividing by the interferometer arm lengthL yields normalization to the GW-induced strain h Eq (55) shows that theSQL falls off with sideband frequency The corresponding equation for aMichelson interferometer that uses arm cavities readsradic

SfmFPSQLx =

radic~

mΩ2

(1

HFP

+HFP

) (56)

with HFP according to Eq (47)Using the expression for the SQL the square root of the total quantum

noise spectral density of a Michelson interferometer in the free-mass approx-imation can be written asradic

SfmFPtotx =

radicSfmFP

SQLx

2

[1

k+ k

] (57)

with the radiation pressure coupling parameter

k(Ω) =2ωP

mc2Ω2 (58)

68

For a fixed light power and fixed reduced mass of the mirrors the quantumnoise limited interferometer reaches the SQL when k = 1 which is realizedat the angular sideband frequency ΩSQL =

radic2ωP(mc2)

Note that neither squeezing the phase quadrature nor squeezing the am-plitude quadrature of the interferometer light leads to sub-SQL performance[Caves (1981)] also confer Ref [Schnabel (2005)] Fig 3 (left) As we willsee in the next sections the standard quantum limit can be surpassed if shotand radiation pressure noise are correlated Then the total quantum noise isnot given by the sum of the variances ie the sum of the squares in Eqs (45)and (51)

55 Squeezed light for surpassing the standard quantum limit

A measurement with sensitivity better than the standard quantum limit(SQL) is also called a lsquoquantum non-demolition (QND)rsquo measurement [Bra-ginsky and Khalili (1995 1996) Kimble et al (2001)] Several QND tech-niques for laser interferometers were proposed in recent decades [Jaekel andReynaud (1990) Kimble et al (2001) Purdue and Chen (2002) Chen (2003)McClelland et al (2011) Danilishin and Khalili (2012) Graf et al (2014)]What they have all in common is they exploit quantum correlations betweenobservable uncertainties

Arguably the most extensive way of introducing quantum correlationsand surpassing the SQL is the injection of squeezed states of light [Jaekeland Reynaud (1990)] If the squeezed quadrature angle of the injected statesis neither 0 nor 90 the quantum uncertainties of the amplitude and phasequadrature amplitudes that describe the differential field in the two interfe-rometer arms become correlatedLet us consider a very simplified setup that just consists of a quasi-mono-chromatic light field that is back-reflected from a quasi-free mirror Thelight power and the mass be such that reflected light in a coherent stateresults in a measurement of the mirror position with a noise spectral den-sity at the SQL at sideband angular frequency ΩSQL At this frequencyquantum measurement noise and back-action noise are of the same sizeie the uncertainty in X produces an equally large additional uncertaintyin Y Upon reflection the quadrature amplitude variances change from∆2X = ∆2Y = 14 to 2∆2X = ∆2Y = 12 This result corresponds tothe situation in Fig 23 at the crossing frequency of shot noise and radiation

69

pressure noise The coupling of the uncertainty variances can be describedby the matrix K = (1 minusk 0 1) where k = 1 at the SQL If the modulationstate at ΩSQL is the ground state its variances are transferred according to

KT

(1 00 1

)K =

(1 0minus1 1

)(1 00 1

)(1 minus10 1

)=

(1 minus1minus1 2

) (59)

In accordance with Fig 23 the variance of YΩSQLis twice as large as the vac-

uum noise varianceNow let the quantum noise of the light field be 10 dB squeezed at 45

(Eq (19)) The projection of the quantum uncertainty onto the X-observableproduces the radiation pressure noise by being transferred with the couplingfactor k = 1 at the SQL into the Y -observable in fact with negative signsince a larger value of X produces a larger optical path length and thus aretardation of the phase Due to the squeezing at 45 the initial uncertaintyin Y cancels with the additional uncertainty that originates from the onein X The following calculation shows that the strength of the cancellationcorresponds to the initial squeezing strength Upon reflection the quantumuncertainties transform in the following way(

1 0minus1 1

)(505 495495 505

)(1 minus10 1

)=

(505 minus01minus01 02

) (60)

The state of light after reflection has a squeezed phase quadrature amplitudeThe improvement in comparison to Eq (59) is exactly 10 dB The quantumnoise improvement corresponds to the input squeeze factor and is also a mea-sure by what factor the SQL is surpassed Squeezed vacuum injection thusallows surpassing the SQL upon measuring the conventional Y -quadrature(which is realized by a single photo diode in the interferometerrsquos output port)as first realized by [Unruh (1983) Yuen (1983) Jaekel and Reynaud (1990)]

In the example above the input squeeze angle is optimized for a sin-gle sideband frequency Injecting a broadband squeezed vacuum field withfrequency-independent squeeze angle of 45 would result in a rather bad inter-ferometer quantum-noise performance at frequencies far smaller or largerthan ΩSQL Fig 24 shows the quantum-noise performance if the input fieldhas squeeze angles that are optimized for every k(Ω) as given in Eq (58)Injected squeezing can thus lead to a broadband sub-SQL performance ifthe quantum measurement noise (shot noise) and the quantum back-actionnoise (radiation pressure noise) are correlated in an optimal way Due to the

70

10 50 100 500 1000 5000Frequency [Hz]

Dis

plac

emen

t noi

se sp

ectra

l den

sity

[mradic

Hz]

10-20

10-19

10-18

Total quantum noise 0dB

Total quantum noise -10dB

SQL

Y

X

Y

X

Y

X

Figure 24 Surpassing the SQL with squeezed-light injection ndash At shot-noise lim-ited sideband frequencies squeezing of the Y -quadrature amplitude improves the noisespectral density of the interferometer At radiation-pressure-noise limited sideband fre-quencies squeezing of the X-quadrature amplitude improves the noise spectral density ofthe interferometer If both kinds of quantum noise contribute equally (at the SQL markedwith a dot) a squeeze angle of 45 results in surpassing the SQL by the full squeeze factorsee Eq (60) In the graph here the squeeze angle is optimized for all frequencies result-ing in a broadband quantum noise reduction [Jaekel and Reynaud (1990)] Measurementsensitivities beyond the SQL (shaded area) are in the so-called quantum non-demolition(QND) regime [Kimble et al (2001)] Dashed horizontal lines represent the (squeezed)shot noise Dashed straight lines with negative slope represent the (squeezed) radiationpressure noise The calculations use 10 dB of squeezing a conventional Michelson inter-ferometer with neither arm resonators nor signal recycling a light power at the beamsplitter of 1 MW at a wavelength of λ = 1550 nm and mirror masses of 1 kg

correlation shot noise and radiation pressure noise can be squeezed simulta-neously

71

Light with a frequency-dependent squeeze angle

The discovery that shot noise and radiation pressure noise can be squeezedsimultaneously and thus a broadband reduction of quantum noise beyond theSQL be achieved required the insight that the spectral analysis of light definesa spectrum of many lsquosideband modulation modesrsquo that all can be in differ-ent quantum states An ordinary squeezing resonator which is on resonancefor light at twice the pump wavelength produces a spectrum of modulationmodes that all have the same squeeze angle A frequency-dependent squeezeangle can be introduced by reflecting such a field from a detuned single-ended filter cavity which was suggested by Kimble and coworkers [Kimbleet al (2001)] They showed that the optimal frequency dependence thatleads to the broadband improvement shown in Fig 24 can be realized byusing altogether two filter cavities as shown in Fig 25 Motivated by this re-sult research and development on filter cavities for optimizing the frequencydependence of broadband squeezed fields has been very active in recent years[Corbitt et al (2004) Chelkowski et al (2005) Dwyer et al (2013) Kweeet al (2014) Straniero et al (2015) Oelker et al (2016)]

Photo diode Squeezed vacuum

Faraday Rotator

Coherent light

YΩΔΩ

Figure 25 Frequency dependent squeezing injection ndash A broadband squeezed fieldwith a frequency-dependent squeeze angle that is optimal for gravitational-wave detectorsis produced by reflecting off an ordinary broadband squeezed field from two detuned opticalfilters [Kimble et al (2001)]

A light field with a frequency-dependent squeeze angle was first demon-strated in Ref [Chelkowski et al (2005)] see Figs 26 and 27 The experi-

72

Detuned filter cavityLockingphoto diode

Isolator

EOM R asymp 1

SHG

EOM

DBSOPA

LaserIsolator

Mode cleaner

LO

Homodyne detector

Figure 26 Generation of a frequency-dependent squeezing ndash A frequency-dependent orientation of the squeeze ellipse was first demonstrated in Ref [Chelkowskiet al (2005)] Initially a conventional spectrum of squeezed vacuum states of light was gen-erated in a squeezing resonator (lsquoOPArsquo) The squeezed vacuum was transmitted throughan optical isolator to a detuned filter cavity After reflection the squeezed vacuum stateswere absorbed in a balanced homodyne detector (BHD) The phase of the BHDrsquos localoscillator (LO) was changed for quantum state tomography of the squeezed states in diffe-rent regions of the spectrum The result showed a frequency-dependent orientation of thesqueeze ellipse see Fig 27 SHG second harmonic generation EOM electro-optical mod-ulator for applying phase modulation sidebands for cavity length control DBS dichroicbeam splitter R mirror reflectivity λ4 quarter wave plate for turning linear polarizedlight into circular polarised light and vice versa

ment consisted of a standing-wave squeezing resonator which produced ans-polarized broadband amplitude quadrature squeezed field accompanied bya dim continuous-wave DC control field with a wavelength of λ = 2πcω =1064 nm The squeeze bandwidth covered sideband frequencies up to aboutΩ(2π) = 30 MHz which corresponded to the linewidth of the squeezing res-onator The optical cavity for producing the frequency dependence of thesqueeze angle was a standing-wave cavity composed of a plane incouplingmirror of reflectivity r1 =

radic097 and a concave end mirror of reflectivity

r2 =radic

09995 The cavity length was L = 50 cm resulting in a linewidthof 147 MHz The squeezed field first passed a Faraday isolator to preventinterference effects between the filter cavity and the squeezing resonator A

73

Phase quadrature

min

max

-2 -1 0 1 2

141 MHz

-2

-1

0

1

2

Am

plit

ud

e q

uad

ratu

re

Figure 27 Frequency-dependent squeezing ndash Picture top right Reconstructed con-tour plot of the Wigner function of the sideband modulation at Ω(2π) = 141 MHz afterreflection from a 1515 MHz detuned filter cavity The state shows quantum correlationsbetween phase and amplitude quadratures ie squeezing at an angle of here about 40The white circle visualizes the standard deviation of the vacuum state uncertainty Thewhite ellipse represents the standard deviation of the squeezed uncertainty Small picturesMeasurement results on the same continuous-wave laser beam at various sideband frequen-cies around 15 MHz For each tomographic picture noise histograms of 100 equidistantquadrature angles were measured In each case the laser beam was phase locked to a ref-erence beam and the quadrature angle stably controlled and stepwise rotated The phasereference was given by a phase modulation at 198 MHz [Chelkowski et al (2005)] Thepicture was first published in Ref [Schnabel (2005)] (copyright ccopy2007 by Imperial CollegePress)

λ4-waveplate turned the s-polarized field into a circularly polarized beamwhich was then mode matched into the detuned cavity The retro-reflectedfield was analyzed by a balanced homodyne detector (BHD) for quantumstate tomography The filter cavity was electro-optically controlled to be de-tuned by 1515 MHz with respect to the DC control field The cavity length

74

control was achieved by the Pound-Drever-Hall (PDH) locking techniqueutilizing a circularly polarized laser beam that carried 15 MHz phase mod-ulation sidebands and was coupled into the filter cavity from the back Theoutput voltage of the BHD was characterized by a spectrum analyser as wellas used to perform quantum state tomography In the latter case the BHDoutput voltage was mixed down with an electronic local oscillator at differentradio-frequencies around 15 MHz and low-pass filtered to set the resolutionbandwidth to ∆Ω(2π) = 100 kHz The final electric signal corresponds to atime series of quadrature amplitude measurements XθΩi∆Ω Quantum statetomography is a method to reconstruct the phase space quasi-probability dis-tribution (Wigner function) of quadrature amplitudes from sets of measuredXθΩ∆Ω distributions when varying the angle θ [U Leonhardt (1997)] Forevery sideband frequency Ωi 100000 quadrature values were measured di-vided up on 100 equidistant quadrature angles Each quadrature angle wasstably controlled with a precision of plusmn1 Fig 27 shows the reconstructedWigner functions which were all measured on the same laser beam but atdifferent sideband frequencies For these measurements the detuned filtercavity was locked to the lower sideband at minus1515 MHz The result clearlyshows the frequency-dependent orientation of the squeeze ellipse In a morerecent experiment a frequency-dependent squeeze angle was also realized inthe kHz regime [Oelker et al (2016)]

56 Optomechanically induced (ponderomotive) squeezing

The radiation pressure of light when acting on a movable mirror re-sults in an intensity dependent phase shift [Pace et al (1993)] The cou-pling produces a so-called lsquoponderomotive effectrsquo [Braginsky and Manukin(1967)] which is of third order optical nonlinearity and which transforms abright coherent state inside an interferometer into a squeezed state of light[Vyatchanin and Matsko (1993)] This type of squeezed-light generationis usually called lsquoponderomotive squeezingrsquo or lsquooptomechanical squeezingrsquoConsequently even if no squeezed field is injected into the interferometercorrelations between the quadrature amplitudes are generated that allow forsurpassing the SQLPonderomotive squeezing as produced by the interferometer itself can onlybe exploited for evading back-action (radiation pressure noise) It can notbe used to squeeze the interferometer shot noise This is why ponderomotivesqueezing is fundamentally less extensive than injecting externally producedsqueezed states of light [Corbitt et al (2006)] suggested an external pon-

75

deromotive squeezing source for gravitational-wave detectors In this casedue to its external generation also the interferometerrsquos shot noise can besqueezed Recently ponderomotive squeezing was observed for the first time[Brooks et al (2012) Purdy et al (2013)] The achieved squeeze factors aremuch smaller than those produced by optical-parametric down-conversion[Vahlbruch et al (2016)]

Let us have a look again at Eq (59) Rotating the covariance matrix onthe right by arctan(minus

radic54minus 12) asymp minus58 indeed reveals squeezing

(cos 58 minussin 58

sin 58 cos 58

)(1 minus1minus1 2

)(cos 58 sin 58

minussin 58 cos 58

)asymp(

262 00 038

) (61)

The vacuum-noise normalized variance of 038 corresponds to about 42 dBof ponderomotive squeezing This is the general value that is produced atthe angular sideband frequency ΩSQL At higher frequencies the squeezefactor gets smaller at lower frequencies higher The squeezing strength of42 dB can be observed if the photo diode in the interferometer output portis replaced by a balanced homodyne detector using a local oscillator phase ofabout minus58 It can be shown however that the optimal signal-to-quantum-noise-ratio at the SQL is achieved for a local oscillator phase of exactly 45At this angle back-action is fully evaded

Full evasion of radiation pressure noise at all frequencies requires an op-timized frequency dependence of the relative local oscillator phase This canbe achieved by reflecting off the interferometer output field from two detunedfilter cavities [Kimble et al (2001)] The scheme was called lsquovariational out-putrsquo In the case of zero optical loss this scheme can fully evade radiationpressure noise just leaving the shot noise as the only quantum noise contri-bution

The variational-output scheme can be used to enhance the frequency-dependent squeezed input scheme The right site of Eq (60) shows thatthe output statersquos squeezing is not optimally detected in the Y -quadratureRather than with a single photo diode the detection should be done witha balanced homodyne detector with optimized phase of its local oscillatorIn this case the output lightrsquos quantum noise is solely given by squeezedshot noise The total quantum noise in Fig 24 would then be given by thelowest (dashed) horizontal line This combined scheme was called lsquosqueezed

76

variationalrsquo [Kimble et al (2001)] It can be realized by reflecting off theinterferometer output light from in total two optical filter cavities placed infront of the balanced homodyne detector

57 Conclusions

The highest quantum-noise-limited sensitivities of high-precision laserinterferometers are achieved by employing a large number of quanta tomaximize the signal strength in combination with strongly squeezed statesto minimize the quantum noise From this perspective it is clear that thequantum-noise-limited sensitivity of future gravitational-wave detectors willbe further improved ndash by increasing the light power and the squeeze factorTo be able to do so the optical loss in these devices needs to be reduced

In principle the optical loss in laser interferometers can be made smallbut never zero Recent theoretical research has shown that for any non-zero loss the sensitivity scales proportional to 1

radicn at best where n is the

average photon number per measurement This scaling is efficiently achievedby combining strongly displaced coherent states with squeezed vacuum statesof light

If a repeated measurement is not only limited by quantum measurementnoise but also by quantum back-action noise squeezed states of light can beused to simultaneously reduce both ie in the case of an interferometer shotnoise and radiation pressure noise

6 The first application of squeezed light in an operating gravita-tional-wave detector

Squeezed states of light have been successfully used to improve the sensi-tivity of the gravitational-wave detector GEO 600 from 2010 up to the pointwhen this Review was written [Abadie (2011) Grote et al (2013)] Afterdecades of proof-of-principle experiments [Xiao et al (1987) Grangier et al(1987) McKenzie et al (2002 2004) Vahlbruch et al (2005 2006 20072008) Goda et al (2008)] the implementation of a squeezed-light source inGEO 600 has resulted in the first sensitivity improvement beyond shot noiseof a measurement device that targets new observations in nature The im-plementation of squeezed states in GEO 600 was not done to provide anotherproof-of-principle demonstration but was realized because it offered a rela-tively cheap way of further improving the measurement sensitivity Of course

77

the sensitivity of GEO 600 can also be further increased by purely classicalmeans however the implementation of arm resonators to enable higher lightpowers without increasing the thermal load on the beam splitter or eventhe realization of longer interferometer arms are much more expensive Inthis respect the sensitivity improvement of GEO 600 with squeezed light canarguably be regarded as the first lsquotruersquo application that developed out of thefield of lsquononclassical (quantum) metrologyrsquo (Note that the term rsquoquantummetrologyrsquo is currently defined in different ways [Giovannetti et al (2006)Gobel and Siegner (2015)] and the term rsquononclassicalrsquo referring to a non-classical P-function gives a distinct description)

61 Gravitational waves

Einsteinrsquos General Theory of Relativity [Einstein (1916)] or simply lsquoGen-eral Relativityrsquo (GR) predicts that accelerating mass distributions producegravitational radiation analogous to electromagnetic radiation from accel-erating charges Experimental evidence of their existence is given by theobservation of the slow spiraling together of two neutron stars caused bythe loss of orbital energy to gravitational waves The inspiral rate exactlymatches the predictions of Einsteinrsquos theory [Weisberg and Taylor (2005)]Recently Advanced LIGO observed gravitational waves for the first time[Abbott (2016)] thereby giving the go-ahead for gravitational-wave astron-omy The gravitational-wave source was the final inspiraling and the mergerof two black holes 13 billion light years away from earth

Gravitational-waves evolve in the far field of the source propagate withthe speed of light and are measurable on earth with laser interferometersFig 28 displays a gravitational wave propagating along a certain directionGravitational waves are dynamical changes of space-time They are transver-sal and quadrupolar in nature and have two polarization states

A variety of known astrophysical and cosmological sources are predictedto emit gravitational radiation that should reach the Earth with a measur-able strength [Sathyaprakash and Schutz (2009)] The first gravitational waveevent detected was produced by two black holes of 36 and 29 solar massesDuring the final 02 seconds of their inspiraling they produced a peak gravi-tational strain in our solar system of 10minus21 covering frequencies up to 250 Hz[Abbott (2016)] Other predicted sources are mergers of neutron stars supernovae and background signals from the Big Bang According to GR GWsfrom complex astrophysical sources carry a plethora of information that willhave a major impact on gravitational physics astrophysics and cosmology

78

L +Δ L

Binary system

fBS

Figure 28 Space-time oscillation ndash Gravitational waves are dynamical deformations ofspace-time that form in the plane perpendicular to the direction of wave propagation Asa result distances between free-falling test masses in a transverse plane will change witha strain h = ∆LL For black hole or neutron star binary systems with orbital frequencyfBS distances will oscillate at frequency fGW = 2fBS The wavelength of this oscillation isgiven by λGW = cfGW where c is the speed of light The wave of orthogonal polarizationwith respect to the one shown is rotated by 45 around the propagation axis

62 Interferometric detection of gravitational waves

Current gravitational wave detectors are kilometre-scale laser interfero-meters [Dooley et al (2016) Aasi (2015) Acernese (2015) Aso et al (2013)]Continuous-wave laser light is split into two beams traveling in orthogonaldirections Both beams are reflected back towards the central beam splitterwhere they interfere Gravitational waves change the optical path lengthdifference and thus the light power directed towards the photo-diode thatis positioned in the signal output port of the beam splitter A gravitationalwave at frequency fGW = ΩGW(2π) reveals itself as a light-power modula-tion at the same frequency The spectral decomposition of the output signalis described by a spectrum of the quadrature amplitude YΩ∆Ω introduced inSec 3 It corresponds to the amplitude quadrature amplitude of the outputlight and relates to the differential phase quadrature of the interferometerarms

79

Photo diode

Squeezed vacuum

Faraday Rotator

Signal

Quantum noise

Coherent light XΩΔΩ

YΩΔΩ

(a) (b)

YΩΔΩFigure 29 Squeezed-light-enhanced interferometric measurement ndash (a) Michelsoninterferometer with arm cavities power recycling and signal recycling (see main text forexplanation) The interferometer is operated close to a dark fringe such that the quantumnoise entering from the dark port is back-reflected The squeezed field is mode-matched tothe signal output field (b) Phase space diagram of the gravitational-wave signal outputat sideband frequency Ω(2π) The quantum noise is squeezed below the ground stateuncertainty and thus the signal to quantum noise ratio improved

The first key ingredient of an interferometric gravitational-wave detectorare suspended heavy mirrors that can be regarded as quasi-free in the di-rection of laser light propagation thereby acting as test masses that probespacetime Being on ground current detectors are located in rather noisy en-vironments that allow the realization of undisturbed quasi-free mirrors onlyabove a sideband frequency of the order of 10 Hz Since sufficiently strongGW signals are expected up to a frequency of 10 kHz todayrsquos gravitationalwave detectors target at signals in the acoustic band from 10 Hz to 10 kHzThe quasi-free motion of the test mass mirrors in this frequency regime isachieved by suspending the mirrors as sophisticated multiple-stage pendulain vacuum chambers [Aasi (2015)] Far above the pendularsquos resonant frequen-cies which are typically around 1 Hz the centre of masses of the mirrors areisolated from vibrations of the ground and they react on frequency compo-nents of small external forces approximately as free masses The mirrors andtheir suspensions are built from materials having exquisitely high mechani-cal quality factors This helps to concentrate the thermal energy that causes

80

displacements of the mirror surface into well-defined vibrational frequencymodes At these particular very sharp frequencies no gravitational wavescan be detected

The second key ingredient of an interferometric gravitational-wave de-tector is laser light with a power of up to hundreds of kilowatts or evenmegawatts The light is quasi-monochromatic and needs to show very lowamplitude (quadrature) noise and phase (quadrature) noise at sideband fre-quencies within the detection band Low amplitude noise is necessary toprovide a shot noise limited output field It is also necessary to avoid classi-cal radiation pressure noise which becomes an issue if the light power or themirror masses in the two arms are not identical Low phase noise is requiredif the storage time of the light in the two arms is not identical This mightaccidentally occur due to different linewidths of the arm cavities or mightbe part of the interferometer design to allow for the length control schemeproposed by Schnupp [Heinzel et al (1998)] To maximize the light power in-side the interferometerrsquos cavities it should be produced in an almost perfecttransversal spatial distribution of a Gaussian TEM00 mode

Light sources of gravitational-wave detectors are ultra-stable NdYAGmaster-slave systems that provide up to 200 W of light at 1064 nm [Winkel-mann et al (2011) Kwee et al (2012)] The high power in the interferometerarms is achieved by cavity built-ups in the so-called power-recycling cavityand in the arm cavities Power recycling uses a partially reflective mirror thatis located between the light source and the interferometer beam splitter Itssurface is matched to the lightrsquos wave front and forms an optical cavity to-gether with the rest of the interferometer Since gravitational-wave detectorsare operated close to a dark fringe large power built-ups can be achievedThe highest power built-up is achieved for a mirror transmission equal to the(given) interferometer round trip loss In this case an impedance-matchedcavity is achieved The power-recycling cavity as well as the arm cavitiesare stabilized on resonance for the input light The difference between theirfunctionality is that the power-recycling cavity does not limit the detectionbandwidth of the interferometer GEO 600 as well as Advanced LIGO em-ploy a third type of cavity the so-called signal-recycling cavity Similarlyto power recycling a partially reflecting mirror that is placed between theoutput port of the beam splitter and the photodiode is used to resonantly en-hance the GW signal [Meers (1988)] The signal-recycling cavity resonantlyenhances the signal modulation fields within its linewidth without furtherenhancing the carrier light power In combination with low linewidth Fabry-

81

Perot arm resonators it can also be used to extract the signal by reducing theeffective finesse of the arm resonators for the signal sidebands This schemeis called resonant sideband extraction [Heinzel et al (1996)] The signal-recycling cavity has also been tested in a detuned setting in which just theupper or lower sideband is extracted or resonantly enhanced respectively[Heinzel et al (2002)] Current gravitational-wave detectors however usecarrier-tuned signal recycling

All these techniques are lsquoclassicalrsquo approaches for maximizing the signal-to-shot-noise ratio At frequencies above a few hundred Hertz howevershot-noise is still the limiting noise source in gravitational-wave detectorsFuture gravitational-wave detectors will therefore use even higher light pow-ers but further increasing the light power becomes more and more challeng-ing Optical materials with less light absorption need to be found to coun-teract an increasing thermal load inside the mirrors Mirror masses need tobe further increased to counteract the increasing radiation pressure noiseNonclassical approaches are superior and become more and more attractivethe farther classical approaches are pushed to the extremes Nonclassicalapproaches allow for simultaneously increasing the signal-to-shot-noise ratioand the signal-to-radiation-pressure-noise ratio without changing light poweror mirror masses see Fig 24 They also allow for a complete evasion of radia-tion pressure noise [Braginsky and Khalili (1995 1996) Kimble et al (2001)]see Subsec 56

63 Squeezed-light enhancement of the gravitational-wave detector GEO 600

In 2010 GEO 600 was equipped with the squeezed-light source shownin Fig 18 The location of the squeezed-light source close to the outputport is shown in Fig 30 It was known that GEO 600 was shot-noise limitedat sideband frequencies above about 700 Hz In this frequency regime thereplacement of the ordinary vacuum states that entered the interferometerfrom the output port by a spectrum of squeezed vacuum states was expectedto reduce the noise spectral density into the nonclassical regime It was notprecisely clear what squeezing factors could be expected since the opticalloss upon mode-matching an external field into the output port propagationalong the arms and the final photo-electric detection was not determined

Critical components were the quantum efficiency of the photo-diode aswell as the optical loss of Faraday rotator for overlapping the squeezed fieldwith the output mode Also the transversal output mode of GEO 600 was an

82

issue since it contained significant contributions from higher order modesto which a good mode matching of the squeezed mode was not possible Theimplementation of the squeezed-light source thus was accompanied with anew custom made InGaAs photo-diode with 3 mm diameter The goal was aquantum efficiency of greater 99 [Vahlbruch et al (2016)] Also the Fara-day rotator was custom-made and optimized for lowest optical loss which

Figure 30 GEO 600 ndash View into the central building of the British-German GW detectorlocated close to Hannover Germany The vacuum chambers contain the suspended beamsplitter power- and signal recycling mirrors additional input and output optics as well asmirrors to realize a double pass of the laser light through the 600 m long interferometerarms By courtesy of the AEI

83

involved a rather precise rotation of the polarisation of 45 plusmn 05 over anaperture of more than 15 mm Finally a ring cavity (output mode cleaner)was placed in front of the photo-diode which acted as a passive filter forhigher transversal modes Since GEO 600 was not limited by radiation pres-sure noise and since it used a carrier-tuned signal-recycling cavity a frequencyindependent orientation of the squeezing angle was optimum After several

100 200 300 500 700 1000 2000 3000 500010-22

10-21

10-20

Obs

erva

tory

noi

se c

alib

rate

d to

str

ain

[1radic

Hz]

10-19

10-18

10-17

Noise calibrated to test m

ass displacement [m

radicHz]

Sideband frequency f

Figure 31 Nonclassical reduction of the GEO 600 instrumental noise ndash Shownare the square roots of the noise spectral densities without squeezed vacuum states (top)and with squeezed vacuum states (bottom) [Abadie (2011)] Their calibrations [Affeldtet al (2014)] to gravitational-wave strain and differential arm length change are shownon the left and right y-axes respectively Note that both traces increase towards higherfrequencies due to the frequency-dependent signal enhancement of the signal-recyclingcavity The injection of squeezed vacuum states leads to a broadband noise reductionof up to 35 dB at shot-noise limited frequencies The spectral features are for instancecaused by excited violin modes of the mirror suspensions (600ndash700 Hz and harmonics)Data by courtesy of the AEI

months the combination of the squeezed-light source and the gravitational-wave detector succeeded The main laser of the squeezed-light source wasphase locked to the main laser of GEO 600 and a stable mode matching be-

84

tween the squeezed field and the interferometer output field was achieved[Abadie (2011)] The effect on GEO 600rsquos (strain normalized) noise spectraldensity is shown in Fig 31 At frequencies above about 700 Hz the squareroot of noise spectral density was reduced by up to one third This value cor-responds to a quantum noise variance of 045 (minus35 dB) with the shot noisevariance normalized to unity For isotropically distributed gravitational-wavesources this factor produces a detection rate increase by a factor of 153 asymp 34After its integration into GEO 600 the squeezed-light source was used in allscientific runs seeking for gravitational waves for instance in the observa-tional run S6eVSR4 that was undertaken from June 3rd to September 5th

in 2011 [Grote et al (2013)]Towards the end of 2011 right before the start of the detector upgrade

to Advanced LIGO a nonclassical sensitivity improvement was also demon-strated in one of the LIGO detectors [Aasi (2013)] In this experiment anonclassical sensitivity improvement corresponding to up to 215 dB abovefrequencies of about 150 Hz was achieved The successful test is a strongmotivation for a squeezed-light upgrade of Advanced LIGO Note that thedesign of Advanced LIGO was completed in 1999 and squeezed-light sourceswere not mature at those times

In the past years the squeezing enhanced GEO 600 detector was not onlyused for observations but also was the control of the injected squeezed modefurther improved Stabilizing the overlap between squeezed mode and brightmode of the interferometer to close to perfect is necessary to reduce theeffective optical loss and to maximize the measurable squeezing factor Re-cently superior methods for stabilizing the longitudinal phase of squeezedvacuum mode were found [Dooley et al (2015)] and the first automatic align-ment system for stabilizing and optimizing the transversal mode overlap wasdemonstrated [Schreiber et al (2016)]

64 Are squeezed states the optimal nonclassical resource ingravitational-wave detectors

For a given number of photons Eq (41) quotes the ultimately smallestphase change that can be measured with a signal-to-noise-ratio of one Thescaling with number of photons per measuring time of this Heisenberg limitseems appealing compared to the scaling achievable with coherent states orsqueezed states according to Eqs (38) and (39) respectively The Heisenberglimit however is only valid for precisely zero photon loss Since the non-classical states required to achieve Eqs (40) and (41) show an exponentially

85

increasing sensitiveness to loss when increasing the photon number the ac-tual scaling can not be deduced from Eq (40) Proposals to use Fock statesand the so-called N00N states for optimizing interferometer sensitivities [Hol-land and Burnett (1993) Dowling (1998) Mitchell et al (2004) Afek et al(2010)] are thus only applicable when the experiment is conditioned on zerophoton loss As discussed in recent publications the correct expression forthe fundamental sensitivity limit needs to consider not only the total photonnumber inside the interferometer but also the total photon loss [Dorner et al(2009) Ko lodynski and Demkowicz-Dobrzanski (2010) Knysh et al (2011)Escher et al (2011) Demkowicz-Dobrzanski et al (2012)]

Based on these earlier works Ref [Demkowicz-Dobrzanski et al (2013)]proved that the nonclassical sensitivity enhancement of GEO 600 reportedin Ref [Abadie (2011)] has been exceedingly close to fundamental quantuminterferometry bound under given energy constraints and photon loss lev-els More than that it was generally proven that the approach of com-bining displaced coherent states and squeezed vacuum states is optimal forgravitational-wave detectors

In Ref [Abadie (2011)] the gravitational-wave detector GEO 600 used aneffective number of photons per second of approximately n = 2 middot 1022 whichcorresponded to a total optical power inside the interferometer arms of P asymp37 kW at a wavelength of 1064 nm The total optical loss was 1minus η asymp 038The injected squeezing factor was eminus2r asymp 01 For these numbers the ratioof Eqs (43) and (44) is calculated to

∆φCSVmin

∆φgenmin

asymp

radicηeminus2r + 1minus η

1minus ηasymp 108 (62)

which is a good approximation within the limit of large coherent state dis-placements α sinh2r The quantum noise of GEO 600 including thesqueezed-light source was just 8 above the fundamental quantum inter-ferometry bound An increased squeezing strength of 16 dB (eminus2r asymp 0025)which is in reach would bring the approach based on coherent states andsqueezed vacuum states to within just 2 above the fundamental bound

Future GW detectors will have significantly reduced optical loss values(1 minus η) lsquoLossrsquo includes scattering and absorption at mirrors non-perfectfringe contrasts and the non-perfect quantum efficiency of the photo detec-tor Optical loss reduction is important for at least four reasons First itleads to an increased signal second it leads to a reduced quantum noise when

86

employing squeezed states third less absorption reduces the thermal load onthe test mass mirrors and fourth less scattering reduces the probability ofback-scattered light which produces disturbance signals [Billing et al (1979)Vahlbruch et al (2007) Punturo et al (2014)] The higher the finesse valuesof the arm and signal-recycling cavities are the more significant is opticalloss at mirror test masses the beam splitter and the signal-recycling mirrorThe finesse value of the power-recycling cavity and the loss of mirrors andlenses that guide the output field to the photo-diode are less critical Suitablephoto detectors of 995 quantum efficiency are available today [Vahlbruchet al (2016)] but achieving a total optical loss of 10 is still challengingThe reason for that is that first of all a measurement device aiming for bestabsolute sensitivity should use as much quanta (photons) as possible Highfinesse values for the enhancement cavities are thus essential but results inan unavoidable scaling-up of the effect of mirror losses A realistic exampleof future gravitational wave detectors thus considers η = 09 with a squeezingfactor of 20 dB (eminus2r = 001) In this case the quantum noise will be about4 above the ultimate fundamental bound for a given photon number

From Eq (62) it can be concluded that there is no need for any moresophisticated nonclassical states than squeezed states In particular non-classical states with a defined photon number such as N00N states are notrequired Within the approximation quoted this result is independent ofthe photon number This result is also independent of the physical systemused for interferometric phase estimation and can also be made for quantum-enhanced atomic clock calibration in the presence of dephasing Here the-oretical results indicate that the precision of Ramsey interferometry withspin-squeezed states is close to the optimal one in the asymptotic regime ofa large number of atoms [Huelga et al (1997) Ulam-Orgikh and Kitagawa(2001) Escher et al (2011)] as already stated in Ref [Demkowicz-Dobrzanskiet al (2013)] More sophisticated nonclassical states with fixed number ofn quanta might still be useful for the exceptional case when the absorptionof one quantum already results in zero measurement sensitivity anyway Anexample is an ensemble measurement where the absorption of a single photondemolishes the source of the phase change to be characterized A typicallyused approach of conditioning the measurement result on n clicks of n singlephoton counters conditions on precisely zero loss and is thus able to use theadvantage of Eq (40) over Eq (39)

87

65 Conclusions

Squeezed states of light will contribute to realizing gravitational-wave ob-servatories with much higher sensitivities than existing or planned ones Tobenefit from squeezed states in a most efficient way optical loss in terms ofabsorption and scattering must be minimized In particular the optical lossof mirror coatings and mirror substrates need to be minimized The rele-vant mirrors include the test masses the balanced beam splitter the signalrecyclingextraction mirror and all optical components between the latterand the photo diode Excellent spatial mode matching between the brightinterferometer field and the squeezed vacuum field is also of great impor-tance Achieving this requires further improvement of the surface figures ofall reflective optical components of the interferometer as well as improvedhomogeneity of all optical components that the light passes through

The quantum noise reduction achieved in a gravitational-wave detectoris of course always smaller than the highest squeeze factor provided by thesqueezed-light source As an example let us consider the observation of 15 dBof nonclassical noise suppression directly at the source If the squeezed fieldsenses an additional loss of 5 when propagating through the interferometerwhich is a very challenging number from todayrsquos point of view the remainingsqueezing level is about 11 dB see Eq (16)

7 The application of 2-mode-squeezed light in laser interferome-ters

71 Quantum Dense Metrology

At first glance the application of bi-partite (two-mode) squeezed statesto a device whose goal is measuring a single observable seems meaninglessSqueezing the uncertainty of that observable should be the optimum one cando This is indeed true when concerning just quantum noise but recentlyit was discovered that in the presence of classical disturbances bi-partitesqueezing can improve such measuring devices [Steinlechner et al (2013)]The concept was named quantum dense metrology (QDM) The potentialimprovement of a gravitational-wave detector with bi-partite squeezed statesis shown in Fig 32 (a) A description is given in the caption The pre-condition for a potential improvement can be best understood within a phasespace diagram Fig 32 (b) contains two different kinds of lsquosignalsrsquo The firstis the actual signal which always shows up as a phase space displacement

88

along the Y axis The second is a disturbance signal that can produce adisplacement in arbitrary direction in phase space A prominent example

Faraday Rotator

Coherent light

Photo detectors Balanced homodyne detectors

Squeezed vacuum

Squeezed vacuum

Quantum noise in

Y (A) and X (B)

XΩΔΩ(B)

YΩΔΩ(A)

(a) (b)

Disturbance projected

onto Y

and X

XΩΔΩ(B)

YΩΔΩ(A)

Example of (unknown) disturbance

Signal

Figure 32 Bi-partite-squeezed-light-enhanced measurement ndash (a) Setup for theapplication of bi-partite (two-mode) squeezed light in a laser interferometer on the basisof QDM Two squeezed vacuum fields are overlapped on a balanced beam splitter withprogrammable squeeze angles for instance with a relative angle of 90 which produces abi-partite state as shown in Fig 12 The beam splitter outputs are entangled for any rela-tive angle greater than zero One part is matched to the interferometer mode The secondpart is kept outside as a reference beam The interference of the interferometer outputand the reference beam is arranged with such a phase difference that it reproduces the twosqueezed inputs on the photo detectors The two squeezed beams are photo-electricallydetected measuring the respective squeezed quadrature (using balanced homodyne detec-tors) Both beams carry half of all interferometer induced modulations which includesignals as well as disturbances A single readout as shown in Fig 21 cannot distinguishbetween the two kinds The double readout shown here provides additional informationand allows for recognition of the disturbance [Steinlechner et al (2013)] as well as in prin-ciple a modeling of the disturbance and with a correct model an improvement of thenoise spectral density of the interferometer [Ast et al (2016)] (b) Phase space diagramdescribing phase quadrature readout A as well as amplitude quadrature readout B Bothshow squeezed quantum noise The amplitude quadrature readout does not contain anygravitational-wave signal ie any feature in this channel must be due to disturbancesThis information can be used to improve the interferometer

89

for such a disturbance is parasitic interference due to back-scattered laserlight [Vahlbruch et al (2007)] Back-scattering is a limiting noise at lowsignal frequencies of gravitational-wave detectors [Billing et al (1979) Vinetet al (1997) Hild (2007) Ottaway et al (2012) Punturo et al (2014)] Notethat all noise that couples in via unwanted motions of the test mass mirrorsso-called lsquodisplacement noisersquo always produces a phase space displacementalong the Y axis and cannot be tackled with QDM

Fig 33 shows measurement results obtained in Ref [Steinlechner et al(2013)] In a table-top experiment one part of a bi-partite squeezed state ofa continuous-wave mode at 1064 nm was mode-matched into the output portof a Michelson laser interferometer operated at its dark fringe in full analogyto Fig 32(a) A lsquosignalrsquo was produced by driving the piezo behind one of theend mirrors at a frequency of 555 MHz The lsquodisturbancersquo was introducedby re-injecting a small amount of light that leaked through the second endmirror with an additional piezo-mounted mirror The piezo was driven ata frequency of 517 MHz to produce a phase modulation An additionalDC voltage defined an arbitrary and unknown optical path length of thelight before being re-injected and as such the phase space orientation of thedisturbance signal This mechanism of a parasitic interference is realizednaturally in any interferometric device due to back-scattering of quanta frommoving surfaces in the environment

The interferometer output consisted of the signal as well as the distur-bance with a quantum uncertainty given by one subsystem of the bi-partiteentanglement It was overlapped with the second subsystem of the entan-gled state on a balanced beam splitter and the two outputs were analysedwith balanced homodyne detectors The phases of the bi-partite entangle-ment and the BHD local oscillators were controlled to resemble Fig 32(b)ie both BHDs measured a squeezed uncertainty regardless of the phase ofthe (generally unknown) disturbance

The beam splitter that combines interferometer output and the entangledreference beam unavoidably splits the signal as well as the disturbance intotwo paths For a balanced beam splitter this generally reduces the signaland disturbance power by 3 dB for both quadrature measurements Fig 33shows however that both BHDs performed about 6 dB below shot noisewhich demonstrates the usefulness of the scheme The squeeze factor can inprinciple be infinite which thus qualifies the lsquo3 dB penaltyrsquo In the abovefigure the additional information from the second BHD output was used torecognize the parasitic interference in the first BHD output providing a lsquovetorsquo

90

50 51 52 53 54 55 56 57

50 51 52 53 54 55 56 57-70

-75

-80

-85

-90

-70

-75

-80

-85

-90

Y (A) (Ω2π)Ω∆Ω

Frequency ( f )

Noi

se p

ower

(dB

m)

Noi

se p

ower

(dB

m)

X (B) (Ω2π)Ω∆Ω

~ ndash6 dB

~ ndash6 dB

Shot noise reference

Shot noise reference

Signal Projected disturbance

Projected disturbance

Vet

o

No signal

Figure 33 Bi-partite-squeezed-light-enhanced measurement ndash The result wasachieved in a table-top setup [Steinlechner et al (2013)] In the two panels the lower

(blue) traces show the squeezed quadrature noise-power spectra ∆2Y(A)Ω∆Ω(Ω2π) (top) and

∆2X(B)Ω∆Ω(Ω2π) (bottom) as simultaneously measured with balanced homodyne detectors

lsquoArsquo and lsquoBrsquo respectively The conventional Y -measurement (top) cannot distinguish be-tween signal and disturbances The additional X-measurement (bottom) does not detectany phase quadrature signal thus any feature in this measurement is a parasitic signal dueto a disturbance The respective projection onto the Y -measurement can thus be lsquovetoedrsquoIn a more sophisticated approach the X-data might be used to model and then to elim-inate the disturbance as well as its projections on both quadrature measurements Theresult is a reduced spectral density of the actual phase quadrature measurement [Ast et al(2016)] Traces shown here are slightly sloped due to the decreasing transfer functions ofthe balanced homodyne detectors The resolution bandwidth was ∆Ω(2π) = 10 kHz thevideo bandwidth was 100 Hz All traces were averaged three times

signal to trigger its removal from the data streamThe question arose whether the additional information can be used to

reduce the actual noise spectral density of the first measurement ie to re-cover signals that were buried by parasitic interferences Very recently it wasshown that this is indeed possible Ref [Ast et al (2016)] reports a table-top proof-of-principle experiment in which the additional information in the

91

QDM approach could be used for improving the sensitivity of an interfero-meter The measurement sensitivity was improved from above-shot-noise tosub-shot-noise (sub-Poissonian) performance This result was possible notbecause the way the parasitic interference arose was known but because theadditional information provided by QDM allowed for fitting a model of theexcess noise to the readout data

Quantum dense metrology (QDM) as shown in Fig 32 improves a mea-surement by simultaneously reading out two conjugate observables Bothreadout observables show a squeezed quantum noise and act as estimatorsof independent physical quantities This situation was recently described aslsquoquantum-mechanics freersquo [Tsang and Caves (2012)] QDM is based on anEinstein-Podolsky-Rosen (EPR) entangled [Einstein et al (1935)] bi-partitesystem as described in Subsec 34 EPR entanglement was previously con-sidered for the quantum-informational task of dense coding which doublesthe capacity of quantum communication channels [Bennett et al (1992)Braunstein and Kimble (2000)] The application of EPR entanglement inmetrology was first proposed by DrsquoAriano et al [DrsquoAriano et al (2001)]

72 Conclusions

A single beam that carries an optimized spectrum of squeezed vacuumstates and that is injected into the interferometerrsquos dark port provides themost efficient and practically optimal approach to reduce the quantum noisein laser interferometers by means of nonclassical states (see Section 5) Theconclusion of the section here is that two entangled beams provide a superiorapproach if the interferometerrsquos sensitivity is limited by classical noise thatis not exclusively restricted to the actual observable which is the phasequadrature amplitude Y Parasitic interferences due to laser light that isbackscattered from vibrating surfaces are an example Current gravitational-wave detectors use light fluxes of about 1024 photons per second [Abbott(2016)] Just a single photon per second and hertz which leaves the mainlight beam and is backscattered from a vibrating surface and in this way getsfrequency shifted into the detection band produces a significant disturbancesignal The lsquoquantum-dense metrologyrsquo approach might provide a powerfultechnique to tackle this problem

Very recently it turned out that QDM is not the only technique that mayexploit EPR entanglement to improve phase measurements Ref [Ma et al(2017)] proposes to use EPR entanglement to simultaneously suppress shotnoise and radiation pressure noise in a gravitational-wave detector without

92

the need for an additional filter cavity (confer subsection 55) In this caseEPR entanglement is exploited that is carried by one broadband squeezedbeam and that is present between quadrature amplitudes defined with re-spect to different optical frequencies ω and ωprime as investigated in Ref [Hageet al (2010)] Such lsquofrequency multiplexedrsquo EPR entanglement might resultin considerably lower costs of building a gravitational-wave detector with abroadband simultaneous squeezing of shot noise and radiation pressure noiseAlso this proposal does not lead to a fundamentally lower quantum noise butrather improves on classical aspects of an interferometer

8 Summary and Outlook

In many cases experiments that involve interference of quantum statescan be described in a semi-classical way This description uses the classicalwave picture for the interference part of the experiment and subsequently theclassical particle picture when the states transfer their energy to a detectoror more generally to a thermal bath This semi-classical description is notpossible when using the specific class of lsquononclassicalrsquo states Squeezed statesof light are a prominent example of these Squeezed states and other nonclas-sical states allow for observations that made Einstein Podolsky and Rosenformulate their critical and seminal paper on quantum theory [Einstein et al(1935)]

In the review here it is argued that after many successful proof-of-principle experiments with nonclassical states in the past decades the routineuse of squeezed-light in observational runs of the gravitational-wave detectorGEO 600 goes beyond proof-of-principle and is a true application of nonclassi-cal light Since 2010 the squeezed-light source has improved the measurementsensitivity of GEO 600 in basically every observational run [Abadie (2011)Grote et al (2013) Dooley et al (2016)] At quantum noise limited frequen-cies ie above a few hundreds of hertz the sensitivity has been improvedcorresponding to a squeezing strength in the noise spectral density of up to37 dB which corresponds to an increase of the average gravitational-wavedetection rate by a factor of 043minus32 = 36 This success is a strong motiva-tion to also equip the Advanced LIGO Virgo and Kagra gravitational-wavedetectors with squeezed light Similar improvement factors even down tolower signal frequencies are expected [LSC (2013)] The achievable improve-ment factors are mainly limited by the optical loss on the squeezed states

93

and much higher factors are achievable in principleUp to now squeezed states have not been used to reduce the radiation-

pressure noise in gravitational-wave detectors The reason is that so far othernoise sources are larger than radiation pressure noise and such an effect can-not be observed It is expected however that future gravitational-wavedetectors will eventually be partly limited by radiation pressure noise Fromthis point on squeezed light will be used to simultaneously reduce shot noiseand radiation pressure noiseSqueezed states are the optimum nonclassical states for gravitational-wavedetectors or more generally for all laser interferometers operating with largeaverage photon numbers per measuring interval [Demkowicz-Dobrzanski et al(2013)] In addition to using higher light powers and heavier test mass mir-rors higher squeeze factors will thus contribute to mitigate the lightrsquos quan-tum noise in laser interferometers

Two-mode (bi-partite) squeezed light has not been used in gravitational-wave detectors so far They are not capable of further reducing the quantumnoise in laser interferometers but they can be used to mitigate classicalnoise that originates from fluctuating phase space displacements A well-known such noise source is back-scattered light Proof-of-principle experi-ments were performed recently [Steinlechner et al (2013) Ast et al (2016)]This new technique could turn out to be valuable in next generations ofgravitational-wave detectors in particular in those targeting high sensitivi-ties at low sub-audio signal frequencies and using high light powers Suchan implementation in gravitational-wave detectors does not require any newtechnology Compared to a squeezed-light enhanced interferometer just asecond squeezed-light source is required

It is certainly remarkable that those quantum states that made EinsteinPodolsky and Rosen falsely think quantum theory incomplete are now ex-ploited as new technologies in measurement devices targeting new observa-tions in nature

Acknowledgements

RS thanks M Ast J Bauchrowitz C Baune S Chelkowski J DiGugliel-mo A Franzen B Hage J Harms A Khalaidovski L Kleybolte NLastzka M Mehmet S Steinlechner and H Vahlbruch for their contri-butions many fruitful discussions and their support with the figures and JFiurasek for many valuable comments on the manuscript Thanks are also

94

due to Y Chen F Khalili and H Miao for fruitful discussions within thequantum noise working group of the LIGO Scientific Collaboration (LSC)Special thanks are due to H Vahlbruch and H Grote together with theGEO 600 team for their pioneering work on the squeezed-light implementa-tion in GEO 600 RS is supported by the Deutsche Forschungsgemeinschaft(Grant No SCHN 757-6) and by the European Research Council (ERC)project lsquoMassQrsquo (Grant No 339897)

References

References

Aasi J et al Jul 2013 Enhanced sensitivity of the LIGO gravitationalwave detector by using squeezed states of light Nature Photonics 7 (8)613ndash619URL httpwwwnaturecomdoifinder101038nphoton2013177

Aasi J et al Apr 2015 Advanced LIGO Classical and Quantum Gravity32 (7) 074001URL httparxivorgabs14114547httpstacks

ioporg0264-938132i=7a=074001key=crossref

20895763c84bce3f8929251031b2475c

Abadie J et al Sep 2011 A gravitational wave observatory operatingbeyond the quantum shot-noise limit Nature Physics 7 (12) 962ndash965URL httparxivorgabs11092295httpwwwnaturecom

doifinder101038nphys2083

Abbott B P et al Feb 2016 Observation of gravitational waves from abinary black hole merger Phys Rev Lett 116 061102URL httplinkapsorgdoi101103PhysRevLett116061102

Acernese F et al 2015 Advanced virgo a second-generation interferome-tric gravitational wave detector Classical and Quantum Gravity 32 (2)024001URL httpstacksioporg0264-938132i=2a=024001

95

If the coherent state is highly excited and thus the average number ofphotons n per detection interval is large the Poissonian distribution canbe approximated by a Gaussian distribution with a standard deviation ofplusmnradicn During the past decades squeezed states of light have attracted a lot

of attention because they can exhibit less quantum noise than a coherent stateof the same coherent excitation ie they can show sub-Poissonian countingstatistic see Fig 1

θ

Shot noise Squeezed noise

(a) (b)

Brightlaser input

Squeezedvacuum input

Faradayrotator

Photo diode

Michelsoninterferometer

Signal5050

(i)Ph

oto

curr

ent [

rel

units

]

Time [ms]

(ii)

0 5 10

Figure 2 Squeezed-light enhanced Michelson interferometer ndash (a) In addition tothe conventional operation of a Michelson laser interferometer with bright coherent lighta broadband squeezed-vacuum field is injected into the signal output port and overlappedwith the bright interferometer mode The interferometer is operated close to a dark fringesuch that most of the bright coherent light as well as most of the squeezed vacuum areback-reflected from the Michelson interferometer respectively Due to interference withthe broadband squeezed vacuum the interferometerrsquos output light on the photo diodeshows reduced variance in the photon number statistic as shown in Fig 1 Overlappingthe two light fields is possible with theoretically zero loss by the combination of a Faradayrotator and a polarizing beam splitter (PBS) A signal is produced by modulating therelative arm length (b) Simulated data for photo diode measurements Without squeezing(i) the signal of the laser interferometer is not visible With squeezing (ii) the shot noiseis reduced and here a sinusoidal signal visible

Squeezed states belong to the class of lsquonon-classicalrsquo states which areconsidered to be at the heart of quantum mechanics These states are de-fined as those that cannot be described as a mixture of coherent states Inthis case their Glauber-Sudarshan P -functions [Sudarshan (1963) Glauber(1963)] do not correspond to (classical) probability density functions ie theyare not positive-valued functions As a lsquoclassicalrsquo example the P -function ofa coherent state corresponds to a δ-function

5

But the question remains what property of coherent states justifies thename lsquoclassicalrsquo even though coherent states are quantum states and showquantum uncertainties My answer to this question is the following Allexperiments which only involve coherent states and mixtures of them allowfor a description that uses a combination of classical pictures As we will seebelow this description swaps between two different classical pictures and isthus not truly classical but semi-classical (A more precise description of thenature of coherent states uses the term lsquosemi-classicalrsquo)

Let us consider a laser interferometer that uses light in a coherent stateFirstly the light beam is split in two halves by a beam splitter The twobeams travel along different paths and are subsequently overlapped on abeam splitter where they interfere exactly as classical waves would do Theelectric fields superimpose thereby producing the phenomenon of interfer-ence Up to this point there is no reason to argue light might be composedof particlesSecondly the new (still coherent) beams that result from the interference areabsorbed for instance by a photo-electric detector In the case of coherentstates the detection process can be perfectly described in the classical parti-cle picture in which the particles appear independently from each other in atruly random fashion yielding the aforementioned Poisson statistic Duringthe detection process no wave feature of the light is present Let us havea closer look A truly random (lsquospontaneousrsquo) event is an event that hasnot been triggered by anything in the past This allows us to make a clearcut between the first part of the experiment described by the classical wavepicture and the second part of the experiment described by the classicalparticle picture Both lsquoworldsrsquo are disconnected The subsequent applicationof two classical pictures is not truly classical but lsquosemi-classicalrsquo It is in-deed the observation that the photons occur individually with truly randomstatistics that allows this semi-classical description In the case of a mixtureof coherent states the photon statistics are super-Poissonian which can beunderstood as a mixture of different Poissonian distributions In the caseof a slowly changing coherent state the mean value n depends on time Inall these cases the semi-classical description is appropriate Let me pointout that in this very reasonable description photons do not exist before theyare detected eg absorbed Further note that the famous double-slit exper-iment with coherent states also allows for the same semi-classical description

For squeezed states [Yuen (1976) Walls (1983)] the situation is different

6

As before the interference can be fully described by the classical wave pic-ture The result of the detection process however is different from that ofmutually independent random events It is also different from any super-Poissonian statistics that could be produced by mixing an arbitrary num-ber of different andor time-dependent Poissonian distributions Insteadthe squeezed probability distribution in Fig 1 suggests that the probabil-ity of detecting a photon decreases with the more photons that are alreadydetected in the same time interval over which a single measurement is in-tegrated From this observation one must conclude that the photons donot individually appear in a random fashion upon detection There mustbe lsquoquantumrsquo correlations between the photons These correlations mustexisted before detection since there is no interaction between the photonsduring their detection Pre-existing correlations between detected photonsseem to imply that the photons themselves existed before detection ie attimes when interference occurred In a semi-classical description howeverphotons are classical particles and cannot interfere for instance on a beamsplitter At this point the semi-classical picture breaks down Squeezedstates are therefor lsquononclassicalrsquoThe failure of the semi-classical model described above generally certifiesnonclassicality

Squeezed states are usually not characterized by counting their pho-tons but by measuring canonical continuous-variable phase-space observ-ables Measurements are performed as usual on an ensemble of identicalstates and quasi-probability density functions are calculated from the dataThe Glauber-Sudarshan P -function is the quasi-probability density distribu-tion over coherent states If the P -function of a state is entirely positivethe state is a coherent state or a (classical) mixture of coherent states Thestate is considered as semi-classical If the P -function is not a positive-valuedfunction the state cannot be expressed as a (classical) mixture of coherentstates and is thus nonclassical [Gerry and Knight (2005) Vogel and Welsch(2006)] A non-positive-valued P -function is the sufficient and necessary con-dition for the failure of the semi-classical model The Wigner function is thequasi-probability phase-space representation over the canonical continuous-variable phase-space observables themselves [Gerry and Knight (2005)] TheWigner functions of squeezed states are entirely positive Although subject todiscussion this fact does not mean that squeezed states are less nonclassicalthan Fock states or cat states which not only have a nonclassical P -function

7

but also a partially negative Wigner function (A cat state is a quantum su-perposition of two macroscopically distinct states [Monroe (2002)] referringto Schrodingerrsquos-cat gedanken experiment [Schrodinger (1935)]) In practicesqueezed states can even be regarded as superior nonclassical states becausethey represent the only nonclassical state that has been produced in a steadystate fashionIn almost all experiments so far the generation of Fock states and cat statesinvolves a probabilistic event such as the detection of a photon in anotherbeam path to herald these states In fact squeezed states provide the non-classical resource for the probabilistic preparation of Fock states as well as catstates But only the squeezed states themselves show a nonclassical effect in astationary way Limited only by the time duration and the frequency span ofthe mode that is in a squeezed state the squeezing effect can be continuouslyobserved independently of the time when the measurement is performed andalso independently of the measurement integration time This fact is of greatimportance for applications of squeezed states in measurement devices sincea squeezed-light-enhanced measurement remains unconditional and the ef-fective measurement time is not reduced

In past decades squeezed states of light were used in many proof-of-principle experiments to research their potential for improving the sensitivityof laser interferometers [Grangier et al (1987) Xiao et al (1987) McKen-zie et al (2002) Vahlbruch et al (2005) Goda et al (2008) Taylor et al(2013)] or the performance of imaging beyond the shot-noise limit [Lugiatoet al (2002) Treps et al (2003)] both accompanied by a huge number oftheoretical works Potential applications in secure optical communication(quantum key distribution) were also proposed and proof-of-principle ex-periments demonstrated [Ralph (1999) Furrer et al (2012) Gehring et al(2015)] This review restricts itself to the improvement of laser interfero-meters since only here has the application of squeezed light gone beyondproof-of-principle The gravitational-wave detector (GWD) GEO 600 hasoperated with squeezed light now for more than seven years starting in 2010[Abadie (2011) Grote et al (2013)] GEO 600 is a 600 m long Michelsonlaser interferometer built for the detection of gravitational waves Thesewaves are audio-band and sub-audio-band changes of space-time curvatureoriginating from cosmic events such as the merger of neutron stars or blackholes as detected recently [Abbott (2016)] In GWDs such as GEO 600 [Doo-ley et al (2016)] Advanced LIGO [Aasi (2015)] Advanced Virgo [Acernese

8

(2015)] and KAGRA [Aso et al (2013)] conventional laser technology hasbeen pushed to extremes over the past decades Noise spectral densities nor-malized to space-time strain of less than 10minus23 Hzminus12 have been measured[Abbott (2016)] Progress will continue and based on the successful appli-cation in GEO 600 squeezed light is now widely accepted to provide a newadditional technology to contribute to the new field of gravitational-waveastronomy It was also successfully tested in one of the LIGO detectors in2013 [LSC (2013)] and is an integral part of the European design study forthe 10 km Einstein-Telescope [Punturo et al (2010)]

GEO 600 has already taken several years of lsquosqueezedrsquo observational datawhich has increased its sensitivity at signal frequencies above 500 Hz Withthe implementation of a squeezed light source in GEO 600 the applicationof nonclassical states in metrology has been pushed beyond merely proof-of-principle

lsquoTwo-mode squeezed statesrsquo show a squeezed uncertainty in at least onejoint continuous variable of two subsystems lsquoArsquo and lsquoBrsquo Examples of jointvariables are differences and sums of phase-space observables of A and BTwo-mode squeezed states not only belong to the class of nonclassical statesbut due to their bi-partite character also to the class of lsquoinseparablersquo orlsquoentangledrsquo states They are the ideal states to demonstrate the Einstein-Podolsky-Rosen paradox [Einstein et al (1935)] as first achieved in [Ouet al (1992)] Apart from fundamental research on quantum mechanicsrecent proof-of-principle experiments demonstrated their usefulness in inter-ferometric measurements that go beyond the application of simple squeezedstates [Steinlechner et al (2013) Ast et al (2016)] This experiment is thefinal topic of this review

2 Observations on light fields in squeezed states

Generally there are two different kinds of observables that can be subjectof a measurement performed on a quantum system The first kind is associ-ated with the systemrsquos wave property In optics it corresponds to the electricfield strength at a given phase angle ϑ The according (dimensionless) opera-tors are called the quadrature amplitudes Xϑ and have a continuous spectrumof eigenvalues Quadrature amplitudes are measured in very good approx-imation with a balanced homodyne detector using the interference with a

9

bright local oscillator beam see Fig 3 (a) In practice any measurement ofXϑ integrates over some sideband (Fourier) spectrum within the angular fre-quencies Ω plusmn ∆Ω2 The sideband information always needs to be quotedA straight forward but rather untypical way is by adding subscripts whichleads to Xϑ

Ω∆Ω The classical analogue of the quadrature amplitude operator

XϑΩ∆Ω is the modulation depth of the optical field at modulation phase angle

ϑ and at angular modulation frequency Ω measured over the band ∆Ω lt 2ΩThe uncertainties of the statersquos quadrature amplitudes at different phases ϑare limited by a Heisenberg uncertainty relation see section 3 The secondkind of measurement is associated with the systemrsquos particle property andis given by the photon number operator n associated with a measuring timeinterval ∆t Its precise measurement requires a photon counter ideally withsingle photon resolution The measurement result obviously has a discretespectrum Continuous as well as discrete observables are usually subject toquantum uncertainties and thus quantum noiseUsually the measurementrsquos integration time and frequency band actuallydefine the physical system that is characterized In quantum optics experi-ments the interrogated physical system is called a lsquomodersquo

21 Definition of a lsquosingle modersquo

Let us define a light field or generally any quantum system to be asingle mode if it corresponds to the lsquosmallest entity of a waversquo In this caseits spectral and temporal distributions as well as waist size and divergenceare at their Fourier limits and all other properties such as optical axis waistposition and polarization are well defined For instance a linearly polarizedlongitudinal resonance of an optical standing-wave cavity defines such a singlemode if the cavity finesse is high and transversal modes are non-degenerateThe complete photo-electrical detection of a cavity mode however is notstraight forward Most quantum optical experiments are instead performedon propagating light In this case single modes are defined by spatial filtersand by temporal-spectral measurement windows both being at the Fourierlimit Examples for single modes are a laser pulse and a spectraltemporalcutout from a continuous observation of a quasi-monochromatic continuous-wave light beam in the spatial TEM00 mode both at the Fourier limits

In classical physics the only remaining free parameter of a given singlemode is its excitation energy In quantum physics the situation is differentFor a given energy a single mode can be in many different quantum states

10

which differ in their quantum statistics Examples are coherent states num-ber (Fock) states and squeezed states

22 Observations on squeezed states using a single PIN photo-diode

An ideal PIN photo-diode absorbs the full energy of a light mode andproduces one photo electron for every absorbed photon energy It uses theinternal photo-electric effect inside a semiconductor such as silicon or In-GaAs In contrast to avalanche photo-diodes PIN photo-diodes operatewith unity gain lsquoPINrsquo stands for lsquopositiversquo lsquointrinsicrsquo and lsquonegativersquo and isdescribing the doping of the semiconductor layers A PIN photo-diode is op-timally suited for the continuous monitoring of a rather bright light field ofup to several tens of milliwatts An example is the photo-diode in the outputport of a gravitational-wave detector as shown in Fig 2 (a) The prominentwavelength of 1064 nm which is emitted by NdYAG lasers has an opticalfrequency of ν = ω(2π) = 282 middot 1014 Hz The period of the field oscillationis a few femtoseconds and cannot be directly resolved with photo-electric de-tectors However variations of the electric field around the averaged opticalfield oscillation on longer time-scales can be resolved Applying an electronicbandpass filter at the sideband angular frequency Ω plusmn ∆Ω2 to the photovoltage provides information about the lsquodepth of the lightrsquos amplitude mod-ulationrsquo which is also called the lsquoamplitude of the amplitude quadraturersquo Itcan also slowly vary in time and reads

Xϑ=0

Ω∆Ω(t) equiv XΩ∆Ω(t) equiv X (1)

The subscript is usually skipped as it is done with the time dependenceas indicated on the right Applying the electronic bandpass filter in factdefines the mode of the light being detected The structure of the defini-tion in Eq (1) forms the basis of interferometric signals and quantum noisealso in the semi-classical case of coherent states Lets take an example Inthe recent observation of gravitational waves [Fig 1 bottom row in Abbott(2016)] the time-frequency representation of the gravitational-wave signalcorresponded to the amplitude quadrature amplitude XΩ∆Ω(t) of the inter-ferometer output light Note that a larger value of ∆Ω allows for changes ofthe quadrature amplitude on shorter time scales

If the light fieldrsquos lsquomodulation modersquo does not contain any quanta simplybecause there are no photons that have a frequency difference of plusmnΩ with

11

respect to the carrier it is in its ground state In this case lsquovacuum noisersquois observed which originates from the ground state uncertainty Since thevacuum noise only becomes measurable as a beat with a bright light field itcan also be seen as the carrierrsquos band-path filtered shot noise A modulationmode in a displaced vacuum state (a coherent state) corresponds to nonzerocoherent modulationThe measured level of the vacuum noise generally depends on the power ofthe bright carrier light and on the electronic amplification In any case itprovides the reference for certifying lsquosqueezingrsquo Observations using a singlePIN photo-diode require an independent measurement to quantify vacuumnoise A necessary condition is that attenuating the total fieldrsquos light powerresults in the same attenuation of the measured XΩ∆Ω values If they showa stronger attenuation a coherent modulation or thermal noise might bepresent If they show a weaker attenuation the photo-diode and its electron-ics might be saturated

Fig 2 (b) illustrates how a broadband squeezed field improves the mea-surement of an amplitude modulation in time domain based on a PIN photo-diode Shown is a simulated time sequence of XΩ∆Ω-data sampled from thephotoelectric voltage In this simulation all sideband frequencies from zero(DC) to the cutoff frequency of the detector electronics (Ωcut) are included(Ω = ∆Ω2 = Ωcut2) No additional band pass filter is applied making it amaximally broadband detection Although the data in Fig 2 (bi) contains aclassical amplitude modulation of the detected light this signal is not visibledue to random noise here representing shot noise Fig 2 (bii) shows thesame situation but with shot noise that is squeezed over the full detectionband The quantum uncertainty of the modulation depth is squeezed andthe classical signal becomes visible

It needs to be noted that a single PIN photo-diode can only measure theamplitude of the amplitude quadrature XΩ∆Ω(t) but not the non-commutingobservable the lsquoamplitude of the phase quadraturersquo

Xϑ=90

Ω∆Ω (t) equiv YΩ∆Ω(t) equiv Y (2)

For values that are small compared to the field strength of the bright fieldthe quantity Y approximately describes the bright fieldrsquos lsquophase modulationdepthrsquo

12

23 Observations on squeezed states using a balanced homodyne detector

-72

-69

-66

-63(i)

(ii)

(iii)

(a)

LO

(Squeezed)signal input

Phaseshifter

PD1

PD2

5050

01 0200 03

-75

-60

-57Balanced homodyne detector

(b)

Time [s]

Noi

se p

ower

[dB

m]

Figure 3 Balanced homodyne detection (BHD) ndash (a) Setup The quadrature atchoosable angle ϑ of the signal field is measured by overlapping the latter with a localoscillator (LO) field of the same mode parameters on a balanced beam splitter and record-ing the difference voltage from two PIN photo-diodes as shown In order to meet theBHD approximation the LO needs to be much more intense than the signal field A closeto perfect mode overlap between LO and signal input field is crucial For a non-perfectoverlap the detector measures the input state with unwanted contributions of the vacuumstate (b) Noise power measurements (i) on an electronically amplified and band-passfiltered quadrature amplitude of the vacuum field (Xvac

Ω∆Ω) (signal input blocked) (ii) on

a squeezed quadrature (XsqzΩ∆Ω) of a squeezed vacuum state (ϑ = 0) and (iii) on re-

spective quadratures of the same squeezed state where the phase angle ϑ was continuouslyshifted by changing the optical path length of the LO The measurement data shows about5 dB of squeezing and was first published in [Chelkowski et al (2007)] Ω2π = 5 MHz∆Ω2π = 100 kHz

In contrast to a single PIN photo diode a balanced homodyne detector(BHD) is suitable to measure the quantum statistic of all types of modu-lations ie for all angles ϑ Such a detector consists of two identical PINphoto-diodes a balanced beam splitter and an external homodyne local os-cillator field that is much brighter than the signal beam and that has anadjustable phase The signal beam corresponds to the squeezed field whichin many experiments is in a squeezed vacuum field having an optical powerthat usually corresponds to just a few photons per mode The two beams areoverlapped on the balanced beam splitter with close to perfect mode match-ing and the two interference outputs are focussed onto the photo diodes seeFig 3 (left) The electric output signal of the BHD is the difference of the

13

photo diode voltages The LO takes over the role of the carrier light fieldbut with the possibility to choose the phase shift ϑ This way eigenvaluesof X Y or Xϑ can be measured where the latter is given by the followinglinear combination of the first two

Xϑ(t) = cos(ϑ) X(t) + sin(ϑ) Y(t) (3)

If the modulation depths of signal and local oscillator beams are weak com-pared to their coherent amplitudes |α| and |αLO| the output voltage of aBHD corresponds to eigenvalues of the following operator

V (t) prop 2cos(ϑ) |αLO||α|+ |αLO| Xϑ(t) + |α| XϑLO(t) (4)

The lsquohomodyne approximationrsquo further involves |αLO| |α| such that theterm on the right can be neglected even if the local oscillator shows someclassical quadrature excitation The output voltage of a BHD is usually spec-trally analysed or at least spectrally filtered which removes the DC part infull analogy to a single photo diode (see previous subsection) Sampling thefiltered voltage provides eigenvalues proportional to the generalized quadra-ture amplitude in Eq (3)

V BHDΩ∆Ω(t) prop |αLO| Xϑ

Ω∆Ω(t) (5)

Fig 3 (a) shows the setup of a balanced homodyne detector for the char-acterization of squeezed states Setting ϑ = 0 eigenvalues of the ampli-tude modulation depths can be sampled from the photo voltage accordingto Eq (5) Setting ϑ = 90 eigenvalues of the phase modulation depths aremeasured The datarsquos expectation values 〈Xϑ〉 provide the coherent displace-ment of the squeezed state The datarsquos variances

∆2Xϑ equiv 〈(Xϑ)2〉 minus 〈Xϑ〉2 (6)

provide the statersquos (quantum) noise A pure squeezed state as well as asqueezed state that experienced photon loss have Gaussian quantum statisticsand are thus fully described by the expectation values and variances (first andsecond moments) of two orthogonal quadratures but only if one quadraturereflects the lowest quadrature variance

14

In most experiments with squeezed light the photo electric voltage ac-cording to Eq (5) is not sampled with a data aquisition system but the signalis directly fed into a spectrum analyser measuring the noise power of the volt-age If the expectation value 〈Xϑ〉 is zero the noise power is proportionalto the variance ∆2Xϑ in Eq (6) The reference for quantifying the squeezefactor is measured by blocking the (squeezed) signal field in Fig 3 (a) Themeasured vacuum noise level corresponds to the LOrsquos (electronically ampli-fied) shot noise level

Traces (ii) and (iii) in Fig 3 (b) show measured noise powers of the mod-ulation mode (Ω2π = 5 MHz ∆Ω2π = 100 kHz) being in a squeezed vac-uum state (i) is proportional to the variance of the ground state uncertainty∆2Xvac

Ω∆Ω (ii) is proportional to the quantum noise variance of the squeezed

quadrature amplitude ∆2XsqzΩ∆Ω (iii) is proportional to the quantum noise

variance of the quadrature amplitude with scanned phase ∆2XΩ∆Ω(ϑ(t))

To fully characterize a quantum state ie to do quantum state tomogra-phy [Vogel and Risken (1989)] a BHD is a prerequisite But also interfero-metric measurements with balanced homodyne detectors instead of singlePIN photo-diodes have several advantages A correctly implemented BHDreadily provides the vacuum noise level when the signal beam is blockedWith a BHD the optimum operating point of the interferometer is preciselyat a dark fringe If a perfect dark fringe can practically be achieved ampli-tude noise of the laser does not couple into the signal port If the interfero-meter has balanced arm length also frequency noise of the laser then does notcouple into the signal port Some quantum non-demolition schemes with theprospect of evading quantum radiation pressure noise require the detection ofa non-canonical quadrature angle [Jaekel and Reynaud (1990) Kimble et al(2001)] Here the adjustable phase of a BHD provides a straight forwardapproach The experimental exploration of BHDs for gravitational-wave de-tectors only has started recently [Steinlechner et al (2015)]

A light field can be analysed with respect to many different modulationfrequencies Ω The result constitutes a spectrum [Breitenbach et al (1998)]where in principle every modulation mode can be in a different quantumstate Fig 4 shows spectra of squeezed states from 5 MHz to 100 MHz with∆Ω2π = 1 MHz The lower curve shows the spectrum of the most strongly

15

-14-12-10-8-6- 4-20 2 4 6 8

10 12 14 16 18

6 8 10 20 40 60 80 1005

Squeezed noise

Anti-squeezed noise

Vacuum noise

Frequency [MHz]

Nor

mal

ized

noi

se v

aria

nce

[dB

]

Figure 4 Spectrum of quadrature amplitude variances ndash Shown are the quantumnoise properties of a large number of modulation modes having a resolution bandwidth of∆Ω(2π) = 1 MHz For all traces the balanced homodyne detector output was analysedwith a spectrum analyser Squeezing of XΩ∆Ω (bottom trace) and anti-squeezing of

YΩ∆Ω (top trace) versus f = Ω(2π) are shown relative to the vacuum noise variance Thespectrum below 5 MHz is not shown since it contained less squeezing due to laser relaxationoscillation in the carrier field as well as disturbances from back-scattered light [Vahlbruchet al (2007)] Disturbances at frequencies above 70 MHz originated from relatively largedetector dark noise which was subtracted from all traces shown here The thin linerepresents a theoretical model that takes into account for the linewidth of the squeezingcavity The data was first presented in in Ref [Mehmet et al (2010)]

squeezed variances in this case the variances of XΩ∆Ω The upper spectrum

shows the variance in the orthogonal quadrature amplitude (YΩ∆Ω) Allvariances are normalized to those of the corresponding vacuum state Thesqueeze factor reduces towards higher frequencies due to the linewidth ofthe squeezing cavity The anti-squeezing is always higher than the absolutevalue of the squeezing due to Heisenbergrsquos uncertainty relation and due to thepresence of optical loss The curves do not represent pure squeezed states butmixed squeezed states with a significant contribution from vacuum statesdue to optical loss Pure squeezed states can only be produced by making

16

the influence of all decoherence processes negligibleThe choice of the resolution bandwidth (RBW ∆Ω) during data taking

and processing defines the spectral-temporal modulation modes includingtheir number within the detected spectrum For any setting of the RBWthe quantum mechanical properties of the quadrature amplitudes XΩ∆Ω and

YΩ∆Ω [Caves (1985)] fully correspond to those introduced for quadratures instandard text books and which are reviewed in Sec 3

24 Observations on two-mode squeezed states using balanced homodyne de-tectors

Two-mode squeezed states are composed of two subsystems lsquoArsquo and lsquoBrsquoand are bi-partite entangled states with a Gaussian quantum statistic Toavoid conflicts with different usage of the term lsquomodersquo they can synony-mously be named lsquobipartite Gaussian entangled statesrsquo or lsquobipartite squeezedstatesrsquo which will be mainly used in this Review In the same way multi-partite Gaussian entangled states correspond to multi-partite squeezed states

The measurement observables that prove or disprove the bi-partite squeez-ing property are XA

Ω∆Ω minus XBΩ∆Ω and Y A

Ω∆Ω + Y BΩ∆Ω where the minus and

plus signs may be swapped Bi-partite squeezed states are precisely thosestates that were discussed by Einstein Podolsky and Rosen (EPR) in theirseminal paper [Einstein et al (1935)] Fig 5 shows a measurement resulton bi-partite squeezed light [Eberle et al (2013)] The variances of bothjoined observables are squeezed as shown in the two lower traces They wererecorded consecutively by adding or subtracting the outputs of two balancedhomodyne detectors But by interfering the subsystems on a beam splitterone could even measure both joined observables simultaneously This pos-sibility is correctly described in quantum theory since their commutator iszero

The so-called EPR paradox arises as follows If we either measure XAΩ∆Ω

and XBΩ∆Ω or Y A

Ω∆Ω and Y BΩ∆Ω it is obvious from the data in Fig 5 that

we can always predict the measurement result at subsystem lsquoBrsquo when know-ing the result at subsystem lsquoArsquo This seems to suggest that both quantitiesat lsquoBrsquo are precisely defined simultaneously before the measurement on lsquoArsquowhich contradicts the rigorous (and correct) interpretation of their non-zerocommutator that they are not precisely defined simultaneously

To solve this paradox EPR conjectured that the wavefunction as definedby quantum theory does not provide the full information This led to a

17

discussion of whether hidden variables existed that needed to be included ina complete theory of quantum mechanics (see also Bell [Bell (1966)]) Theexperimentally observed violation of Bellrsquos inequality [Bell (1964) Aspectet al (1981) Giustina et al (2013) Hensen et al (2015)] however ruled outthe existence of (local) hidden variables

Based on that the EPR paradox needs to be solved in a different wayContrary to what EPR assumed it is in fact possible to predict the valueof an arbitrary observable of a physical system A with certainty via a mea-surement on system B although this observable was not defined before themeasurement Without any interaction a measurement on subsystem lsquoArsquo notonly creates lsquorealityrsquo of eg XA

Ω∆Ω simultaneously lsquorealityrsquo is also created

regarding the observable XBΩ∆Ω describing subsystem lsquoBrsquo Here the term

lsquorealityrsquo has the meaning as defined by EPR [Einstein et al (1935)] Simi-larly the detection of one photon of a two photon entangled number statenot only produces the reality of this photon but also that of a second oneA discussion of Einstein-Podolsky-Rosen entanglement can also be found in[Schnabel (2015)] Note that the EPR paradox can also be described aslsquoquantum steeringrsquo [Schrodinger (1935) Cavalcanti et al (2009) Handchenet al (2012)] It should also be mentioned that two-mode squeezing beingdetected with BHDs and not with photon counters cannot be used to violatea Bell inequality The latter topic is outside the scope of this Review

Bi-partite squeezed states were first characterized with balanced homo-dyne detectors by the group of J Kimble in 1992 [Ou et al (1992)] Gener-ally the EPR paradox becomes more pronounced the stronger the bi-partitesqueezing is A measure of the strength of EPR entanglement was introducedby M Reid [Reid and Walls (1985)] According to this measure the resultin Fig 5 can be quantified to ε2 = 00309 where the critical value is one Itcorresponds to the strongest Gaussian EPR entangled state generated so far

For a long time it looked like that two-mode squeezed states are notuseful for laser interferometers The reason for that belief was that a laserinterferometer as any other measurement device too is built to measureone observable It seems to be ideal already if the quantum noise in thissingle observable is squeezed The increased quantum noise in the orthog-onal observable is not harmful in this case and squeezing in two differentobservables useless Only recently realistic scenarios were discussed in whichtwo-mode squeezing in fact does improve the performance of a laser inter-

18

-12

-10

-8

-6

-4

-2

0

0 1 2 3 4 5 6 7 8 9 10

100

dB

109

dB

Δ (XAvac + X B

vac ) = Δ (Y

Δ (XA + X B)

Avacminus Y B

vac )2 2

2

Δ (YAminus Y B)2

Nor

mal

ized

noi

se v

aria

nce

[dB

]

Times [s]

Figure 5 Two-mode squeezing measurement ndash For this measurement the outputs oftwo balanced homodyne detectors are added or subtracted and the variances (noise pow-ers) of the results recorded The upper trace was measured with modes lsquoArsquo and lsquoBrsquo beingin their ground states This measurement served as a reference level Strong two-modesqueezing was observed as shown by the lower two traces The sideband frequency wasΩ(2π) = 8 MHz and the resolution bandwidth was ∆Ω(2π) = 200 kHz The measure-ment results were first published in Ref [Eberle et al (2013)]

ferometer [Steinlechner et al (2013)] The proof-of-principle experiment isreviewed in Sec 7

25 Observations using photon counters

Alternatively to field quadratures an optical mode in a squeezed statecan also be characterized at least partly by detecting its photon numberdistribution For a pure squeezed vacuum state such a measurement wouldreveal the existence of solely even photon numbers including a large prob-ability for zero photons The average photon numbers of squeezed vacuumstates with feasible squeeze factors are very small of the order of one persecond and bandwidth in hertz see Fig 13 (a) ndash (c) A distribution with closeto zero probability of odd photon numbers however has not been measuredso far The reason is the lack of ideal photon counters First of all theefficiency of these detectors ie their probability of converting one photoninto one click and no photon into no click must be almost perfect lsquoLostrsquophotons as well as dark counts wash out the oddeven oscillations Further-more most detectors available can only distinguish between zero and one

19

photon This problem can be solved by distributing the squeezed mode ontoa large number of single photon detectors using an array of beam splitterssuch that all paths have a low probability of carrying more than one pho-ton Photon number measurements on squeezed vacuum states neverthelessplay an extremely important role in quantum optics When the squeezingstrength is very low the probability of detecting more than 2 photons can beneglected and the detection of a photon heralds the existence of a second one

0

300

600

900

1200

-40 -20 0 20 400

5

10

15

20

25

Two-

fold

coi

ncid

ence

s [1

(4s

)](a) (b)

Non-degeneratetwo-mode squeezing

5050

APDA1

APDA2

APDHerald

Photon coincidencedetection

Delay [ns]

Thr

ee-f

old

coin

cide

nces

[1

(4s)

]1500

Figure 6 Coincidence clicks from non-degenerate photon pairs ndash The first suchexperiment was reported in Ref [Hong and Mandel (1986)] (a) shows a setup with threeavalanche photo-diodes (APDs) for proving the successful heralding of a single photonnumber state (b) Histograms of the two-fold coincidence detections at APDHerald andAPDA1 (red) and at APDHerald and APDA2 (yellow) with theoretical models (solid lines)If the two-mode squeezing just carried one photon in each spatial subsystem the three-foldcoincidence detection should be zero Indeed the according histogram (grey points righty axis) shows only a few events These are produced by false (dark) counts of the APDsThe delay for the three-fold coincidences is defined as the time between counts at lsquoA1rsquo andlsquoA2rsquo given that the trigger APDHerald detected a photon (within a 100 ns time window)The data was taken on photons that were up-converted from 1550 nm to 532 nm and itwas first published in Ref [Baune et al (2014)]

If a mode of light is always excited by either zero or two photons lsquocondi-tionalrsquo or lsquoheraldedrsquo one-photon Fock states can be realized (Measurementson an ensemble of the n-photon Fock state would always produce the mea-surement result n ie Fock states have a zero photon number uncertaintyThey are also called lsquonumber statesrsquo) The above concept of producing aone-photon Fock state obviously requires the deterministic and balanced dis-tribution of the down-converted signal and idler fields into two different paths

20

In order to achieve this the signal and idler fields need to be non-degenerateUsually a mode in a squeezed state is composed of degenerate signal andidler fields and this degeneracy thus needs to be removed Possible waysare producing the down-converted fields at well separated wavelengths [Vil-lar et al (2005) Su et al (2006) Li et al (2010) Samblowski et al (2011)]separating the upper and lower sidebands belonging to an ordinary squeezedmode by frequency filters [Schori et al (2002) Hage et al (2010)] and us-ing spatial filters [Hong et al (1987)] A frequently used approach is usingtype II parametric down-conversion where the photons within a pair are al-ways orthogonally polarized [Ou et al (1992) Kiess et al (1993) Kwiat et al(1995)]The list of experiments with conditional or heralded photon number states islong They showed for instance nonclassical g(2)-functions [Hong et al (1987)]and violations of Bell inequalities [Weihs et al (1998)] Fig (6) shows a re-sult from a more recent experiment in which a bipartite-squeezed state withsubsystems at 1550 nm and 810 nm was produced the subsystem at 1550 nmsubsequently up-converted to 532 nm and the lsquoquantum non-Gaussianityrsquo ofheralded up-converted single photons demonstrated [Baune et al (2014)]Squeezed states are also the resource for the conditional generation of super-positions of coherent states [Ourjoumtsev et al (2006) Neergaard-Nielsenet al (2006)] and so-called N00N-states [Afek et al (2010)]

The generation of nonclassical states mentioned in the paragraph aboveis not stationary but relies on a probabilistic trigger event The produc-tion of squeezed states themselves usually happens in a stationary fashionThis distinction has an important consequence for applications of nonclassi-cal states in measurement devices Only (stationary) squeezed states allowfor a continuous improvement of a measurement Avoiding any loss of mea-suring time is generally of high relevance for the detection of short-livedsignals with unknown arrival time as well as for the detection of long-livedquasi-monochromatic signals since the signal-to-noise-ratio (SN) improveswith measuring time

26 Conclusions

The detection of squeezed light produces measurement results that canbe considered as remarkable Let us focus on experiments where a mode in abright coherent state is overlapped with a mode in a squeezed vacuum state

21

as shown in Figs (1) and (3) In both setups the squeezed vacuum field caneasily be blocked which allows us to compare the measurement results ona bright coherent state with and without the interference with the squeezedvacuum state Without squeezing the photo-electric detectors measure alarge number of photon events with a large quantization noise (shot noise)The large noise reflects the fact that all photon events were independent fromeach other as shown in Fig 2 (bi) With squeezing the photo-electric de-tectors again measure a large number of photon events with an expectationvalue that is even slightly higher but nevertheless the quantization noise ofall detected photons is significantly reduced Fig 2 (bii)Based on the discussion of EPR entanglement in Subsec 24 the photo-electric detection of the output light of a squeezing-enhanced laser inter-ferometer (with αlowastα1) produces the reality of photons This way we cankeep the lsquowave picturersquo in which no photons exist when light travels alongthe interferometer arms and when it interferes at the beam splitter Whenthe energy of the beam is elevating electrons to the conductance band ofthe photo-diodersquos semi-conductor n photon events simultaneously appearwithin the measuring interval with probability P (n) What conclusion hasto be drawn if the probabilities resemble a sub-poissonian statistic ndash Theoccurrence of photon events is still truly random but in this case not forindividual photons The occurrence of photons is correlated in such a waythat the probability of detecting an additional photon in the same time in-terval reduces the larger the number of already detected photons is Whatfollows from the discussion of EPR entanglement for a photon counting ex-periment with pure squeezed vacuum and ideal photon counters Here theprobabilistic detection of one photon entails the detection of a second onewith certainty With some smaller probability a third photon is detectedwhich entails the detection of a fourth photon with certainty and so on

If a photon of a mode that was not interrogated by the environment beforeis absorbed its reality is created in this very moment If the photon belongsto a squeezed state this process instantaneously influences the probability ofother photons becoming reality

Of course a more general statement can be made based on the insightthat interaction with the environment creates the reality of any kind ofquanta including electrons atoms and molecules

22

3 Theoretical description of squeezed states

31 The quadrature amplitude operators

Consider a single mode of light at optical frequency ω Its Hamiltonoperator reads

Hω = ~ω(n+

1

2

)= ~ω

(adaggerωaω +

1

2

)= ~ω

(X2ω + Y 2

ω

) (7)

where n is the photon number operator and aω and adaggerω are the annihilationand creation operators which obey the commutation rule

[aω a

daggerω

]= 1 The

operator aω has a complex-valued dimensionless eigenvalue spectrum andcorresponds to the complex amplitude αω in classical optics Xω and Yω arethe hermitian amplitude and phase quadrature operators The eigenvaluesof the quadrature operators are also dimensionless and proportional to theelectric fields at the oscillationrsquos antinode and at the oscillationrsquos node Inthe above equation they are defined such that their variances are ∆2Xω =∆2Yω = 14 if the oscillator is in its ground state ie if 〈n〉 = 0

Although Eq (7) simply describes the energy of an harmonic oscillator itis the essence of quantum theory since it mathematically describes the wave-particle dualism Whereas the eigenvalues of n have a discrete spectrum theeigenvalues of Xω and Yω have a continuous spectrum In classical opticsthe phase quadrature is zero In quantum optics its expectation value is alsozero but its uncertainty contributes to the overall energy

Eq (7) describes a cavity mode as well as a section that is cut from apropagating quasi-monochromatic light beam The latter example is of highrelevance in actual experiments By setting the sectionrsquos time window ie themeasuring time interval the time-frequency (lsquomodulationrsquo) mode is defined

The quadrature operators introduced in Eq (7) and displayed in Fig 7 donot correspond to lsquoXrsquo and lsquoY rsquo that are of relevance in laser interferometry andin optical communication and which were already discussed in Subsec 22and 23 The optical frequency of visible and near-infrared light is far too highto be transferred to an oscillation of photoelectric voltage Quite general alaser interferometer targets signals at audio or radio band frequencies Ωi ωSuch a measurement is achieved as stated before by decomposing the photo-electric voltage from the photo diode at the interferometer output into asingle-sided spectrum (positive frequencies only) of intervals of Ωplusmn∆Ω2

23

(a)

(b)

(c)

(d)

2π 4π

2π 4π

2π 4π

2π 4π

Figure 7 Phase spaces and electric field oscillations of monochromatic lightndash Top Left Monochromatic light in a coherent state is represented by a phasor (whitearrow) including its quantum uncertainty (white dashed circle and fuzzy area) located inthe phase-space spanned by the quadratures Xω and Yω When the phase space rotateswith optical frequency ω2π the projection of the quantum phasor onto a fixed (vertical)axis corresponds to the electric field E(t) as shown on the right side (a) Weakly displacedcoherent state (b) Corresponding amplitude squeezed state The electric field uncertaintyaround the zero average field region is anti-squeezed (c) Vacuum state at the same opticalfrequency (d) Corresponding squeezed vacuum state The meaning of the uncertaintycould be carved out by supplementing them with monochromatic waves all having theoptical frequency ω2π Changing amplitudes then display amplitude quadrature noiseChanging shifts along the time axis model the electric field uncertainty at the expectedzero crossing They are not implemented in the graphics here however since any of thosewaves does not exist due to Heisenbergrsquos uncertainty relation

24

The signals as well as the quantum uncertainties carried by a beam oflight are thus described by a spectrum of pairs of non-commuting quadratureoperators Mathematically every such operator is defined by an integral overthe Fourier components within the bandwidth The spectral weighting of theFourier components is called the lsquowindow functionrsquo By going to sidebandintervals a spectrum of a new type of optical mode is defined which describesthe modulation of the electric field in the respective frequency interval Ω plusmn∆Ω2 In this Review we call it a lsquomodulation modersquo

The quadrature operators that are defined around a modulation frequencyΩ with a bandwidth of ∆Ω are the quadrature amplitude operators thatare relevant in laser interferometry Whenever they are not related to aspecific band we use the short form XΩ∆Ω(t) equiv X and YΩ∆Ω(t) equiv Y cfEqs (1) and (2) These operators can slowly vary with time where thetime dependence is limited by ∆Ω (The time dependence is not due toquantum uncertainty which usually is time independent but for instancedue to the time dependence of the signal eg a passing gravitational wave)Let us consider now a pair of quadrature operators for a particular sidebandΩ plusmn ∆Ω2 The Hamilton operator of the corresponding modulation modeis found by switching to the frame rotating at optical frequency ω Thetransition is done by applying the unitary transformation U = exp(iωadaggerat)generating a new Hamiltonian H = U daggerHωU minus i~UpartU daggerpartt The Hamiltonianof the modulation mode reads

H = ~Ω

(nΩ +

1

2

)= ~Ω

(adaggera+

1

2

)= ~Ω

(X2 + Y 2

) (8)

where nΩ is the (occupation) number operator for the modulation modeand a and adagger its annihilation and creation operators The commutation rule[a adagger

]= 1 is unchanged X and Y are the amplitude and phase quadra-

ture amplitude operators respectively They correspond to the depth of theamplitude modulation and for weak excitations to the depth of the phasemodulation respectively They are the conventional hermitian field operatorsin experimental quantum optics Note that modulation modes at angularfrequency Ω can be described by a superposition of three optical frequenciesa carrier at ω an upper sideband at ω + Ω and a lower sideband at ω minus ΩThe quantum mechanical description of modulation states in connection tooptical carrier and upper and lower sidebands is known as the lsquoTwo-PhotonFormalismrsquo [Caves and Schumaker (1985) Schumaker and Caves (1985)]

The quadrature amplitude operators in Eq (8) are again defined such

25

that the variances of the uncertainty of a modulation field in its ground stateor in a coherent state are

∆2Xvac = ∆2Yvac = 14 (9)

Generally quadrature operators X and Y as defined in Eqs (7) and (8) arethe real and imaginary parts of the annihilation operator

a = X + iY hArr adagger = X minus iY (10)

hArr X =1

2

(a+ adagger

) Y =

1

2i

(aminus adagger

) (11)

They satisfy the commutation relation[X Y

]=i

2 (12)

and their variances are limited by a Heisenberg uncertainty relation of thefollowing form

∆2X∆2Y ge 1

16 (13)

A quantum state is called a ldquosqueezed staterdquo [Bachor and Ralph (2004)]if ∆2Xϑ lt 14 for an arbitrary field quadrature Xϑ = X cosϑ + Y sinϑ seeEq (3) The angle of the lowest variance below 14 is called the squeeze angleθ The largest factor by which the variance is below 14 is called the squeezefactor often given on a decibel (dB) scale using the following transformation

minus 10 middot log10

(∆2Xθ

∆2Xvac

) (14)

The squeeze factor can also be described by the squeeze parameter r

eminus2r =∆2Xθ

∆2Xvac

(15)

When a squeezed state experiences optical loss it remains squeezed butthe squeeze factor is reduced Also the statersquos purity is reduced ie theproduct of the quadrature uncertainties increases above the minimum valueOptical loss corresponds to mixing the state with the vacuum state Let∆2Xϑ be the variance of a quadrature amplitude ∆2Xvac the variance of the

26

(quadrature angle independent) ground state uncertainty and (1 minus η2) therelative energy loss Then the resulting quadrature variance reads

∆2Xprime

ϑ = η2∆2Xϑ + (1minus η2)∆2Xvac (16)

To maximize the benefit from squeezed states in applications stronglysqueezed states need to be generated and optical loss minimized Opticalloss occurs due to absorption and scattering in the optical components inthe path of the squeezed beam including the squeezing resonator itself anddue to non-perfect matching to the interferometer mode non-perfect inter-ference contrast of the interferometer and non-perfect quantum efficiency ofthe photo diodes The sum of all losses including those outside the inter-ferometer need to be less then 10 to allow a nonclassical quantum noisesuppression of a factor of 10 in power ie 10 dB

32 Phase space representations of squeezed states

The Wigner function ndash The properties of squeezed states are nicely dis-played by the Wigner function W (X Y ) [Wigner (1932)] An example interms of a squeezed vacuum state is shown in Fig 8 It is a quasi-probabilitydistribution which contains the statersquos full information including its quan-tum statistic There are two ways how a Wigner function provides a sufficientcriterion for nonclassicality First by containing negative values second byfeatures that have a smaller (squeezed) width compared with the Wignerfunction of the ground state Integrating the Wigner function over Y pro-vides the probability density of measurement results ie of the eigenvaluesof the observable X and vice versa

infinintminusinfin

W (X Y )dY = p(X)

infinintminusinfin

W (X Y )dX = p(Y ) (17)

where p(X) and p(Y ) are the observed probability distributions also exem-plarily shown in Fig 8

The ground state coherent states as well as (quadrature) squeezed stateshave quadrature eigenvalue probability densities that are Gaussian TheirWigner functions are also Gaussian and thus entirely positive Wigner func-tions of other nonclassical states for instance Fock states exhibit negativevalues For this reason the Wigner function is called a quasi -probabilityfunction

27

X Y XY

Figure 8 Wigner function and its projections ndash Displayed is the full information ofa squeezed vacuum state The quasi probability density of the Wigner function (centre)is plotted along a third dimension and is color-coded Also shown are the actual mea-surement results from which the Wigner function is reconstructed They are representedby the squeezed and anti-squeezed Gaussian projections onto the X and Y axes Theirprobability densities p(X) and p(Y ) are given on the vertical axes The Gaussian measure-ment statistic of the first clearly shows squeezing compared to the ground state statistic(dashed) The squeeze factor is 116 dB and the squeeze parameter r = 1335 [Mehmetet al (2010)]

Fig 9 shows the Wigner functions for (a) a coherent state (b) the ground(vacuum) state (c) a displaced squeezed state and (d) a squeezed vacuumstate All Wigner functions describe a modulation of the carrier light atsideband frequency Ω integrated over the frequency interval ∆Ω The car-rier light is not part of these Wigner functions The displacement in (a)represents a classical amplitude modulation (b) corresponds to the absenceof any photons with a frequency offset of plusmnΩ from the local oscillator field(c) and (d) represent states whose amplitude modulation depth is more pre-cisely defined than that of the ground state Fig 10 shows Wigner functionspectrum for a broadband squeezed vacuum field Every Wigner functiondescribes the modulation field at some modulation frequency Ωi integrated

28

X

Y

X

(a) (b)

X X

(c) (d)

Y

Y Y

Figure 9 Simplified representation of Wigner functions ndash The darker the shadedareas the larger is the phase-space quasi-probability Shown are four different (time-independent) states of a modulation mode at frequency Ω for a specific resolution bandwidth ∆Ω Panel (a) represents a coherent state the displacement (α) corresponds to aclassical amplitude modulation Panel (b) represents the ground (vacuum) state (c) adisplaced squeezed state and (d) a squeezed vacuum state both with squeeze angle θ = 0The latter is in analogy to Fig 8 Again the light field that carries the modulation is notpart of the pictures

over the resolution bandwidth (RBW) of ∆Ω

The Glauber-Sudarshan P -function ndash The P -function [Glauber (1963)Sudarshan (1963)] is calculated by de-convoluting the Wigner function fromthe ground state uncertainty [Gerry and Knight (2005)] For displaced vac-uum states (coherent states) the P -function corresponds to a displaced δ-function The mathematical expression of the P -function of a squeezed statecontains infinitely high orders of derivatives of the δ-function [Vogel andWelsch (2006)] Such a function contains negativities but cannot be dis-played It is possible however to define a phase-space quasi probabilityfunction for squeezed states that can be displayed and that does show neg-

29

0

Ω1

Ω2

Ω

Single-sided spectrum

Y

X

Figure 10 Hint of a Wigner function spectrum ndash A single-sided spectrum (positivefrequencies only) with respect to the carrier field can be used to visualize a broadbandsqueezed field Shown are two examples displaying a squeezed vacuum state at Ω1 anda displaced amplitude squeezed state at Ω2 The individual Wigner functions cover theresolution bandwidth ∆Ω gt 0 (not shown) In general the squeezing strength as well asthe squeeze angle and the displacement are a function of sideband frequency

ativities as a sufficient and necessary condition for certifying the squeezingeffect This lsquononclassicality functionrsquo is calculated by de-convoluting theWigner function from an uncertainty distribution that is steeper than theGaussian distribution A pronounced negativity of a squeezed vacuum stateof up to 69 standard deviations was found [Kiesel et al (2011)]

The double-sided phasor picture ndash This phasor picture links quantumstates of modulations with the quantum states of the contributing opticalfields [Bachor and Ralph (2004)] and is mathematically described by thetwo-photon-formalism [Caves and Schumaker (1985) Schumaker and Caves(1985)] Generally a weak amplitude or phase modulation at frequency Ω ofa carrier field at optical frequency ω can be understood as the carrierrsquos beatwith two optical frequencies at ωplusmnΩ The double-sided phasor picture is ableto display a spectrum of different and independent modulation frequenciesin the rotating frame of the carrier field The carrier light field is time-independent but the upper and lower sidebands are not They rotate withplusmnΩi(2π) respectively around the frequency axis

30

ω0

Yω

Xω

Upper sideband

Lower sideband

ω0+Ω1

ω0+Ω2

ω0-Ω2

ω0-Ω1

ω

Upper sideband

Lower sideband

Yω

Xω

ω0+Ω1

ω0+Ω2

ω0-Ω2

ω0-Ω1

ω

ω0

Figure 11 Double-sided phasor pictures ndash Phase spaces at optical frequency ω0 plusmnΩirotate around the frequency axis with sideband frequency plusmnΩi Its sign determines thedirection of rotation with respect to the local oscillator in the rotating frame at ω0 Apair of phase spaces need to be superposed to provide a description of a modulation fieldat |Ωi| Top Amplitude quadrature squeezed field with a coherent displacement at |Ω2|The displacement corresponds to a classical amplitude modulation The uncertaintiesof all optical frequencies are circular and larger than that of the ground state (dashed)Quantum correlations are indicated by additional symbols More details are given in themain text Bottom Corresponding spectrum of (displaced) vacuum states which do nothave any quantum correlations

31

Fig 11 shows such a double-sided phase space picture where the carrierrsquosmodulation at Ω1 is in a squeezed vacuum state and where the modulationat Ω2 is in a displaced squeezed state The picture shows how a classicalamplitude modulation as well as the quantum statistic of a modulation fieldis decomposed into contributions from upper and lower sidebands For asqueezed modulation field the upper and lower sidebands show no squeezedbut circular thermally excited quantum uncertainties The uncertainties of apair of sidebands however show correlations as well as anti-correlations InFig 11 these (anti-) correlations are marked with times and + for the modulationfrequency Ω1 and with N and bull for the modulation frequency Ω2

33 Covariance matrix representation of (single-party) squeezed states

Since squeezed states have a Gaussian quantum statistic four numbersare sufficient for their full description These numbers are the second mo-ment of the quadrature amplitude showing the strongest squeezing and thesecond moment of its orthogonal quadrature amplitude as well as their firstmoments describing the displacement These four numbers are sufficientto calculate the Wigner function shown in Fig 8 In general the quadra-ture of strongest squeezing is not perfectly aligned with one of the axesof the measurementrsquos coordinate system The so-called covariance matrix(VXXVXY VY XVY Y ) [Simon et al (1994)] accounts for phase space rotationsand enables the calculation of how these states evolve within an interfero-metric arrangement Their components are normalized to the vacuum noisevariance ∆2Xvac = ∆2Yvac and read

VXY =

langXY + Y X

rangminus 2

langXranglang

Yrang

2∆2Xvac

(18)

The following examples represent the ground state a pure 10 dB am-plitude quadrature squeezed state and a pure 10 dB squeezed state with asqueeze angle of 45

Vvac =

(1 00 1

) V0

01 =

(01 00 10

) V45

01 =

(505 495495 505

) (19)

with V4501 =RT

45V001R45 where Rα = (cosα minussinα sinα cosα) is the rota-

tion matrix

32

34 Phase space representation of two-mode (bi-partite) squeezed states

XA

YA

XB

YB

Figure 12 Bi-partite squeezed vacuum state ndash Shown is a Wigner-function-basedphase space representation in close analogy to the single party version in Fig 9 (d) Thepicture describes a single modulation at frequency Ω with bandwidth ∆Ω Measurementsat party A and B reveal local Wigner functions that correspond to thermal states sincethe uncertainties (indicated by the color and by the large circles) are symmetric and largerthan that of the ground state (indicated by the small dashed circles) The uncertaintieshowever show correlations and anti-correlations here indicated by + and times respectivelyThe strength of these (anti-)correlations are indicated by ellipses Bi-partite squeezingie entanglement is present if the short axes of the ellipses are shorter than the diameterof the ground state uncertainty The picture in fact represents Einstein-Podolsky-Rosenentanglement [Einstein et al (1935)] From a measurement of XA or YA the correspondingmeasurement result at party B can always be inferred with an uncertainty that is smallerthan the ground state uncertainty

A bi-partite state enables a measurement on subsystem A and simulta-neous a measurement on subsystem B For a large number of simultaneousensemble measurements of the same quadrature amplitude Xϑ the followingtwo joint quadrature variance can be calculated

∆2(XAϑ plusmn XB

ϑ ) (20)

A state that is symmetrically shared between two parties (A and B) is calleda two-mode squeezed state if the variances of joint quadrature measurementsfulfill the following inequality [Duan et al (2000)] ie

∆2(XA minus XB)

∆2(XAvac minus XB

vac)+

∆2(Y A + Y B)

∆2(Y Avac + Y B

vac)lt 2 (21)

33

with ∆2(XAvac plusmn XB

vac) = ∆2(Y Avac plusmn Y B

vac) = 2∆2Xvac A lsquotwo-mode squeezedstatersquo reveals entanglement in the second moments of the measurement statis-tics It is thus a lsquobi-partite Gaussian entangled statersquo

Fig 12 displays a (pure) bi-partite squeezed vacuum state distributedbetween A and B The state shows full symmetry regarding its subsystemsat the two sites The large circles and the colored area represent Wignerfunctions of the subsystems Measurements of the quadrature amplitudesXA YA XB and YB show identical variances and the correlations and anti-correlations have identical strength since ∆2(XA minus XB) = ∆2(YA + YB) lt12 for our normalization of quadrature amplitudes having a ground statevariance of 14

Generally a symmetric bi-partite squeezed state fulfills another quantita-tive (Gaussian) entanglement criterion if less than 50 of the vacuum state issymmetrically mixed into the initially pure state Bi-partite squeezed statesare always entangled but in this case they are even Einstein-Podolsky-Rosen(EPR) entangled [Reid (1989)] allowing the demonstration of the quantumsteering effect [Einstein et al (1935) Schrodinger (1935) Reid (1989) Cav-alcanti et al (2009)] The first such experiment was performed by Ou etal [Ou et al (1992)] using type II parametric down-conversion (PDC) Laterexperiments produced bi-partite squeezed vacuum states by overlapping twosqueezed vacuum states each produced with type I PDC on a balanced beamsplitter and used the entangled output for the demonstration of quantumteleportation [Furusawa et al (1998) Bowen et al (2003ca)] The criterionin Eq (21) and the EPR criterion from [Reid (1989)] was experimentallycompared in Ref [Bowen et al (2003b)] The steering effect in asymmet-ric bi-partite squeezed states were recently experimentally characterized inRef [Handchen et al (2012)]

Fig 12 shows features similar to those in the top part of Fig 11 Thisis not a coincidence and shows that a bi-partite squeezed state can also begenerated by spatially splitting the upper and lower sideband of a (single-party) squeezed state This was first experimentally demonstrated by thegroup of E Polzik [Schori et al (2002)] and later used for EPR multiplex-ing of a single longitudinal mode of a squeezing resonator [Hage et al (2010)]

35 Covariance matrix representation of bi-partite squeezed states

Also the full information of bi-partite states including the entanglementcan be cast by the covariance matrix [Simon et al (1994)] which can be used

34

to calculate the propagation of these states in laser interferometers Againall variances are normalized to the vacuum noise variance in full analogy toEq (18) The generic bi-partite covariance matrix has dimension 4times4 andreads

Vbp =

VXAXA

VXAYA VXAXBVXAYB

VYAXAVYAYA VYAXB

VYAYBVXBXA

VXBYA VXBXBVXBYB

VYBXAVYBYA VYBXB

VYBYB

with

VXAYB =

langXAYB + YBXA

rangminus 2

langXA

ranglangYB

rang2∆2Xvac

ϑ

(22)

Due to the symmetry in Eq (22) the 4times4 covariance matrix is fully spec-ified by just ten independent coefficients If the phase spaces at A and Bare aligned along the strongest correlations and anti-correlations the ma-trix components referring to different quadrature amplitudes eg VXAYA arezero Such entangled states can be produced by overlapping two squeezedfields with a squeeze angle difference of 90 on a balanced beam splitter

A symmetric bi-partite squeezed vacuum state which is also called anlsquoS-classrsquo [DiGuglielmo et al (2007)] bi-partite squeezed vacuum state shows(anti-)correlations in two joint quadratures as defined in Eq (21) For a puresuch state of 10 dB squeezing the covariance matrix reads

Vbp10|10 =

505 0 495 0

0 505 0 minus495495 0 505 0

0 minus495 0 505

The following covariance matrix describes a so-called lsquoV-classrsquo 10 dB bi-partite squeezed vacuum state Here only one joint quadrature shows 10 dBsqueezing whereas the orthogonal joint quadrature shows vacuum noise Thestate is obtained by overlapping one 10 dB squeezed state with a vacuumstate on a balanced beam splitter

35

Vbp0|10 =

055 0 045 0

0 55 0 minus45045 0 055 0

0 minus45 0 55

The first measurement of all elements of such a covariance matrix was achievedin [DiGuglielmo et al (2007)]

36 Photon numbers of squeezed states

In contrast to the ground state squeezed vacuum states do have photonexcitations As said earlier quantum theory links the wave and the particlepictures Indeed the squeeze factor of a modulation mode is directly con-nected to a certain photon number excitation Squeezed states of light areproduced via spontaneous photon pair generation eg by parametric down-conversion The following operator S is called the lsquosqueeze operatorrsquo [Gerryand Knight (2005)] It creates and annihilates photon pairs

|r θ〉 = S(r θ) |0〉 (23)

where |r θ〉 is a squeezed vacuum state with squeeze parameter r and squeezeangle θ and |0〉 is the vacuum state The definition of the squeeze operatoris

S(r θ) = exp

[1

2

(reminusiθa2 minus reiθadagger 2

)] (24)

The following shows that this definition indeed results in a state with squeezedquadrature amplitude variances Lets set θ = 0

〈0| Sdagger(r 0) X S(r 0) |0〉 =1

2〈0| Sdagger(r 0)

(a+ adagger

)S(r 0) |0〉 (25)

〈0| Sdagger(r 0) Y S(r 0) |0〉 =i

2〈0| Sdagger(r 0)

(aminus adagger

)S(r 0) |0〉 (26)

Using the Baker-Hausdorff formula we get

Sdagger(r θ) a S(r θ) = a cosh r minus adaggereiθsinh r (27)

Sdagger(r θ) adagger S(r θ) = adaggercosh r minus aeminusiθsinh r (28)

36

Since 〈0| X |0〉 = 〈0| Y |0〉 = 0 also Eqs (25) and (26) are zero To finallycalculate the variances we need

〈0| Sdagger(r 0) X2 S(r 0) |0〉 =1

4〈0| Sdagger(r 0)

(a2 + adaggera+ aadagger + adagger2

)S(r 0) |0〉

〈0| Sdagger(r 0) Y 2 S(r 0) |0〉 = minus1

4〈0| Sdagger(r 0)

(a2 minus adaggeraminus aadagger + adagger2

)S(r 0) |0〉

Given that SSdagger is the identity and using again Eqs (27) and (28) we get theexpected variances

∆2X =1

4

[cosh2 r minus 2cosh r sinh r + sinh2 r

]=

1

4eminus2r

∆2Y =1

4

[cosh2 r + 2cosh r sinh r + sinh2 r

]=

1

4e2r

Since the squeeze operator can only create and annihilate photon pairsa squeezed vacuum state without photon loss must correspond to an evennumber of photons But not only photon loss also a coherent displacementleads to flattening out the odd-even oscillations The probability of detectingN photons in a pure displaced squeezed state are derived for instance in[Gerry and Knight (2005)] and read

P (N) = |〈N |α r θ〉|2 =(05 tanh r)N

N cosh rexp

[minus|α|2 minus 1

2(αlowast2eiθ + α2eminusiθ)tanh r

]times∣∣∣HN

[(αcosh r + αlowasteiθsinh r)

radiceiθsinh(2r)

]∣∣∣2 (29)

where HN is the N th Hermite polynomialFig 13 shows the photon number distributions for 5 different pure squeezed

states according to Eq (29) Panels (a) to (c) show squeezed vacuum stateswith 43 dB 86 dB and 172 dB of squeezing Panel (d) shows the moregeneral case of a squeezed state with a coherent displacement α 6= 0 Dueto θ = 0 the state is amplitude quadrature squeezed Panel (e) refers to thecorresponding phase quadrature squeezed state For comparison panel (f)shows the photon number distribution of the coherent state with the samedisplacement

The panels in Fig 13 represent the diagonal elements of the statersquos den-sity matrix in number basis Only the latter also contains the coherences

37

002

004

006

000 000

002

004

006

008

403020100 Photon number n

403020100Photon number n

Pro

babi

lity

403020100000

005

010

015

100 2 4 6 8100 2 4 6 800

04

06

02

100 2 4 6 800

03

02

01Pro

babi

lity

Pro

babi

lity

08

00

04

06

02

08

(a) (b)

(c) (d)

(e) (f)

Figure 13 Photon number distributions ndash All panels represent pure states (a) 43 dBsqueezed vacuum state (r = 05 α = 0) (b) 86 dB squeezed vacuum state (r = 1 α = 0)(c) 172 dB squeezed vacuum state (r = 2 α = 0) (d) displaced 86 dB squeezed state(r = 1 θ = 0 α = 4) (e) displaced 86 dB squeezed state (r = 1 θ = π2 α = 4) (f)Coherent state (r = 0 α = 4) The average photon numbers are about 027 14 131174 174 and exactly 16 see Eq (30)

between photon numbers [Gerry and Knight (2005)] Figures as shown heregenerally do not give full descriptions of the states

38

A squeezed vacuum state (r 6= 0) always has a non-zero photon numberand can not be the ground state The average photon number of a puresqueezed vacuum state can be calculated using Eq (8) With the maximallysqueezed quadrature variance ∆2Xsqz the average photon number is given by

n = 〈n〉|α=0θr〉 = ∆2Xsqz +(∆2Xsqz)

minus1

16minus 1

2=eminus2r

4+e2r

4minus 1

2 (30)

with the vacuum noise variance normalized to one quarter A coherent dis-placement further adds |α|2 photons on average

4 Squeezed-light generation

41 Overview

Squeezed light was first produced in 1985 by Slusher et al using four-wave-mixing in sodium atoms in an optical cavity [Slusher et al (1985)]Shortly after squeezed light also was generated by four-wave-mixing in an op-tical fibre [Shelby et al (1986)] and by degenerate parametric down-conversion(PDC) in a 2nd-order nonlinear crystal placed in an optical cavity [Wu et al(1986)] The pumped cavity was operated below its oscillation thresholdie the parametric gain did not fully compensate the round trip losses whichis also called lsquocavity-enhanced optical-parametric amplification (OPA)rsquo

The early day experiments achieved squeeze factors of a few percent up toabout 3 dB Today squeeze factors of more than 10 dB are directly observedin several experiments [Vahlbruch et al (2008) Eberle et al (2010) Stefszkyet al (2012) Vahlbruch et al (2016)] All of them are based on cavity-enhanced OPA (below threshold) The parametrically amplified mode isdegenerate ie signal and idler modes are identical In particular the down-conversion process is of lsquotype Irsquo which means that the amplified mode has awell-defined polarization Squeezed states can also be generated above oscil-lation threshold In Refs [Villar et al (2006) Jing et al (2006)] bi-partitesqueezing was generated with above-threshold PDC Both experiments usedtype II PDC which provides orthogonally polarized signal and idler fieldsType II PDC below threshold was also used to generate squeezed and bi-partite squeezed fields [Grangier et al (1987) Ou et al (1992)] All these

39

experiments were performed in the continuous-wave regime which is also thefocus of this Review Squeezed states of modulations of trains of laser pulseshowever have been also generated since the 1980s using either PDC or theoptical Kerr effect [Slusher et al (1987) Bergman and Haus (1991) Our-joumtsev et al (2006) Dong et al (2008)] For an overview of the develop-ments in squeezed-light generation in the continuous-wave as well as pulsedregime see Ref [Bachor and Ralph (2004)] Squeezed-light generation inopto-mechanical setups [Aspelmeyer et al (2014)] which use the intensitydependent phase shift from radiation pressure was discussed in Refs [Paceet al (1993) Rehbein et al (2005) Corbitt et al (2006)] and recently ex-perimentally achieved by several groups [Brooks et al (2012) Safavi-Naeiniet al (2013) Purdy et al (2013)]

42 Degenerate type I optical-parametric amplification (OPA)

This section provides a graphical description of how degenerate type IOPAPDC turns a vacuum state into a squeezed vacuum state and a coher-ent state into a displaced squeezed state The process requires a bright pumpfield and a 2nd-order nonlinear crystal For simplicity we set all nonlinearitiesabove 2nd-order to zero

Let us consider a short segment of the second-order nonlinear crystalpumped with light of optical frequency 2ν All other modes that enter thecrystal shall not contain any photons ie are in their vacuum states Ofthese the only mode of interest is that at optical frequency ν which spatiallyoverlaps with the pump mode Fig 14 shows the total electric field of theoptical input E in and the 2nd-order nonlinear dielectric polarisation of thecrystal P The latter is proportional to the total electric field of the outputEout The pump field at 2ν periodically drives the vacuum field at ν betweenregions of low and high polarisation This process transforms the vacuumstate into a squeezed vacuum state in the output [Bauchrowitz et al (2013)]The output further contains the hardly depleted pump field and frequencydoubled parts of the pump field at 4ν It is again emphasized that Fig 14displays OPA in a small segment of the crystal In reality the nonlinear effectaccumulates over the crystal length or even over several passages since thecrystal is usually put into an optical resonator A noticeable effect is achievedif all infinitesimal contributions constructively interfere This is achieved incase of phase matching ie if the wave fronts of the modes at 2ν and νpropagate with the same speed and thus do not run out of phase Note that

40

t

t

Squeezing

Figure 14 Optical parametric amplification of a vacuum state ndash The upper leftcorner shows the crystal polarization P(E) = ε0

(χ(1)E + χ(2)E2

) ie the separation of

charge carriers by the electric component of an optical field E The graph illustrates howan input quantum field (from below) is projected into an output quantum field (towardsthe right) In the example shown here the input field is composed of a classical pumpfield E in

2ν at frequency 2ν and zero-point fluctuations E inν of a field at frequency ν cf

Fig 7(c) The superposition E in of these two fields is transferred into a time-dependentdielectric polarization that is the source of (and thus directly proportional to) the elec-tric component of the output field Eout The quantum uncertainty of the output fieldshows a phase-dependent (parametric) amplification at frequency 2ν Spectral decompo-sition of the output field Eout reveals coherent amplitudes at frequencies 2ν and 4ν and asqueezed vacuum state Eout

sqzν cf to Fig 7(d) The concept of this figure was published inRef [Bauchrowitz et al (2013)]

41

t

t Amplitudequadraturesqueezing

Figure 15 Optical parametric amplification of a coherent state ndash The pictureshows how a displaced vacuum state is transformed into a displaced squeezed state Thepumprsquos electric field is maximal when the input field at fundamental frequency ν shows azero crossing The phase relation described results in an output state that is amplitudequadrature squeezed If the pump field at the input in phase was shifted by half of itswavelength the squeezed output field were a phase quadrature squeezed The squeezinggeneration displayed here corresponds to the transition from Fig 7(a) to Fig 7(b) but withan additional reduction of the displacement The concept of this figure was published inRef [Bauchrowitz et al (2013)]

in actual squeezing experiments the 4ν component is usually suppressed byphase miss-matching

Fig 15 shows the same process but now for an input field at frequencyν in a coherent state In this case the relative phase between the two inputstates ∆ϕ = ϕ2ν minus 2ϕν is relevant In Fig 15 the relative phase is set suchthat the expectation value of the field at frequency ν is zero when the pump

42

(a) (b)

(c) (d)

X

Y

X

YΔϕ = ndash50deg

X

Y

Δϕ = 0deg

X

Y

Δϕ = 90deg

Figure 16 Phase-space illustration of degenerate OPA ndash The (displaced) dashedcircle in each diagram represents the uncertainty of the initial state at optical frequencyν The (displaced) shaded area represents the state after degenerate optical parametricamplification All quantum uncertainties shown correspond to pure states The boldgreen arrow describes the bright second-harmonic pump field whose uncertainty can beneglected The phase between the 2nd harmonic pump and the initial state (∆ϕ = ϕ2ν minus2ϕν) determines the result of the parametric amplification

field reaches its maximum (∆ϕ = 90) The output at the fundamentalfrequency is then an amplitude squeezed state with a deamplified coherentamplitude

Fig 16 summarizes the squeezing operation on the vacuum state as wellas on displaced vacuum states for different phase relations ∆ϕ between thetwo input fields

43 Cavity-enhanced OPA

Placing the nonlinear crystal inside a cavity can greatly enhance thedown-conversion efficiency but not only that A cavity introduces a thresholdfor the pump power above which the parametric gain is infinite just limited

43

by the finite pump power In this case the vacuum uncertainty of the inputfield at frequency ν is amplified to a bright laser field at frequency ν Thedevice is then called an optical-parametric oscillator (OPO) For the gener-ation of squeezed states however the pump power is usually kept (slightly)below threshold Due to nonzero optical loss there exists a pump powersmaller than the threshold above which the tiny improvement of squeezingis not noticeable anymore Getting the pump power closer to the thresholdcould even reduce the observed squeeze factor if a fluctuating squeeze an-gle projects anti-squeezing into the observed quadrature amplitude [Franzenet al (2006) Suzuki et al (2006) Dwyer et al (2013)] The cavity has an-other important purpose It confines the transverse spatial mode usuallyto TEM00 This mode confinement is crucial for any efficient application ofthe squeezed state in laser interferometry since it allows the suppression ofanti-squeezing from other transversal modes The squeezing process requiresa nonlinear material that should show negligible absorption at both opticalfrequencies involved in particular at the wavelength of the squeezed modeIn Refs [Vahlbruch et al (2008) Mehmet et al (2009)] 10 dB and 116 dBof squeezing were achieved using MgOLiNbO3 The highest squeeze factorstoday are produced in (quasi phase matched) periodically poled KTP [Eberleet al (2010) Mehmet et al (2011) Stefszky et al (2012) Vahlbruch et al(2016)]

The optical cavity that is built around the nonlinear crystal is vital forsqueezed-light generation and it deserves a detailed consideration Gener-ally the mode propagating away from a cavity is the result of interferenceat the cavity coupling mirror One contribution is given by the intra-cavityfield attenuated by the amplitude transmission coefficient t of the outcouplingmirror The second contribution is given by the outside field that is reflectedby the same mirror with amplitude reflectivity r =

radic1minus t2 and spatially

overlapped with the first Also the mode from a squeezing resonator is suchan interference product

The impedance matched resonator

Let us consider first an empty optically stable and loss-less Fabry-Perotresonator built from two identical mirrors each with amplitude reflectivityr = r1 = r2 lt 1 A propagating field be perfectly mode-matched to one ofthe cavity resonances In this setup the resonator shows zero reflection andthe resonator is said to be impedance matched (for all such input fields)

44

|α〉

r1α

r12

= +

|α〉 r22 = r1

2

minusr1α equiv t1 αcav = 1minus r12 minusα sdot r1

1minus r12

0

XY

Figure 17 Empty impedance-matched resonator ndash Mode-matched and resonant lightthat enters the cavity from the left here displayed by a quantum phasor for a coherentstate is fully transmitted including its quantum uncertainty The back-reflected statedestructively interferes with itself for all frequencies well within the cavity linewidth The180 phase shift of the transmitted cavity field amplitude minusr1α is a necessary conditionin order to fulfill energy conservation on cavity resonance Impedance matching is forinstance achieved for a resonant cavity with matched mirror reflectivities (r2

2 = r21) and

zero optical loss The complex amplitude of the field traveling towards left inside the cavityis readily deduced from the figure and reads αcav = minusαr1(1minus r2

1)minus12 (highlighted by thedashed circle) In the displayed setup zero field uncertainties are reflected however alsothe vacuum state that enters the cavity from the right is fully transmitted (not shown)

r1α

XY

r12 r2

2 =1

= + |α〉

minus 1+ r1( )α equiv t1 αcav = 1minus r12 minus 1+ r1( )α

1minus r12

|α〉

Figure 18 Empty maximally overcoupled resonator ndash Maximal overcoupling isachieved for a resonant cavity with a perfect end-mirror reflectivity (r2

2 = 1) and zerooptical loss For a given input-mirror reflectivity r2

1 the intracavity light power is maximalMode-matched and resonating light entering the resonator from the left is fully reflectedThe complex amplitude of the field traveling towards left inside the cavity is readilydeduced from energy conservation to αcav = minusα(1 + r1)(1 minus r2

1)minus12 (dashed circle) Inthis setup no uncertainty from the right couples to the left side of the cavity

45

Obviously the interference described in the previous paragraph is fullydestructive The same resonator also shows zero reflection of the input fieldrsquosquantum uncertainty since the interference happens between parts of thesame quantum state The mode propagating away from such a resonatorhowever is not in a nonclassical but in a vacuum state because the vacuumstate that enters the cavity through the opposite site is also fully transmittedThe interference at the coupling mirror of an impedance matched resonatoris displayed in Fig 17

r1α

r12

+

=

OPA PDC

Xcavg r1(1+r1) Xcav

Δ2Xg 0 Δ2Yg

XY

r22 =1

|α〉

Figure 19 Squeezing resonator ndash Shown is the interference at the zero-loss squeezingresonator operated at threshold The lower line represents the perfectly squeezed modepropagating away from the cavity towards the left The parametric gain medium inside thecavity deamplifies the X quadrature of the cavity mode (Xcav) by the factor r1(1 + r1)which is the ratio of the intra-cavity field amplitudes of the two previous figures The Xquadrature of the field that is back-reflected towards the left destructively interferes withitself similar to the situation of the impedance matched cavity in Fig 17 The parametricpower gain per resonator round-trip (G) needs to mimic the effect of an end mirror withreflectivity r2

2 = r21 For this reason the deamplification of Xcav corresponds to a round-

trip deamplification factor of r1 equivradic

1G The round-trip amplification factor for Ycav

then is 1r1 equivradicG which exactly compensates for the outcoupling and thus determines

the parametric oscillation threshold (threshold for bright lasing) The variances of thequantum uncertainties ∆2Ycav and ∆2Y are thus infinite In this setup no field uncertaintyfrom the right couples to the left of the cavity and a perfectly X-quadrature-squeezed fieldoutside the squeezing resonator is produced

46

The perfectly over-coupled single-ended resonator

We now increase the reflectivity of the far mirror lsquo2rsquo to being perfect (r2 = 1)This way the counter-propagating vacuum state can not enter the cavityAgain a propagating field be perfectly mode-matched through mirror lsquo1rsquo toone of the cavity resonances For frequencies well inside the cavity linewidththe situation is displayed in Fig 18 The setup protects the left side of thecavity against vacuum fluctuations entering through mirror lsquo2rsquo but of coursedoes not squeeze quantum noise The intra-cavity built-up factor is too highfor achieving destructive interference below the vacuum uncertainty on theleft side of the resonator

The impedanced-matched single-ended squeezing resonator

Building on the two previous concepts the straight forward approach now isto start from the perfectly over-coupled single-ended resonator and insert anattenuator into the cavity that does not couple the cavity mode to any bathbut still results in a roundtrip efficiency of precisely r1(lt 1) in amplitudeOptical loss is not appropriate since it increases the coupling of the cavitymode to a thermal bath neither would any phase-insensitive atenuator beappropriate It is easy to show that a phase-insensitive attenuator adds ad-ditional uncertainty since otherwise the commutation relation [a adagger] = 1 isviolated The amplification process that matches our requirement is OPA Toachieve infinite squeezing in X on cavity resonance a second-order nonlinearcrystal needs to be put into the cavity and pumped such that the intra-cavityamplitude quadrature is attenuated by the factor (1 + r1)r1 (on cavity reso-nance) with respect to the empty cavity This factor is readily deduced fromFigs 17 and 18 Due to the symmetry in parametric amplification the intra-cavity phase quadrature is then amplified by (1 + r1)r1 and the round-tripgain has a value of 1r1 in amplitude In this situation not only infinitesqueezing but also the (laser) threshold of the resonator is achieved sincethe round-trip gain of the intra-cavity phase quadrature equals its roundtriploss here fully given by the incoupling mirror

The physical descriptions in Figs 17 to 19 are fully consistent with ob-servations in squeezing experiments The consideration above in particularshows that the intra-cavity field shows a finite squeezing strength while theexternal field shows infinite squeezing The strongest intra-cavity squeezefactor possible is (1 + r1)2r2

1 In the high reflectivity limit this factor corre-

47

sponds to 6 dB Averaged over the full cavity mode the squeeze factor of thecavity mode is in this limit even limited to 3 dB [Walls and Milburn (2008)]Higher intra-cavity squeeze factors are possible for lower mirror reflectivities

44 The generation of squeezed light for laser interferometry

With the insights gained in the previous subsection we now turn to ac-tual experiments The application of squeezed states in laser interferometrycertainly requires large squeeze factors (idealy accompanied with the highestpossible purity) to maximize the impact in terms of sensitivity improvementIn cavity-enhanced OPA the highest parametric gain is achieved on cavityresonance ie at zero sideband frequency But this is not the main reasonwhy this Subsection focusses on the generation of squeezed states at lowsideband frequencies The application of squeezed states in a laser inter-ferometer requires that their sideband frequencies cover the devicersquos signalband Ground-based gravitational wave (GW) detectors have a detectionband from about 10 Hz to 10 kHz frequencies which can be considered aslsquolowrsquo compared to typical frequencies in quantum optics experiments

Squeezing at MHz sideband frequencies is easier to observe than at acous-tic frequencies because the latter are often polluted with excess noise fromlight beams that serve as control beams [Bowen et al (2002) McKenzie et al(2004)] and parasitic interferences from back-scattered light [Vahlbruch et al(2007)] Furthermore the observation of squeezing at low sideband frequen-cies requires a more stable setup since larger measuring times are necessaryThe observation of strong squeezing at MHz frequencies however alreadysets an upper limit to the optical loss of the setup At least the same squeezefactor can be observed at lower frequencies

There are two different main topologies for squeezing resonators TheFabry-Perot-type standing-wave resonator consists of a minimum number ofmirror surfaces and has the advantage of being compact and thus robustagainst mechanical vibrations Usually one or even two mirror coatings aredirectly placed on the spherical and polished surfaces of the nonlinear crys-tal itself [Wu et al (1986) Grangier et al (1987) Breitenbach et al (1998)Vahlbruch et al (2008) Eberle et al (2010) Vahlbruch et al (2016)] TheBowtie traveling-wave resonator has the advantage of providing a separa-tely accessible counter propagating mode for cavity length control [Ou et al(1992) Takeno et al (2007)] It shows no direct back-reflection of incoupledlight which helps reducing parasitic interferences [Stefszky et al (2012)]

48

(a)

(b)

(c)

(d)Squeezingresonator

To inter-ferometer

OPA

SHG

BHD

Mode cleaner

LO

Laser

DBS

DBS

Figure 20 Generation of squeezed light ndash (a) Example of a 2nd-order nonlinearcrystal for the squeezed-light generation at 1064 nm Shown is a bi-convex 65 mm long7MgOLiNbO3 crystal whose polished surfaces also carry the mirror coatings of the res-onator The crystal thus realizes a monolithic squeezing resonator as it was used forthe first demonstration of 10 dB squeezing [Vahlbruch et al (2008)] (b) Optical con-figuration of a half-monolithic (hemilithic) standing-wave squeezing resonator Here thecavity length can be adjusted by displacing the coupling mirror The crystal surface insidethe cavity is anti-reflection coated The photograph shows a 10 mm long PPKTP crystalsqueezing resonator as used for the GEO 600 squeezed-light source [Abadie (2011)] (c)Mechanically stable housing of a standing-wave squeezing resonator The crystalrsquos temper-ature is stabilized at its phase matching condition using Peltier elements (d) Schematicfor the squeezed-light generation After spatial filtering of continuous-wave laser lighttwo hemilithic standing-wave resonators are employed The first generates second har-monic pump light (SHG) The second (OPA) generates a squeezed vacuum field at theinitial wavelength The squeezed states are observed by a balanced homodyne detector(BHD) or alternatively sent and mode-matched to the optical mode of an interferometerbeforehand LO local oscillator DBS dichroic beam splitter

49

Fig 20 (a) and (b) show photographs of typical nonlinear crystals used forsqueezed-light generation at near infra-red wavelengths The crystals shownhere form a monolithic standing-wave squeezing resonator (a) or are partof a half-monolithic standing-wave squeezing cavity (c) shows a tempera-ture stabilized and mechanically stable housing of the squeezing resonator(d) shows a schematic of a full setup for the generation of squeezed vacuumstates of light for an application in a laser interferometer The only brightinput required for the squeezing resonator (OPA) is the second-harmonicpump field The resonator mode at fundamental frequency is thus initiallynot excited by photons ie it is in its ground state characterized by vacuumfluctuations due to the zero point energy see Fig 7 (c) [Gerry and Knight(2005)] The pump field spontaneously decays in the degenerate pair of sig-nal and idler fields The combined down-converted field leaving the resonatorexhibits quantum correlations which give rise to a squeezed photon countingnoise when overlapped with a bright coherent local oscillator beam Thedetection is done either in a balanced homodyne detector (BHD) or with asingle photo diode The squeeze factor increases the closer the pump powerof the squeezing resonator gets to the oscillation threshold and the lower theoptical loss on down-converted photon pairs is

441 High squeeze factors ndash minimizing decoherence

Squeezed states of light have significant impact on the sensitivity of laserinterferometers if large squeeze factors can be produced Squeezing of 3 dBimproves the signal-normalized quantum-noise spectral density by a factorof 2 This factor corresponds to doubling the (coherent state) light powercirculating inside the interferometer Squeezing of 10 dB corresponds to aten-fold power increase The experimentally demonstrated squeeze factorswere considerably improved in recent years [Takeno et al (2007) Vahlbruchet al (2008) Polzik (2008) Eberle et al (2010) Stefszky et al (2012)]culminating in a value of as large as 150 dB [Vahlbruch et al (2016)] Thisvalue corresponds to the same reduction of signal-normalized quantum noisethat is achieved by increasing the light power by a factor of 32 (At this pointit is already noted that squeezing the quantum noise can simultaneouslyreduce quantum measurement noise (shot noise) as well as quantum backaction noise (radiation pressure noise) This is not possible with scaling thelight power of coherent states see Subsec 55)

50

Ideally a parametric squeezed-light source can produce an infinite squeez-ing level see Fig 19 fundamentally just limited by the energy provided bythe pump field In practice the limit is set by decoherence mechanismsThe by far most important one is optical loss Optical loss occurs duringsqueezed-light generation its propagation through the interferometric setupincluding imperfect mode matchings and finally the photo-electric detec-tion Also detector dark noise [Schneider et al (1998)] phase noise [Takenoet al (2007)] and excess noise [Bowen et al (2002)] impair the observablesqueezing strength

Optical loss is usually understood as coupling the squeezed mode to a zerotemperature bath ie overlapping it with a vacuum mode For any amountof loss the resulting state is still squeezed But to be able to directly observesay 10 dB of squeezing the total loss on the state needs to be less than 10in this example cf Eq (16) To minimize optical loss the nonlinear crys-tal as well as lenses and beam splitters in the interferometric path need toshow very low absorption and scattering at the wavelength of the squeezedlight PPKTP shows absorption of about 10minus4cm and below at near-infraredwavelengths Low OH content fused silica is a suitable material for all otheroptics Absorptions of less than 10minus6cm were measured [Hild (2007)] Coat-ings on crystal surfaces and on all other optical components should also showlowest optical loss Total loss of the 10minus6 level are available today Superpol-ished surfaces which show roughnesses with less than 1 A root mean square(integrated over spatial scales from approximately 1 micron to 100 microns)and thus very low scattering are necessary to achieve these low numbersMinimizing the total number of optical components is essential From thisperspective a monolithic squeezing resonator as shown in Fig 20 (a) is theoptimum choice The squeezed mode needs to be matched to the mode ofthe laser interferometer or to the mode of the balanced homodyne detectorVisibilities of up to 998 have been achieved [Eberle et al (2010)] whichcorresponds to a loss of about 04 Of great importance also is the quantumefficiency of the photo-diodes used for detecting the squeezed field (togetherwith the interferometric signal) Recently a quantum efficiency of photo-diodes in a squeezing experiment of (995plusmn 05) was measured [Vahlbruchet al (2016)] To minimize photon loss the photo-diodes had no protectionwindow an anti-reflection coating on the semi-conductor material and theremaining reflection was re-focussed with an external mirror

Also the dark-noise spectral density of the detection electronics reducesthe observable squeezing and needs to be as low as possible Similar to optical

51

noise it also provides a contribution to the observed variance The dark noiseof the detection electronics needs to be much lower than the detected photoncounting noise In [Vahlbruch et al (2016)] it was 28 dB below shot noisebut still reduced the observable squeeze factor from 153 dB to 150 dB

Excess noise emerges if the squeezed mode couples to a nonzero tem-perature bath or to a mode whose excitation is strongly fluctuating (Thecoupling process can always be understood as a beam splitter coupling andis physically described by overlapping electric fields Coupling to a zero tem-perature bath leads to Eq (16)) The captured excess noise variance thenneeds to be added to the initial squeezing variance which deteriorates theobserved squeezing stronger than just mixing in the vacuum mode Excessnoise is less likely to occur at MHz frequencies but can be significant ataudio-band sideband frequencies and below and is thus a serious issue ingravitational-wave detectors [Chua et al (2014)] The reason for that isthat acoustically or thermally excited motions of surfaces produce frequencyshifts of back-scattered light mainly at these low frequencies [Vahlbruch et al(2007)]

Phase noise corresponds to stochastic phase fluctuations between thesqueezed field and the local oscillator within the measuring time It cor-responds to mixing the squeezed mode with itself with a fluctuating squeezeangle [Suzuki et al (2006) Franzen et al (2006)] Phase noise in squeez-ing experiments typically is less of an issue than optical loss [Dwyer et al(2013) Oelker et al (2016) Vahlbruch et al (2016)] The setuprsquos phasenoise can be reduced by making the squeezing resonator more compact andthus mechanically more stable against acoustic and thermal fluctuations ofthe environment and by improving the quality of the servo loops for cavitylength and propagation length controls Operating a squeezed-light resonatorin vacuum might also be beneficial The ability to run a high performancesqueezed-light generator in vacuum was demonstrated in [Wade et al (2015)]

442 Squeezing in the gravitational-wave (GW) detection band

High squeeze factors have been first demonstrated at sideband frequen-cies of a few MHz and above where excess noise is generally negligible whenworking with visible or near-infra-red light Today we know that extendingthe squeezing spectrum towards the audio-band and even below is technicallynot always easy but straight forward once a high squeeze factor is achieved

52

at MHz frequencies In most squeezing experiments the main laser light

Figure 21 Photograph of the GEO 600 squeezed-light source ndash The breadboarddimensions are 135 cm times 113 cm The squeezing resonator is high-lighted by the white ar-row and is set up as a standing-wave hemilithic cavity containing a plano-convex PPKTPcrystal of about 10 mm length (see also Fig 20 b) It is pumped with continuous-wave532 nm light that is produced by frequency doubling of light (at angular frequency ω)from a commercial NdYAG laser Two more laser fields at about 1064 nm having fre-quency offsets of more than 10 MHz with respect to ω(2π) serve as optical control fieldsBoth fields are mode-matched and injected into the squeezing resonator together with thesecond-harmonic pump field

at the squeezing wavelength is accompanied by significant noise up to thelaser relaxation oscillation For this reason laser control fields at the opticalcarrier-frequency in the optical path of the squeezed mode need to be avoided[Bowen et al (2002) Schnabel et al (2004) McKenzie et al (2004)] and thesqueezing resonator length and the optical path stabilized by other means[McKenzie et al (2005) Vahlbruch et al (2006)] Furthermore and mostimportantly excess noise due to back-scattering is an issue Back-scattering(also called lsquoparasitic interferencesrsquo) is produced if DC light scatters out ofthe optical path hits a vibrating surface and re-scatters back into the opticalpath [Vahlbruch et al (2007)] Significant back-scattering can be produced

53

in interferometers for the detection of gravitational waves since light powersof several hundreds of kilowatts are used Even back-scattering from the mil-liwatt local oscillator of balanced homodyne detectors is an issue at acousticsideband frequencies and below The recipe for avoiding parasitic interfer-ences turns out to be threefold (i) avoiding scattering by using ultra-cleansuperpolished optics with close to perfect anti-reflex coatings (ii) avoidingback-scattering by carefully blocking all residual (faint) light fields and (iii)reduce the vibrationally and thermally excited motion of all mechanical andoptical parts that could potentially act as a re-scattering surface with re-spect to the optical path [Vahlbruch et al (2007) McKenzie et al (2007)]The insights described above led to the first demonstration of audio-bandsqueezing at frequencies down to 200 Hz [McKenzie et al (2004)] and laterto the first demonstration of squeezing over the full gravitational-wave de-tection band even from as low as 1 Hz [Vahlbruch et al (2007)] Whilea standing-wave squeezing resonator [Ou et al (1992)] can be built in avery compact way that is rather insensitive against mechanical vibrations[Chelkowski et al (2007)] a traveling-wave bow-tie squeezing resonator [Wuet al (1986)] is more tolerant against back-scattered light [Chua et al (2011)]The strongest squeezing in the audio-band of up to 116 dB was reported inRef [Stefszky et al (2012)]

443 The first squeezed-light source for GW detection

The first squeezed-light source for the continuous operation in GW de-tectors had been designed and completed between 2008 and 2010 [Vahlbruch(2008) Vahlbruch et al (2010)] Since then this source has been producingsqueezed vacuum states in a fully phase controlled way using co-propagatingfrequency-shifted bright control beams [Vahlbruch et al (2006)] as an inte-gral part of the GW detector GEO 600 The source is a turn-key device witha fully automated re-lock system [Vahlbruch et al (2010) Khalaidovski et al(2012)] Re-locking is required if the temperature of the environment changessignificantly which drives the actuators outside their dynamic ranges

Up to 9 dB of squeezing over the entire GW detection band was observedusing a balanced homodyne detector (BHD) located close to the squeezingresonator The squeeze factor has been limited by optical loss due to ab-sorption in the PPKTP crystal transmission of the back-surface and thenon-perfect AR-coating of the crystalsrsquos intra-cavity surface The adjustableair gap has been necessary to allow for an easy way to apply length control

54

-12

-8

-4

0

4

8

12

16

20

10 100 1k 10k

Rel

ativ

e no

ise

pow

er [

dB]

Frequency [Hz]

shot noise (a)

squeezed noise (b)

anti-squeezed noise (c)

~ 9dB

Figure 22 Broadband squeezing spectrum ndash Noise power spectra measured on theoutput of the GEO 600 squeezed-light source shown in Fig 21 with a balanced homodynedetector The traces correspond to the spectra of quadrature amplitude variances normal-ized to vacuum noise The resolution bandwidth used increases towards higher frequenciesto reduce the measurement time (a) Shot noise normalized to unity which serves as thereference level (0 dB) (b) Squeezed noise covering the complete detection band of ground-based GW detectors (c) Anti-squeezed noise Peaks at 50 Hz and 100 Hz are the electricmains frequency and its first harmonic The data was published in Ref [Vahlbruch et al(2010)]

Additional optical loss in the path to the balanced homodyne detector mainlyarose due to a Faraday isolator that eliminated parasitic interferences Fi-nally the mode missmatch to the BHD as well as its non-perfect quantumefficiency provided additional loss Inferring the squeeze factor without theBHD detection loss more than 10 dB of squeezing are provided by the sourceSince 2010 it has been used in basically all observational runs of the GEO 600GW detector see Section 6

444 Generation of two-mode (bi-partite) squeezing

lsquoTwo-mode squeezed lightrsquo or lsquobi-partite squeezed lightrsquo is light that allowsfor joint measurements at two locations A and B These joint quadraturemeasurements reveal correlations and anti-correlations with a remaining un-certainty smaller than the ground-state uncertainty which certifies the pres-

55

ence of entanglement cf Subsec 34 Bi-partite squeezed light has been gen-erated by type I and by type II parametric down-conversion In case of type Ithe squeezed fields from two squeezing resonators as described in Subsec 44are overlapped on a balanced beam splitter with a 90 phase shift The twooutput fields together represent the entangled mode [Furusawa et al (1998)Bowen et al (2003c) Eberle et al (2013)] In case of type II signal and idlerfields are non-degenerate regarding polarisation and a single cavity contain-ing an appropriate crystal and a polarising beam splitter are sufficient for theproduction of bi-partite squeezing Also in this case the measurements ofthe quadrature amplitudes of signal and idler fields show large uncertaintiestogether with bi-partite correlations and anti-correlations that are strongerthan the ground state uncertainty of individual subsystems [Ou et al (1992)Villar et al (2006) Jing et al (2006)]

To date the strongest entanglement of bi-partite squeezed light has beenproduced based on type I parametric down-conversion [Eberle et al (2013)]The requirements of producing strong entanglement are identical to those ofproducing strong squeezing outlined above The strength of bi-partite en-tanglement can be given in decibels in full analogy to the squeeze factorPractically the strength of bi-partite squeezing is always somewhat smallerthan that of single party squeezing since it requires an additional mode-matching that results in additional optical loss

45 Conclusions

The first observation of squeezed light was achieved in 1985 [Slusher et al(1985)] Shortly after cavity-enhanced optical parametric amplification forsqueezed-light generation was demonstrated [Wu et al (1986)] which todayenables the observation of up to 15 dB of squeezing [Vahlbruch et al (2016)]Quite generally the maximum squeezing level that is observed does not de-pend on the strength of the optical nonlinearity Squeezing cavities can easilybe operated at their oscillation threshold where they should provide infinitesqueezing if decoherence is zero The main limiting factor is optical lossincluding that of the photo-electric detection

Dedicated experimental research and development towards a squeezed-light source for applications in gravitational-wave detectors can be tracedback to 2002 [McKenzie et al (2002) Bowen et al (2002)] Since then a sur-prising amount of progress has been made culminating in the first squeezed-light source specifically built for the integration into a gravitational-wave

56

detector For the future squeeze factors above 15 dB will certainly be possi-ble by further reducing optical loss This statement is supported by the highdegree of matching between experimental data and a theoretical loss modelas presented in Fig 3 of [Vahlbruch et al (2016)]

5 Quantum noise in laser interferometers

51 Interferometric measurements

The purpose of a laser interferometer is the precise measurement of smallchanges of an optical path length with respect to a reference path Forthis the interferometer transfers the change of the phase difference betweentwo light fields into an amplitude quadrature change of the interferometerrsquosoutput light The latter can easily be detected by a single photo diode Ofgeneral interest are differential changes of the optical path length that aremuch smaller than the laser wavelength ie differential phase changes thatare much smaller than 2π In this case the differential phase change can bedescribed in very good approximation as a differential change of the phasequadrature instead

In order to transfer the phase quadrature signal with minimum loss ahigh interference contrast at the interferometerrsquos beam splitter is essentialAdditionally instrumental noise in terms of unwanted excitations of the out-putrsquos amplitude quadrature needs to be reduced to a minimum Noise arisesdue to power fluctuations of the input laser light back-scattered laser lightinside the interferometer thermally driven displacements of mirror surfacesand in many more ways The important measure of the sensitivity of aninterferometer obviously is its signal-to-noise-ratio The most useful measureis given in terms of the noise spectral density S(f=Ω2π) that is normalizedto the physical unit of the signal S(f) is in fact a lsquonoise-to-signal-ratiorsquoand can be seen as the signal-normalized variance of the photo diode out-put decomposed into spectral components versus sideband frequency f withthe resolution bandwidth of 1 Hz As an example S(100 Hz) = 10minus39 m2Hzmeans that the instrumental noise in the one hertz band around 100 Hz equalsa signal that would be produced if the mirror of one interferometer arm os-cillates with an amplitude of just

radic10 middot 10minus20 m in the very same band Such

small spectral densities are achieved by gravitational-wave detectors [Abbott(2016)]

57

52 Quantum measurement noise and shot noise

The most fundamental noise source in laser interferometers is due to thequantum noise of light which is in fact two-fold [Caves et al (1980)] Firstof all there is lsquoquantum measurement noisersquo which arises in the processof photo-electric detection For coherent states the quantum measurementnoise is the lsquophoton counting noisersquo of mutually independent photons andusually simply called lsquoshot-noisersquo Fig 2 (b i) shows a time series of suchnoise hiding the actual signal The frequency components of the shot noiseare well described by the quantum uncertainty of the output fieldrsquos amplitudequadratures XΩ∆Ω see Subsec 22 (Recall this quantity corresponds to the

differential phase quadrature YΩ∆Ω of the light beams in the interferometerarms) The photon counting noise has a white Fourier spectrum howeverthe lsquoshot noisersquo of an interferometer is usually normalized to the signal whosetransfer function is usually not white for instance due to the presence of armcavities or a signal-recycling cavity

All current and planned gravitational-wave detectors are Michelson-typelaser interferometers with operating points very close to a dark fringe Thelight power in the output port is just a couple of tens of mW which canbe handled by a single photo diode In this configuration the signal-to-shot-noise-ratio is actually maximized which can be shown in three steps [Bachorand Ralph (2004)] For the first step we use plane waves to describe theelectric field in the output port of a Michelson interferometer For perfectinterference contrast at the balanced beam splitter ie for perfect modematching and for defining φ = 0 as the dark port condition we get

Eout(t φ) =1

2E0 sin(ωt+ φ)minus 1

2E0 sin(ωt) (31)

where E0 is the amplitude of the total internal field whose two parts hasaccumulated a differential phase It directly follows for the squared fields

E2out(t φ) =

(sin

φ

2

)2

(E0 cos(ωt+ φ2))2 (32)

We now turn to a light beam with a localized transversal mode that can befocussed onto a photo-electric detector The photo diode has perfect quantumefficiency ie the rate of photo electrons is not only proportional to the rateof output field photons but also has a unity slope efficiency Since the optical

58

frequency is too high to be resolved we consider the averaged light power

P out(φ) =

(sin

φ

2

)2

P (33)

The next step is a formulation of the signal being the derivative of detectedphoton number versus phase Let n be the average value of the photonnumber per measuring time interval Eq (33) can then be rewritten as

nout(φ) =

(sin

φ

2

)2

n (34)

rArr dnout(φ) = n sinφ

2cos

φ

2dφ (35)

The final step is the calculation of the signal-to-shot-noise-ratio Shot noiserefers to coherent states which have a standard deviation of the photonnumber of σ(n) =

radicn

dnout(φ)

σ(nout)=n sinφ

2cosφ

2dφ

radicn sinφ

2

(36)

and find for a signal-to-noise ratio of unity for coherent states and for anon-zero but still small phase difference ∆φCoh 2π

1 =radicn cos

φ

2∆φCoh with φ 6= 0 (37)

In this equation the smallest measurable phase difference is given for φrarr 0

∆φCohmin =

1radicn (38)

This is the well-known shot-noise limit of high-precision phase sensing ∆φCohmin

is the smallest phase shift that can be measured with a signal-to-noise ra-tio of one when using n mutually independent photons per measuring time(those of a coherent state) when the loss of photons is assumed to be zeroThe typical purpose of a laser interferometer is the continuous sensing (moni-toring) of a continuously changing phase An illustrative example is the phasesignal produced by the black hole merger measured by Advanced LIGO onSept 14 in 2015 (Fig 1 in [Abbott (2016)]) The measuring interval should

59

be short to be able to resolve the time-evolution of the signal Generally themeasurement of an arbitrary signal that lasts for a finite time thus needs tobe understood as l subsequent measurement intervals using n photons eachIt can be shown that Eq (38) is also valid for interferometers operated athalf fringe ie when each output port contains the same light power In thiscase photo diodes need to be placed in both output ports and the actualsignal is provided by their difference voltage

Due to its importance the shot-noise limit deserves some remarksThe phase φ in Eq (31) is the phase difference of two mode-matched fieldsand might be accumulated by a single pass along the length L such as ina Mach-Zehnder interferometer or in a double pass such as in a (simple)Michelson interferometer or in four passes as realized in a Michelson inter-ferometer with folded arms [Grote (2005)] The shot-noise limit in Eq (38)and its scaling therefore holds independent of the number of passes Theclaim in Ref [Higgins et al (2007)] that the scaling according to Eq (38) canbe surpassed by multiple passes is not justifiedThe fact that Eq (38) is derived by approaching φ rarr 0 correctly describesthe actual operation point of gravitational-wave detectors which is close tobut not exactly at a dark port In practice a tiny offset from dark port ischosen at which the shot noise is well above the photo diodersquos electronic darknoiseEq (38) solely depends on the number of quanta but not on the lightrsquos wave-length λ Of course the shot-noise limit for the change of an optical pathlength ∆L does depend on the wavelength and ∆φCoh

min needs to be replacedby ∆φCoh

min = 2π∆LCohminλ

Finally an essential result of the shot-noise limit is that the ideal precisemeasurement should use lsquoas much quanta as possible per measuring inter-valrsquo which translates to lsquoas much light power in the interferometer armsas possiblersquo Eq (38) is indeed the one and only reason why gravitational-wave detectors use high power lasers power-recycling and arm resonatorsExtending the measuring time for a given light power can also improve thesensitivity but only if the signal repeats ie is periodic Let us assume thatone period of the signal is resolved by l intervals using n photons each Inthis case repeating the overall measurement k times improves Eq (38) by1radick The fundamental statement of Eq (38) however does not change

since the actual photon number n may then simply incorporate the factor k

60

For a given average photon number the shot-noise limit in Eq (38) canonly be surpassed by using photons that are quantum correlated ie by usingnonclassical states of light How is the shot-noise limit surpassed with thehelp of squeezed states A Michelson interferometer that is operated closeto a dark fringe acts like an almost perfect mirror for both input ports Allthe input light is back-reflected towards the laser source This also accountsfor the quantum uncertainty of the input light The quantum uncertaintythat impinges onto the photo diode thus (mainly) enters the interferometerthrough its (almost) dark port An interferometer that uses displaced co-herent states entering from one port can thus be improved by replacing theordinary vacuum entering the signal output port by a squeezed vacuum stateThis was the proposal by CM Caves in 1981 [Caves (1981)] which is labeledhere with lsquoCSVrsquo If the differential phase quadrature of the interferometer issqueezed Eq (38) then within the limit of large coherent state displacementα sinh2r improves to

∆φCSVmin asymp

eminusrradicn (39)

(The above expression is an approximation since the squeezing operationproduces a small number of photons that are not accounted for here) Ofcourse the mode of the squeezed vacuum needs to be precisely matchedto the mode of the interferometer The first experimental demonstrations ofsqueezed phase measurements used a Mach-Zehnder [Xiao et al (1987)] and apolarization interferometer [Grangier et al (1987)] Fig 2 shows how spatialdegeneracy between an externally generated squeezed mode and the signalmode in a Michelson interferometer is achieved using a polarizing beam split-ter and a Faraday rotator Again the limit in Eq (39) can only be achievedif optical loss is zero Optical loss not only reduces the signal but here alsoreduces the squeeze parameter see Eqs (15) and (16)

Let us consider an example The sensitivity of a laser interferometer thatuses coherent states with an excitation of 1023 photons per second can beimproved by a factor of

radic10 by either adding 09 middot 1024 photons per second

or by adding about just 2 photons per second and bandwidth in hertz thatbelong to the 10 dB squeezed vacuum confer Eq (30) Since the full sig-nal band of ground-based GW detectors covers sideband frequencies up to10 kHz just 2 middot 104 photons per second are necessary At a wavelength ofλ = 1064 nm these values correspond to a power increase by 168 kW and

61

37 fW respectively

The question arises whether a scaling of the sensitivity better than propradic1n is possible It was theoretically shown that in principle the scaling

can indeed considerably be improved yielding the so-called Heisenberg limitor Heisenberg scaling [Bondurant and Shapiro (1984) Yurke et al (1986)Braunstein (1992) Holland and Burnett (1993)]

∆φHLmin prop

1

n (40)

The Heisenberg scaling requires nonclassical states that have a certain num-ber of quanta similar to Fock states ie n = n The theoretically optimalstates describe a superposition of n (N) indistinguishable photons in oneinterferometer arm while having zero (0) photons in the second arm and viceversa and were named lsquoN00Nrsquo-states [Dowling (2008)] A specific propertyof these states is lsquosuper-resolutionrsquo The output ports of the interferometershow an n-times faster oscillation of the interference fringes when changingthe phase between the two interferometer arms Super-resolution correspondsto an n-times improved signal transfer function and was demonstrated forinstance in Refs [Rarity et al (1990) Kuzmich and Mandel (1998) Mitchellet al (2004) Afek et al (2010)] The presence of this nonclassical phe-nomenon however does not prove a sensitivity better than the semi-classicalbound according to Eq (38) Sensitivity is rather related to the signal-to-noise-ratio and needs to take into account all imperfections as well as theprobability of a successful detection of the sensing state [Thomas-Peter et al(2011)] All experiments so far used post-selection on particular measure-ment outcomes and neglected the typically large probability that nothingwas detected

Super-resolution was demonstrated with up to n = 5 [Afek et al (2010)]In addition to the fact that super-resolution does not prove a sensitivity bet-ter than the semi-classical bound photon numbers in state-of-the-art super-resolution experiments are extremely small compared to the photon numberof about 1023 (within a measuring interval of one second) of coherent statesused in Ref [Abbott (2016)] and of about 1022 using coherent states plussqueezed vacuum states used in Ref [Abadie (2011)]( which did prove a sen-sitivity better than the semi-classical bound)

Another interesting and related question is what the smallest phase is

62

that can be estimated in a single measurement again using a given numberof quanta Taking into account that no prior information about the phaseshift exists still a scaling proportional to 1n is possible In the limit of largen however an additional factor of π is required in the nominator of Eq (40)[Sanders and Milburn (1995) Berry and Wiseman (2000)] yielding

∆φHLmin asymp πn (41)

The state that can actually achieve this bound is different from the N00Nstate and was found in [Summy and Pegg (1990) Luis and Perina (1996)Berry and Wiseman (2000)] A N00N state is not the optimum state forphase estimation (via a single measurement) since it only provides one bit ofinformation A recent review on generell aspects on phase measurements isgiven by Ref [Demkowicz-Dobrzanski et al (2015)]

It is important to note that Eqs (38-40) do not consider photon lossExperiments that demonstrated super-resolution and aimed for achievingthe scaling in Eq (40) were conditioned on zero photon loss Let η gt 0 bethe average efficiency of detecting (all) photons Eq (38) then reads

∆φηCohmin =

radic1

η n (42)

Eq (39) turns into

∆φηCSVmin asymp

radicηeminus2r + 1minus η

η n(43)

and Eq (40) turns into [Demkowicz-Dobrzanski et al (2012 2013)]

∆φηmin =

radic1minus ηη n

for 0 lt η lt 1 (44)

For non-zero photon loss most interestingly the ultimate sensitivity of aphase measurement for a given photon number also shows a 1

radicn -scaling

The difference between the CSV strategy of using bright coherent states incombination with squeezed vacuum states which is bounded by Eq (43) andthe strategy of using the optimal nonclassical state which is bounded byEq (44) is marginal in practice [Demkowicz-Dobrzanski et al (2013)] Forgravitational-wave detectors and for any other laser interferometer using in-tense light there is no need for an alternative to the CSV strategy

63

We now turn back to the shot noise according to Eq (38) Generally noisecan be decomposed into its spectral contributions For a simple Michelsoninterferometer without arm resonators and without a signal-recycling cavitythe square-root of the single-sided shot-noise spectral density normalized tothe differential arm length change x in units of m

radicHz is given by [Saulson

(1994)] radicSMI

SNx =

radic~c2

2ωPprop 1radic

P (45)

where ω is the optical angular frequency of the quasi-monochromatic carrierlight and P the total light power in both arms including the built-ups fromcavities In combination with a squeezed vacuum whose relative phase gen-erates squeezing of the output lightrsquos amplitude quadrature the right handside reduces according to the factor eminusr Note that the single-sided spectraldensity is only defined for positive sideband frequencies and thus twice aslarge as the double-sided spectral density

The spectral density of the measurement of a GW induced strain is givenby the same expression but normalized to h = xL (If the gravitational waveis oriented in an optimal way with respect to the Michelson interferometerone arm is squeezed while the other is expanded by the same amount of ∆L =x2 and h then corresponds to the actual gravitational-wave amplitude)The square-root of the single-sided shot-noise spectral density normalized tostrain in units 1

radicHz is given by

radicSMI

SNh =

radic~c2

2L2ωP (46)

Equations (45) and (46) show that the smallest measurable signal (corre-sponding to unity signal-to-shot-noisendashratio) is inversely proportional to thesquare root of the laser power and has a white spectrum for sideband frequen-cies much smaller than the carrier frequency see horizontal line in Fig 23

All first- and second-generation GW detectors use power-recycling andadditional cavities to improve their sensitivities Fabry-Perot arm resonatorsdo not only increase the light power but additionally also increase the signalfor signal frequencies inside the resonator linewidth For lossless Fabry-Perot

64

arm resonators the spectral densities in Equations (45) and (46) need to bemultiplied by the following factor [Kimble et al (2001)]

HFP =

radicL2(γ2

FP + Ω2)

c2 (47)

where γFP = cTFP(4L) is the Fabry-Perot arm resonatorrsquos half bandwidthand TFP is the light power transmission of the input mirror The end mirrorsare assumed to have perfect reflectivity A similar expression can be derivedfor describing the improvement due to signal-recycling [Buonanno and Chen(2001)]

In summary shorter laser wavelengths higher light powers and squeezingof the amplitude quadrature of the interferometer output reduce shot noise ina broadband way ie for all signal frequencies Fabry-Perot arm resonatorsas well as signal-recycling provide improvements mainly for frequencies insidethe resonator linewidths

53 Quantum back-action and quantum radiation pressure noise

In laser interferometers quantum back-action noise results from the un-certainty of the lightrsquos radiation pressure force on the interferometer mirrorsand is also called lsquo(quantum) radiation pressure noisersquo (RPN) Its origin isthe quantum uncertainty of the differential amplitude quadrature XΩ∆Ω ofthe fields in the interferometer arms It results in an uncertain momentumtransfer to the mirrors and thus in an position uncertainty of the mirrors atfuture times with respect to their differential mode of motion [Caves et al(1980)] The physical mechanism of radiation pressure corresponds to anintensity dependent phase shift [Pace et al (1993)]The higher the light power in the arms of a laser interferometer the loweris its shot-noise spectral density see Eq (45) Unfortunately the spectraldensity of quantum back-action noise increases with light power The single-sided force noise spectral density reads

radicSRPNF =

radic8~ωPc2

(48)

Whereas the force noise of the quantum radiation pressure has a white spec-trum the RPN does not since the mirrorrsquos reaction to external periodicforces depends on frequency The link between the Fourier component of an

65

external force F (Ω) and the Fourier component of the displacement x(Ω) isgiven be the mechanical susceptibility HM It reads for an harmonic oscillatorwith mass M

HM(Ω) =1

M | minus Ω2 + Ω2M + iΩΩMQ|

(49)

where ΩM is the oscillatorrsquos resonance frequency and Q its quality factorThe square root of the single-sided spectral density of the RPN normalized

to the displacement of an harmonic oscillator with mass M is then given by

radicSRPNx = HM(Ω)

radic8~ωPc2

(50)

In GW detectors the test mass mirrors are suspended as pendula with highmechanical Q-factors and their centre of mass motion corresponds to that ofa harmonic oscillator The resonance frequencies of the pendula are lowerthan the detection band of interest The mechanical susceptibility is thereforeoften approximated for the so-called free-mass regime as H fm

M (Ω) = (mΩ2)minus1The square root of the single-sided spectral density of the RPN normalizedto differential displacement of two mirrors with each of mass M in a simpleMichelson interferometer is given by [Saulson (1994)]radic

SfmMIRPNx =

radic2~ωPc2m2Ω4

propradicP (Ω ΩM) (51)

where m = M2 is the mirrorsrsquo reduced mass In case of a simple Michel-son interferometer that is enhanced with arm cavities the spectral density inEq (51) needs to be multiplied with the expression given in Eq (47) In com-bination with a squeezed vacuum whose relative phase generates squeezing ofthe output lightrsquos phase quadrature the right hand side reduces according tothe factor eminusr Note if the radiation pressure noise is squeezed the shot noisemust be anti-squeezed or vice versa The radiation pressure noise calibratedto strain of space time is given by the right side of Eq (51) divided by theinterferometer arm length LIn summary heavier masses longer laser wavelengths lower light powersand squeezing of the amplitude quadrature in the interferometer arms re-duce radiation pressure noise

66

1 10 100 1000Frequency [Hz]

Radiation pressure noise

Shot noise

SQL

10-21

10-18

10-15

10-12

radicSx

[mradic

Hz]

ndashndash

ndashndash

Figure 23 Displacement-normalized quantum noise spectral densities ndash Consid-ered is a simple Michelson interferometer with neither arm cavities nor signal recyclingThe two end mirrors (m = 100 g) of the interferometer arms are suspended as pendulahaving a resonance frequency of ΩM2π = 1 Hz and a Q-factor of 107 The interferometeruses quasi-monochromatic light (in coherent states) with a total power of 4 kW Opticalloss and the offset from a dark output fringe is assumed to be negligible Wavelengthλ = 1550 nm The standard quantum limit (SQL) corresponds to the lowest noise achiev-able at a given sideband frequency when varying the light power without using quantumcorrelations

54 Interferometer total quantum noise and the standard quantum limit

Both shot noise and radiation pressure noise contribute to the total quan-tum noise of a given interferometer If they are not quantum correlatedwhich is the case for a conventional Michelson interferometer when detectingthe output lightrsquos amplitude quadrature their variances add up (The resultis not shown in Fig 23) It can easily be deduced from the previous sectionsthat changing the laser power will shift the two quantum noise contributionsHowever the total quantum noise never goes below the standard quantumlimit (SQL) [Braginsky and Manukin (1967)]

Let us consider Fig 23 for sideband frequencies much greater than thependulum resonance Here the test mass mirrors react as free masses whenexerted to external forces The SQL in this free-mass regime is calculated

67

by minimizing the sum of the squares of Eqs (45) and (51) [Saulson (1994)]

Sfmtotx =

~c2

2ω

[1

P+

4ω2

c4m2Ω4P

] (52)

Its derivative reads

dSfmtotx

dP=minus1

P 2+

4ω2

c4m2Ω4 (53)

Setting the above equation to zero provides the optimum laser power versussideband frequency in order to achieve the lowest total quantum noise

P fmopt =

c2mΩ2

2ω (54)

Inserting the optimal light power into Eq (57) provides the square root ofthe single-sided noise spectral density of the free-mass SQL in m

radicHzradic

SfmSQLx =

radic2~mΩ2

(55)

Again m is the reduced mass and dividing by the interferometer arm lengthL yields normalization to the GW-induced strain h Eq (55) shows that theSQL falls off with sideband frequency The corresponding equation for aMichelson interferometer that uses arm cavities readsradic

SfmFPSQLx =

radic~

mΩ2

(1

HFP

+HFP

) (56)

with HFP according to Eq (47)Using the expression for the SQL the square root of the total quantum

noise spectral density of a Michelson interferometer in the free-mass approx-imation can be written asradic

SfmFPtotx =

radicSfmFP

SQLx

2

[1

k+ k

] (57)

with the radiation pressure coupling parameter

k(Ω) =2ωP

mc2Ω2 (58)

68

For a fixed light power and fixed reduced mass of the mirrors the quantumnoise limited interferometer reaches the SQL when k = 1 which is realizedat the angular sideband frequency ΩSQL =

radic2ωP(mc2)

Note that neither squeezing the phase quadrature nor squeezing the am-plitude quadrature of the interferometer light leads to sub-SQL performance[Caves (1981)] also confer Ref [Schnabel (2005)] Fig 3 (left) As we willsee in the next sections the standard quantum limit can be surpassed if shotand radiation pressure noise are correlated Then the total quantum noise isnot given by the sum of the variances ie the sum of the squares in Eqs (45)and (51)

55 Squeezed light for surpassing the standard quantum limit

A measurement with sensitivity better than the standard quantum limit(SQL) is also called a lsquoquantum non-demolition (QND)rsquo measurement [Bra-ginsky and Khalili (1995 1996) Kimble et al (2001)] Several QND tech-niques for laser interferometers were proposed in recent decades [Jaekel andReynaud (1990) Kimble et al (2001) Purdue and Chen (2002) Chen (2003)McClelland et al (2011) Danilishin and Khalili (2012) Graf et al (2014)]What they have all in common is they exploit quantum correlations betweenobservable uncertainties

Arguably the most extensive way of introducing quantum correlationsand surpassing the SQL is the injection of squeezed states of light [Jaekeland Reynaud (1990)] If the squeezed quadrature angle of the injected statesis neither 0 nor 90 the quantum uncertainties of the amplitude and phasequadrature amplitudes that describe the differential field in the two interfe-rometer arms become correlatedLet us consider a very simplified setup that just consists of a quasi-mono-chromatic light field that is back-reflected from a quasi-free mirror Thelight power and the mass be such that reflected light in a coherent stateresults in a measurement of the mirror position with a noise spectral den-sity at the SQL at sideband angular frequency ΩSQL At this frequencyquantum measurement noise and back-action noise are of the same sizeie the uncertainty in X produces an equally large additional uncertaintyin Y Upon reflection the quadrature amplitude variances change from∆2X = ∆2Y = 14 to 2∆2X = ∆2Y = 12 This result corresponds tothe situation in Fig 23 at the crossing frequency of shot noise and radiation

69

pressure noise The coupling of the uncertainty variances can be describedby the matrix K = (1 minusk 0 1) where k = 1 at the SQL If the modulationstate at ΩSQL is the ground state its variances are transferred according to

KT

(1 00 1

)K =

(1 0minus1 1

)(1 00 1

)(1 minus10 1

)=

(1 minus1minus1 2

) (59)

In accordance with Fig 23 the variance of YΩSQLis twice as large as the vac-

uum noise varianceNow let the quantum noise of the light field be 10 dB squeezed at 45

(Eq (19)) The projection of the quantum uncertainty onto the X-observableproduces the radiation pressure noise by being transferred with the couplingfactor k = 1 at the SQL into the Y -observable in fact with negative signsince a larger value of X produces a larger optical path length and thus aretardation of the phase Due to the squeezing at 45 the initial uncertaintyin Y cancels with the additional uncertainty that originates from the onein X The following calculation shows that the strength of the cancellationcorresponds to the initial squeezing strength Upon reflection the quantumuncertainties transform in the following way(

1 0minus1 1

)(505 495495 505

)(1 minus10 1

)=

(505 minus01minus01 02

) (60)

The state of light after reflection has a squeezed phase quadrature amplitudeThe improvement in comparison to Eq (59) is exactly 10 dB The quantumnoise improvement corresponds to the input squeeze factor and is also a mea-sure by what factor the SQL is surpassed Squeezed vacuum injection thusallows surpassing the SQL upon measuring the conventional Y -quadrature(which is realized by a single photo diode in the interferometerrsquos output port)as first realized by [Unruh (1983) Yuen (1983) Jaekel and Reynaud (1990)]

In the example above the input squeeze angle is optimized for a sin-gle sideband frequency Injecting a broadband squeezed vacuum field withfrequency-independent squeeze angle of 45 would result in a rather bad inter-ferometer quantum-noise performance at frequencies far smaller or largerthan ΩSQL Fig 24 shows the quantum-noise performance if the input fieldhas squeeze angles that are optimized for every k(Ω) as given in Eq (58)Injected squeezing can thus lead to a broadband sub-SQL performance ifthe quantum measurement noise (shot noise) and the quantum back-actionnoise (radiation pressure noise) are correlated in an optimal way Due to the

70

10 50 100 500 1000 5000Frequency [Hz]

Dis

plac

emen

t noi

se sp

ectra

l den

sity

[mradic

Hz]

10-20

10-19

10-18

Total quantum noise 0dB

Total quantum noise -10dB

SQL

Y

X

Y

X

Y

X

Figure 24 Surpassing the SQL with squeezed-light injection ndash At shot-noise lim-ited sideband frequencies squeezing of the Y -quadrature amplitude improves the noisespectral density of the interferometer At radiation-pressure-noise limited sideband fre-quencies squeezing of the X-quadrature amplitude improves the noise spectral density ofthe interferometer If both kinds of quantum noise contribute equally (at the SQL markedwith a dot) a squeeze angle of 45 results in surpassing the SQL by the full squeeze factorsee Eq (60) In the graph here the squeeze angle is optimized for all frequencies result-ing in a broadband quantum noise reduction [Jaekel and Reynaud (1990)] Measurementsensitivities beyond the SQL (shaded area) are in the so-called quantum non-demolition(QND) regime [Kimble et al (2001)] Dashed horizontal lines represent the (squeezed)shot noise Dashed straight lines with negative slope represent the (squeezed) radiationpressure noise The calculations use 10 dB of squeezing a conventional Michelson inter-ferometer with neither arm resonators nor signal recycling a light power at the beamsplitter of 1 MW at a wavelength of λ = 1550 nm and mirror masses of 1 kg

correlation shot noise and radiation pressure noise can be squeezed simulta-neously

71

Light with a frequency-dependent squeeze angle

The discovery that shot noise and radiation pressure noise can be squeezedsimultaneously and thus a broadband reduction of quantum noise beyond theSQL be achieved required the insight that the spectral analysis of light definesa spectrum of many lsquosideband modulation modesrsquo that all can be in differ-ent quantum states An ordinary squeezing resonator which is on resonancefor light at twice the pump wavelength produces a spectrum of modulationmodes that all have the same squeeze angle A frequency-dependent squeezeangle can be introduced by reflecting such a field from a detuned single-ended filter cavity which was suggested by Kimble and coworkers [Kimbleet al (2001)] They showed that the optimal frequency dependence thatleads to the broadband improvement shown in Fig 24 can be realized byusing altogether two filter cavities as shown in Fig 25 Motivated by this re-sult research and development on filter cavities for optimizing the frequencydependence of broadband squeezed fields has been very active in recent years[Corbitt et al (2004) Chelkowski et al (2005) Dwyer et al (2013) Kweeet al (2014) Straniero et al (2015) Oelker et al (2016)]

Photo diode Squeezed vacuum

Faraday Rotator

Coherent light

YΩΔΩ

Figure 25 Frequency dependent squeezing injection ndash A broadband squeezed fieldwith a frequency-dependent squeeze angle that is optimal for gravitational-wave detectorsis produced by reflecting off an ordinary broadband squeezed field from two detuned opticalfilters [Kimble et al (2001)]

A light field with a frequency-dependent squeeze angle was first demon-strated in Ref [Chelkowski et al (2005)] see Figs 26 and 27 The experi-

72

Detuned filter cavityLockingphoto diode

Isolator

EOM R asymp 1

SHG

EOM

DBSOPA

LaserIsolator

Mode cleaner

LO

Homodyne detector

Figure 26 Generation of a frequency-dependent squeezing ndash A frequency-dependent orientation of the squeeze ellipse was first demonstrated in Ref [Chelkowskiet al (2005)] Initially a conventional spectrum of squeezed vacuum states of light was gen-erated in a squeezing resonator (lsquoOPArsquo) The squeezed vacuum was transmitted throughan optical isolator to a detuned filter cavity After reflection the squeezed vacuum stateswere absorbed in a balanced homodyne detector (BHD) The phase of the BHDrsquos localoscillator (LO) was changed for quantum state tomography of the squeezed states in diffe-rent regions of the spectrum The result showed a frequency-dependent orientation of thesqueeze ellipse see Fig 27 SHG second harmonic generation EOM electro-optical mod-ulator for applying phase modulation sidebands for cavity length control DBS dichroicbeam splitter R mirror reflectivity λ4 quarter wave plate for turning linear polarizedlight into circular polarised light and vice versa

ment consisted of a standing-wave squeezing resonator which produced ans-polarized broadband amplitude quadrature squeezed field accompanied bya dim continuous-wave DC control field with a wavelength of λ = 2πcω =1064 nm The squeeze bandwidth covered sideband frequencies up to aboutΩ(2π) = 30 MHz which corresponded to the linewidth of the squeezing res-onator The optical cavity for producing the frequency dependence of thesqueeze angle was a standing-wave cavity composed of a plane incouplingmirror of reflectivity r1 =

radic097 and a concave end mirror of reflectivity

r2 =radic

09995 The cavity length was L = 50 cm resulting in a linewidthof 147 MHz The squeezed field first passed a Faraday isolator to preventinterference effects between the filter cavity and the squeezing resonator A

73

Phase quadrature

min

max

-2 -1 0 1 2

141 MHz

-2

-1

0

1

2

Am

plit

ud

e q

uad

ratu

re

Figure 27 Frequency-dependent squeezing ndash Picture top right Reconstructed con-tour plot of the Wigner function of the sideband modulation at Ω(2π) = 141 MHz afterreflection from a 1515 MHz detuned filter cavity The state shows quantum correlationsbetween phase and amplitude quadratures ie squeezing at an angle of here about 40The white circle visualizes the standard deviation of the vacuum state uncertainty Thewhite ellipse represents the standard deviation of the squeezed uncertainty Small picturesMeasurement results on the same continuous-wave laser beam at various sideband frequen-cies around 15 MHz For each tomographic picture noise histograms of 100 equidistantquadrature angles were measured In each case the laser beam was phase locked to a ref-erence beam and the quadrature angle stably controlled and stepwise rotated The phasereference was given by a phase modulation at 198 MHz [Chelkowski et al (2005)] Thepicture was first published in Ref [Schnabel (2005)] (copyright ccopy2007 by Imperial CollegePress)

λ4-waveplate turned the s-polarized field into a circularly polarized beamwhich was then mode matched into the detuned cavity The retro-reflectedfield was analyzed by a balanced homodyne detector (BHD) for quantumstate tomography The filter cavity was electro-optically controlled to be de-tuned by 1515 MHz with respect to the DC control field The cavity length

74

control was achieved by the Pound-Drever-Hall (PDH) locking techniqueutilizing a circularly polarized laser beam that carried 15 MHz phase mod-ulation sidebands and was coupled into the filter cavity from the back Theoutput voltage of the BHD was characterized by a spectrum analyser as wellas used to perform quantum state tomography In the latter case the BHDoutput voltage was mixed down with an electronic local oscillator at differentradio-frequencies around 15 MHz and low-pass filtered to set the resolutionbandwidth to ∆Ω(2π) = 100 kHz The final electric signal corresponds to atime series of quadrature amplitude measurements XθΩi∆Ω Quantum statetomography is a method to reconstruct the phase space quasi-probability dis-tribution (Wigner function) of quadrature amplitudes from sets of measuredXθΩ∆Ω distributions when varying the angle θ [U Leonhardt (1997)] Forevery sideband frequency Ωi 100000 quadrature values were measured di-vided up on 100 equidistant quadrature angles Each quadrature angle wasstably controlled with a precision of plusmn1 Fig 27 shows the reconstructedWigner functions which were all measured on the same laser beam but atdifferent sideband frequencies For these measurements the detuned filtercavity was locked to the lower sideband at minus1515 MHz The result clearlyshows the frequency-dependent orientation of the squeeze ellipse In a morerecent experiment a frequency-dependent squeeze angle was also realized inthe kHz regime [Oelker et al (2016)]

56 Optomechanically induced (ponderomotive) squeezing

The radiation pressure of light when acting on a movable mirror re-sults in an intensity dependent phase shift [Pace et al (1993)] The cou-pling produces a so-called lsquoponderomotive effectrsquo [Braginsky and Manukin(1967)] which is of third order optical nonlinearity and which transforms abright coherent state inside an interferometer into a squeezed state of light[Vyatchanin and Matsko (1993)] This type of squeezed-light generationis usually called lsquoponderomotive squeezingrsquo or lsquooptomechanical squeezingrsquoConsequently even if no squeezed field is injected into the interferometercorrelations between the quadrature amplitudes are generated that allow forsurpassing the SQLPonderomotive squeezing as produced by the interferometer itself can onlybe exploited for evading back-action (radiation pressure noise) It can notbe used to squeeze the interferometer shot noise This is why ponderomotivesqueezing is fundamentally less extensive than injecting externally producedsqueezed states of light [Corbitt et al (2006)] suggested an external pon-

75

deromotive squeezing source for gravitational-wave detectors In this casedue to its external generation also the interferometerrsquos shot noise can besqueezed Recently ponderomotive squeezing was observed for the first time[Brooks et al (2012) Purdy et al (2013)] The achieved squeeze factors aremuch smaller than those produced by optical-parametric down-conversion[Vahlbruch et al (2016)]

Let us have a look again at Eq (59) Rotating the covariance matrix onthe right by arctan(minus

radic54minus 12) asymp minus58 indeed reveals squeezing

(cos 58 minussin 58

sin 58 cos 58

)(1 minus1minus1 2

)(cos 58 sin 58

minussin 58 cos 58

)asymp(

262 00 038

) (61)

The vacuum-noise normalized variance of 038 corresponds to about 42 dBof ponderomotive squeezing This is the general value that is produced atthe angular sideband frequency ΩSQL At higher frequencies the squeezefactor gets smaller at lower frequencies higher The squeezing strength of42 dB can be observed if the photo diode in the interferometer output portis replaced by a balanced homodyne detector using a local oscillator phase ofabout minus58 It can be shown however that the optimal signal-to-quantum-noise-ratio at the SQL is achieved for a local oscillator phase of exactly 45At this angle back-action is fully evaded

Full evasion of radiation pressure noise at all frequencies requires an op-timized frequency dependence of the relative local oscillator phase This canbe achieved by reflecting off the interferometer output field from two detunedfilter cavities [Kimble et al (2001)] The scheme was called lsquovariational out-putrsquo In the case of zero optical loss this scheme can fully evade radiationpressure noise just leaving the shot noise as the only quantum noise contri-bution

The variational-output scheme can be used to enhance the frequency-dependent squeezed input scheme The right site of Eq (60) shows thatthe output statersquos squeezing is not optimally detected in the Y -quadratureRather than with a single photo diode the detection should be done witha balanced homodyne detector with optimized phase of its local oscillatorIn this case the output lightrsquos quantum noise is solely given by squeezedshot noise The total quantum noise in Fig 24 would then be given by thelowest (dashed) horizontal line This combined scheme was called lsquosqueezed

76

variationalrsquo [Kimble et al (2001)] It can be realized by reflecting off theinterferometer output light from in total two optical filter cavities placed infront of the balanced homodyne detector

57 Conclusions

The highest quantum-noise-limited sensitivities of high-precision laserinterferometers are achieved by employing a large number of quanta tomaximize the signal strength in combination with strongly squeezed statesto minimize the quantum noise From this perspective it is clear that thequantum-noise-limited sensitivity of future gravitational-wave detectors willbe further improved ndash by increasing the light power and the squeeze factorTo be able to do so the optical loss in these devices needs to be reduced

In principle the optical loss in laser interferometers can be made smallbut never zero Recent theoretical research has shown that for any non-zero loss the sensitivity scales proportional to 1

radicn at best where n is the

average photon number per measurement This scaling is efficiently achievedby combining strongly displaced coherent states with squeezed vacuum statesof light

If a repeated measurement is not only limited by quantum measurementnoise but also by quantum back-action noise squeezed states of light can beused to simultaneously reduce both ie in the case of an interferometer shotnoise and radiation pressure noise

6 The first application of squeezed light in an operating gravita-tional-wave detector

Squeezed states of light have been successfully used to improve the sensi-tivity of the gravitational-wave detector GEO 600 from 2010 up to the pointwhen this Review was written [Abadie (2011) Grote et al (2013)] Afterdecades of proof-of-principle experiments [Xiao et al (1987) Grangier et al(1987) McKenzie et al (2002 2004) Vahlbruch et al (2005 2006 20072008) Goda et al (2008)] the implementation of a squeezed-light source inGEO 600 has resulted in the first sensitivity improvement beyond shot noiseof a measurement device that targets new observations in nature The im-plementation of squeezed states in GEO 600 was not done to provide anotherproof-of-principle demonstration but was realized because it offered a rela-tively cheap way of further improving the measurement sensitivity Of course

77

the sensitivity of GEO 600 can also be further increased by purely classicalmeans however the implementation of arm resonators to enable higher lightpowers without increasing the thermal load on the beam splitter or eventhe realization of longer interferometer arms are much more expensive Inthis respect the sensitivity improvement of GEO 600 with squeezed light canarguably be regarded as the first lsquotruersquo application that developed out of thefield of lsquononclassical (quantum) metrologyrsquo (Note that the term rsquoquantummetrologyrsquo is currently defined in different ways [Giovannetti et al (2006)Gobel and Siegner (2015)] and the term rsquononclassicalrsquo referring to a non-classical P-function gives a distinct description)

61 Gravitational waves

Einsteinrsquos General Theory of Relativity [Einstein (1916)] or simply lsquoGen-eral Relativityrsquo (GR) predicts that accelerating mass distributions producegravitational radiation analogous to electromagnetic radiation from accel-erating charges Experimental evidence of their existence is given by theobservation of the slow spiraling together of two neutron stars caused bythe loss of orbital energy to gravitational waves The inspiral rate exactlymatches the predictions of Einsteinrsquos theory [Weisberg and Taylor (2005)]Recently Advanced LIGO observed gravitational waves for the first time[Abbott (2016)] thereby giving the go-ahead for gravitational-wave astron-omy The gravitational-wave source was the final inspiraling and the mergerof two black holes 13 billion light years away from earth

Gravitational-waves evolve in the far field of the source propagate withthe speed of light and are measurable on earth with laser interferometersFig 28 displays a gravitational wave propagating along a certain directionGravitational waves are dynamical changes of space-time They are transver-sal and quadrupolar in nature and have two polarization states

A variety of known astrophysical and cosmological sources are predictedto emit gravitational radiation that should reach the Earth with a measur-able strength [Sathyaprakash and Schutz (2009)] The first gravitational waveevent detected was produced by two black holes of 36 and 29 solar massesDuring the final 02 seconds of their inspiraling they produced a peak gravi-tational strain in our solar system of 10minus21 covering frequencies up to 250 Hz[Abbott (2016)] Other predicted sources are mergers of neutron stars supernovae and background signals from the Big Bang According to GR GWsfrom complex astrophysical sources carry a plethora of information that willhave a major impact on gravitational physics astrophysics and cosmology

78

L +Δ L

Binary system

fBS

Figure 28 Space-time oscillation ndash Gravitational waves are dynamical deformations ofspace-time that form in the plane perpendicular to the direction of wave propagation Asa result distances between free-falling test masses in a transverse plane will change witha strain h = ∆LL For black hole or neutron star binary systems with orbital frequencyfBS distances will oscillate at frequency fGW = 2fBS The wavelength of this oscillation isgiven by λGW = cfGW where c is the speed of light The wave of orthogonal polarizationwith respect to the one shown is rotated by 45 around the propagation axis

62 Interferometric detection of gravitational waves

Current gravitational wave detectors are kilometre-scale laser interfero-meters [Dooley et al (2016) Aasi (2015) Acernese (2015) Aso et al (2013)]Continuous-wave laser light is split into two beams traveling in orthogonaldirections Both beams are reflected back towards the central beam splitterwhere they interfere Gravitational waves change the optical path lengthdifference and thus the light power directed towards the photo-diode thatis positioned in the signal output port of the beam splitter A gravitationalwave at frequency fGW = ΩGW(2π) reveals itself as a light-power modula-tion at the same frequency The spectral decomposition of the output signalis described by a spectrum of the quadrature amplitude YΩ∆Ω introduced inSec 3 It corresponds to the amplitude quadrature amplitude of the outputlight and relates to the differential phase quadrature of the interferometerarms

79

Photo diode

Squeezed vacuum

Faraday Rotator

Signal

Quantum noise

Coherent light XΩΔΩ

YΩΔΩ

(a) (b)

YΩΔΩFigure 29 Squeezed-light-enhanced interferometric measurement ndash (a) Michelsoninterferometer with arm cavities power recycling and signal recycling (see main text forexplanation) The interferometer is operated close to a dark fringe such that the quantumnoise entering from the dark port is back-reflected The squeezed field is mode-matched tothe signal output field (b) Phase space diagram of the gravitational-wave signal outputat sideband frequency Ω(2π) The quantum noise is squeezed below the ground stateuncertainty and thus the signal to quantum noise ratio improved

The first key ingredient of an interferometric gravitational-wave detectorare suspended heavy mirrors that can be regarded as quasi-free in the di-rection of laser light propagation thereby acting as test masses that probespacetime Being on ground current detectors are located in rather noisy en-vironments that allow the realization of undisturbed quasi-free mirrors onlyabove a sideband frequency of the order of 10 Hz Since sufficiently strongGW signals are expected up to a frequency of 10 kHz todayrsquos gravitationalwave detectors target at signals in the acoustic band from 10 Hz to 10 kHzThe quasi-free motion of the test mass mirrors in this frequency regime isachieved by suspending the mirrors as sophisticated multiple-stage pendulain vacuum chambers [Aasi (2015)] Far above the pendularsquos resonant frequen-cies which are typically around 1 Hz the centre of masses of the mirrors areisolated from vibrations of the ground and they react on frequency compo-nents of small external forces approximately as free masses The mirrors andtheir suspensions are built from materials having exquisitely high mechani-cal quality factors This helps to concentrate the thermal energy that causes

80

displacements of the mirror surface into well-defined vibrational frequencymodes At these particular very sharp frequencies no gravitational wavescan be detected

The second key ingredient of an interferometric gravitational-wave de-tector is laser light with a power of up to hundreds of kilowatts or evenmegawatts The light is quasi-monochromatic and needs to show very lowamplitude (quadrature) noise and phase (quadrature) noise at sideband fre-quencies within the detection band Low amplitude noise is necessary toprovide a shot noise limited output field It is also necessary to avoid classi-cal radiation pressure noise which becomes an issue if the light power or themirror masses in the two arms are not identical Low phase noise is requiredif the storage time of the light in the two arms is not identical This mightaccidentally occur due to different linewidths of the arm cavities or mightbe part of the interferometer design to allow for the length control schemeproposed by Schnupp [Heinzel et al (1998)] To maximize the light power in-side the interferometerrsquos cavities it should be produced in an almost perfecttransversal spatial distribution of a Gaussian TEM00 mode

Light sources of gravitational-wave detectors are ultra-stable NdYAGmaster-slave systems that provide up to 200 W of light at 1064 nm [Winkel-mann et al (2011) Kwee et al (2012)] The high power in the interferometerarms is achieved by cavity built-ups in the so-called power-recycling cavityand in the arm cavities Power recycling uses a partially reflective mirror thatis located between the light source and the interferometer beam splitter Itssurface is matched to the lightrsquos wave front and forms an optical cavity to-gether with the rest of the interferometer Since gravitational-wave detectorsare operated close to a dark fringe large power built-ups can be achievedThe highest power built-up is achieved for a mirror transmission equal to the(given) interferometer round trip loss In this case an impedance-matchedcavity is achieved The power-recycling cavity as well as the arm cavitiesare stabilized on resonance for the input light The difference between theirfunctionality is that the power-recycling cavity does not limit the detectionbandwidth of the interferometer GEO 600 as well as Advanced LIGO em-ploy a third type of cavity the so-called signal-recycling cavity Similarlyto power recycling a partially reflecting mirror that is placed between theoutput port of the beam splitter and the photodiode is used to resonantly en-hance the GW signal [Meers (1988)] The signal-recycling cavity resonantlyenhances the signal modulation fields within its linewidth without furtherenhancing the carrier light power In combination with low linewidth Fabry-

81

Perot arm resonators it can also be used to extract the signal by reducing theeffective finesse of the arm resonators for the signal sidebands This schemeis called resonant sideband extraction [Heinzel et al (1996)] The signal-recycling cavity has also been tested in a detuned setting in which just theupper or lower sideband is extracted or resonantly enhanced respectively[Heinzel et al (2002)] Current gravitational-wave detectors however usecarrier-tuned signal recycling

All these techniques are lsquoclassicalrsquo approaches for maximizing the signal-to-shot-noise ratio At frequencies above a few hundred Hertz howevershot-noise is still the limiting noise source in gravitational-wave detectorsFuture gravitational-wave detectors will therefore use even higher light pow-ers but further increasing the light power becomes more and more challeng-ing Optical materials with less light absorption need to be found to coun-teract an increasing thermal load inside the mirrors Mirror masses need tobe further increased to counteract the increasing radiation pressure noiseNonclassical approaches are superior and become more and more attractivethe farther classical approaches are pushed to the extremes Nonclassicalapproaches allow for simultaneously increasing the signal-to-shot-noise ratioand the signal-to-radiation-pressure-noise ratio without changing light poweror mirror masses see Fig 24 They also allow for a complete evasion of radia-tion pressure noise [Braginsky and Khalili (1995 1996) Kimble et al (2001)]see Subsec 56

63 Squeezed-light enhancement of the gravitational-wave detector GEO 600

In 2010 GEO 600 was equipped with the squeezed-light source shownin Fig 18 The location of the squeezed-light source close to the outputport is shown in Fig 30 It was known that GEO 600 was shot-noise limitedat sideband frequencies above about 700 Hz In this frequency regime thereplacement of the ordinary vacuum states that entered the interferometerfrom the output port by a spectrum of squeezed vacuum states was expectedto reduce the noise spectral density into the nonclassical regime It was notprecisely clear what squeezing factors could be expected since the opticalloss upon mode-matching an external field into the output port propagationalong the arms and the final photo-electric detection was not determined

Critical components were the quantum efficiency of the photo-diode aswell as the optical loss of Faraday rotator for overlapping the squeezed fieldwith the output mode Also the transversal output mode of GEO 600 was an

82

issue since it contained significant contributions from higher order modesto which a good mode matching of the squeezed mode was not possible Theimplementation of the squeezed-light source thus was accompanied with anew custom made InGaAs photo-diode with 3 mm diameter The goal was aquantum efficiency of greater 99 [Vahlbruch et al (2016)] Also the Fara-day rotator was custom-made and optimized for lowest optical loss which

Figure 30 GEO 600 ndash View into the central building of the British-German GW detectorlocated close to Hannover Germany The vacuum chambers contain the suspended beamsplitter power- and signal recycling mirrors additional input and output optics as well asmirrors to realize a double pass of the laser light through the 600 m long interferometerarms By courtesy of the AEI

83

involved a rather precise rotation of the polarisation of 45 plusmn 05 over anaperture of more than 15 mm Finally a ring cavity (output mode cleaner)was placed in front of the photo-diode which acted as a passive filter forhigher transversal modes Since GEO 600 was not limited by radiation pres-sure noise and since it used a carrier-tuned signal-recycling cavity a frequencyindependent orientation of the squeezing angle was optimum After several

100 200 300 500 700 1000 2000 3000 500010-22

10-21

10-20

Obs

erva

tory

noi

se c

alib

rate

d to

str

ain

[1radic

Hz]

10-19

10-18

10-17

Noise calibrated to test m

ass displacement [m

radicHz]

Sideband frequency f

Figure 31 Nonclassical reduction of the GEO 600 instrumental noise ndash Shownare the square roots of the noise spectral densities without squeezed vacuum states (top)and with squeezed vacuum states (bottom) [Abadie (2011)] Their calibrations [Affeldtet al (2014)] to gravitational-wave strain and differential arm length change are shownon the left and right y-axes respectively Note that both traces increase towards higherfrequencies due to the frequency-dependent signal enhancement of the signal-recyclingcavity The injection of squeezed vacuum states leads to a broadband noise reductionof up to 35 dB at shot-noise limited frequencies The spectral features are for instancecaused by excited violin modes of the mirror suspensions (600ndash700 Hz and harmonics)Data by courtesy of the AEI

months the combination of the squeezed-light source and the gravitational-wave detector succeeded The main laser of the squeezed-light source wasphase locked to the main laser of GEO 600 and a stable mode matching be-

84

tween the squeezed field and the interferometer output field was achieved[Abadie (2011)] The effect on GEO 600rsquos (strain normalized) noise spectraldensity is shown in Fig 31 At frequencies above about 700 Hz the squareroot of noise spectral density was reduced by up to one third This value cor-responds to a quantum noise variance of 045 (minus35 dB) with the shot noisevariance normalized to unity For isotropically distributed gravitational-wavesources this factor produces a detection rate increase by a factor of 153 asymp 34After its integration into GEO 600 the squeezed-light source was used in allscientific runs seeking for gravitational waves for instance in the observa-tional run S6eVSR4 that was undertaken from June 3rd to September 5th

in 2011 [Grote et al (2013)]Towards the end of 2011 right before the start of the detector upgrade

to Advanced LIGO a nonclassical sensitivity improvement was also demon-strated in one of the LIGO detectors [Aasi (2013)] In this experiment anonclassical sensitivity improvement corresponding to up to 215 dB abovefrequencies of about 150 Hz was achieved The successful test is a strongmotivation for a squeezed-light upgrade of Advanced LIGO Note that thedesign of Advanced LIGO was completed in 1999 and squeezed-light sourceswere not mature at those times

In the past years the squeezing enhanced GEO 600 detector was not onlyused for observations but also was the control of the injected squeezed modefurther improved Stabilizing the overlap between squeezed mode and brightmode of the interferometer to close to perfect is necessary to reduce theeffective optical loss and to maximize the measurable squeezing factor Re-cently superior methods for stabilizing the longitudinal phase of squeezedvacuum mode were found [Dooley et al (2015)] and the first automatic align-ment system for stabilizing and optimizing the transversal mode overlap wasdemonstrated [Schreiber et al (2016)]

64 Are squeezed states the optimal nonclassical resource ingravitational-wave detectors

For a given number of photons Eq (41) quotes the ultimately smallestphase change that can be measured with a signal-to-noise-ratio of one Thescaling with number of photons per measuring time of this Heisenberg limitseems appealing compared to the scaling achievable with coherent states orsqueezed states according to Eqs (38) and (39) respectively The Heisenberglimit however is only valid for precisely zero photon loss Since the non-classical states required to achieve Eqs (40) and (41) show an exponentially

85

increasing sensitiveness to loss when increasing the photon number the ac-tual scaling can not be deduced from Eq (40) Proposals to use Fock statesand the so-called N00N states for optimizing interferometer sensitivities [Hol-land and Burnett (1993) Dowling (1998) Mitchell et al (2004) Afek et al(2010)] are thus only applicable when the experiment is conditioned on zerophoton loss As discussed in recent publications the correct expression forthe fundamental sensitivity limit needs to consider not only the total photonnumber inside the interferometer but also the total photon loss [Dorner et al(2009) Ko lodynski and Demkowicz-Dobrzanski (2010) Knysh et al (2011)Escher et al (2011) Demkowicz-Dobrzanski et al (2012)]

Based on these earlier works Ref [Demkowicz-Dobrzanski et al (2013)]proved that the nonclassical sensitivity enhancement of GEO 600 reportedin Ref [Abadie (2011)] has been exceedingly close to fundamental quantuminterferometry bound under given energy constraints and photon loss lev-els More than that it was generally proven that the approach of com-bining displaced coherent states and squeezed vacuum states is optimal forgravitational-wave detectors

In Ref [Abadie (2011)] the gravitational-wave detector GEO 600 used aneffective number of photons per second of approximately n = 2 middot 1022 whichcorresponded to a total optical power inside the interferometer arms of P asymp37 kW at a wavelength of 1064 nm The total optical loss was 1minus η asymp 038The injected squeezing factor was eminus2r asymp 01 For these numbers the ratioof Eqs (43) and (44) is calculated to

∆φCSVmin

∆φgenmin

asymp

radicηeminus2r + 1minus η

1minus ηasymp 108 (62)

which is a good approximation within the limit of large coherent state dis-placements α sinh2r The quantum noise of GEO 600 including thesqueezed-light source was just 8 above the fundamental quantum inter-ferometry bound An increased squeezing strength of 16 dB (eminus2r asymp 0025)which is in reach would bring the approach based on coherent states andsqueezed vacuum states to within just 2 above the fundamental bound

Future GW detectors will have significantly reduced optical loss values(1 minus η) lsquoLossrsquo includes scattering and absorption at mirrors non-perfectfringe contrasts and the non-perfect quantum efficiency of the photo detec-tor Optical loss reduction is important for at least four reasons First itleads to an increased signal second it leads to a reduced quantum noise when

86

employing squeezed states third less absorption reduces the thermal load onthe test mass mirrors and fourth less scattering reduces the probability ofback-scattered light which produces disturbance signals [Billing et al (1979)Vahlbruch et al (2007) Punturo et al (2014)] The higher the finesse valuesof the arm and signal-recycling cavities are the more significant is opticalloss at mirror test masses the beam splitter and the signal-recycling mirrorThe finesse value of the power-recycling cavity and the loss of mirrors andlenses that guide the output field to the photo-diode are less critical Suitablephoto detectors of 995 quantum efficiency are available today [Vahlbruchet al (2016)] but achieving a total optical loss of 10 is still challengingThe reason for that is that first of all a measurement device aiming for bestabsolute sensitivity should use as much quanta (photons) as possible Highfinesse values for the enhancement cavities are thus essential but results inan unavoidable scaling-up of the effect of mirror losses A realistic exampleof future gravitational wave detectors thus considers η = 09 with a squeezingfactor of 20 dB (eminus2r = 001) In this case the quantum noise will be about4 above the ultimate fundamental bound for a given photon number

From Eq (62) it can be concluded that there is no need for any moresophisticated nonclassical states than squeezed states In particular non-classical states with a defined photon number such as N00N states are notrequired Within the approximation quoted this result is independent ofthe photon number This result is also independent of the physical systemused for interferometric phase estimation and can also be made for quantum-enhanced atomic clock calibration in the presence of dephasing Here the-oretical results indicate that the precision of Ramsey interferometry withspin-squeezed states is close to the optimal one in the asymptotic regime ofa large number of atoms [Huelga et al (1997) Ulam-Orgikh and Kitagawa(2001) Escher et al (2011)] as already stated in Ref [Demkowicz-Dobrzanskiet al (2013)] More sophisticated nonclassical states with fixed number ofn quanta might still be useful for the exceptional case when the absorptionof one quantum already results in zero measurement sensitivity anyway Anexample is an ensemble measurement where the absorption of a single photondemolishes the source of the phase change to be characterized A typicallyused approach of conditioning the measurement result on n clicks of n singlephoton counters conditions on precisely zero loss and is thus able to use theadvantage of Eq (40) over Eq (39)

87

65 Conclusions

Squeezed states of light will contribute to realizing gravitational-wave ob-servatories with much higher sensitivities than existing or planned ones Tobenefit from squeezed states in a most efficient way optical loss in terms ofabsorption and scattering must be minimized In particular the optical lossof mirror coatings and mirror substrates need to be minimized The rele-vant mirrors include the test masses the balanced beam splitter the signalrecyclingextraction mirror and all optical components between the latterand the photo diode Excellent spatial mode matching between the brightinterferometer field and the squeezed vacuum field is also of great impor-tance Achieving this requires further improvement of the surface figures ofall reflective optical components of the interferometer as well as improvedhomogeneity of all optical components that the light passes through

The quantum noise reduction achieved in a gravitational-wave detectoris of course always smaller than the highest squeeze factor provided by thesqueezed-light source As an example let us consider the observation of 15 dBof nonclassical noise suppression directly at the source If the squeezed fieldsenses an additional loss of 5 when propagating through the interferometerwhich is a very challenging number from todayrsquos point of view the remainingsqueezing level is about 11 dB see Eq (16)

7 The application of 2-mode-squeezed light in laser interferome-ters

71 Quantum Dense Metrology

At first glance the application of bi-partite (two-mode) squeezed statesto a device whose goal is measuring a single observable seems meaninglessSqueezing the uncertainty of that observable should be the optimum one cando This is indeed true when concerning just quantum noise but recentlyit was discovered that in the presence of classical disturbances bi-partitesqueezing can improve such measuring devices [Steinlechner et al (2013)]The concept was named quantum dense metrology (QDM) The potentialimprovement of a gravitational-wave detector with bi-partite squeezed statesis shown in Fig 32 (a) A description is given in the caption The pre-condition for a potential improvement can be best understood within a phasespace diagram Fig 32 (b) contains two different kinds of lsquosignalsrsquo The firstis the actual signal which always shows up as a phase space displacement

88

along the Y axis The second is a disturbance signal that can produce adisplacement in arbitrary direction in phase space A prominent example

Faraday Rotator

Coherent light

Photo detectors Balanced homodyne detectors

Squeezed vacuum

Squeezed vacuum

Quantum noise in

Y (A) and X (B)

XΩΔΩ(B)

YΩΔΩ(A)

(a) (b)

Disturbance projected

onto Y

and X

XΩΔΩ(B)

YΩΔΩ(A)

Example of (unknown) disturbance

Signal

Figure 32 Bi-partite-squeezed-light-enhanced measurement ndash (a) Setup for theapplication of bi-partite (two-mode) squeezed light in a laser interferometer on the basisof QDM Two squeezed vacuum fields are overlapped on a balanced beam splitter withprogrammable squeeze angles for instance with a relative angle of 90 which produces abi-partite state as shown in Fig 12 The beam splitter outputs are entangled for any rela-tive angle greater than zero One part is matched to the interferometer mode The secondpart is kept outside as a reference beam The interference of the interferometer outputand the reference beam is arranged with such a phase difference that it reproduces the twosqueezed inputs on the photo detectors The two squeezed beams are photo-electricallydetected measuring the respective squeezed quadrature (using balanced homodyne detec-tors) Both beams carry half of all interferometer induced modulations which includesignals as well as disturbances A single readout as shown in Fig 21 cannot distinguishbetween the two kinds The double readout shown here provides additional informationand allows for recognition of the disturbance [Steinlechner et al (2013)] as well as in prin-ciple a modeling of the disturbance and with a correct model an improvement of thenoise spectral density of the interferometer [Ast et al (2016)] (b) Phase space diagramdescribing phase quadrature readout A as well as amplitude quadrature readout B Bothshow squeezed quantum noise The amplitude quadrature readout does not contain anygravitational-wave signal ie any feature in this channel must be due to disturbancesThis information can be used to improve the interferometer

89

for such a disturbance is parasitic interference due to back-scattered laserlight [Vahlbruch et al (2007)] Back-scattering is a limiting noise at lowsignal frequencies of gravitational-wave detectors [Billing et al (1979) Vinetet al (1997) Hild (2007) Ottaway et al (2012) Punturo et al (2014)] Notethat all noise that couples in via unwanted motions of the test mass mirrorsso-called lsquodisplacement noisersquo always produces a phase space displacementalong the Y axis and cannot be tackled with QDM

Fig 33 shows measurement results obtained in Ref [Steinlechner et al(2013)] In a table-top experiment one part of a bi-partite squeezed state ofa continuous-wave mode at 1064 nm was mode-matched into the output portof a Michelson laser interferometer operated at its dark fringe in full analogyto Fig 32(a) A lsquosignalrsquo was produced by driving the piezo behind one of theend mirrors at a frequency of 555 MHz The lsquodisturbancersquo was introducedby re-injecting a small amount of light that leaked through the second endmirror with an additional piezo-mounted mirror The piezo was driven ata frequency of 517 MHz to produce a phase modulation An additionalDC voltage defined an arbitrary and unknown optical path length of thelight before being re-injected and as such the phase space orientation of thedisturbance signal This mechanism of a parasitic interference is realizednaturally in any interferometric device due to back-scattering of quanta frommoving surfaces in the environment

The interferometer output consisted of the signal as well as the distur-bance with a quantum uncertainty given by one subsystem of the bi-partiteentanglement It was overlapped with the second subsystem of the entan-gled state on a balanced beam splitter and the two outputs were analysedwith balanced homodyne detectors The phases of the bi-partite entangle-ment and the BHD local oscillators were controlled to resemble Fig 32(b)ie both BHDs measured a squeezed uncertainty regardless of the phase ofthe (generally unknown) disturbance

The beam splitter that combines interferometer output and the entangledreference beam unavoidably splits the signal as well as the disturbance intotwo paths For a balanced beam splitter this generally reduces the signaland disturbance power by 3 dB for both quadrature measurements Fig 33shows however that both BHDs performed about 6 dB below shot noisewhich demonstrates the usefulness of the scheme The squeeze factor can inprinciple be infinite which thus qualifies the lsquo3 dB penaltyrsquo In the abovefigure the additional information from the second BHD output was used torecognize the parasitic interference in the first BHD output providing a lsquovetorsquo

90

50 51 52 53 54 55 56 57

50 51 52 53 54 55 56 57-70

-75

-80

-85

-90

-70

-75

-80

-85

-90

Y (A) (Ω2π)Ω∆Ω

Frequency ( f )

Noi

se p

ower

(dB

m)

Noi

se p

ower

(dB

m)

X (B) (Ω2π)Ω∆Ω

~ ndash6 dB

~ ndash6 dB

Shot noise reference

Shot noise reference

Signal Projected disturbance

Projected disturbance

Vet

o

No signal

Figure 33 Bi-partite-squeezed-light-enhanced measurement ndash The result wasachieved in a table-top setup [Steinlechner et al (2013)] In the two panels the lower

(blue) traces show the squeezed quadrature noise-power spectra ∆2Y(A)Ω∆Ω(Ω2π) (top) and

∆2X(B)Ω∆Ω(Ω2π) (bottom) as simultaneously measured with balanced homodyne detectors

lsquoArsquo and lsquoBrsquo respectively The conventional Y -measurement (top) cannot distinguish be-tween signal and disturbances The additional X-measurement (bottom) does not detectany phase quadrature signal thus any feature in this measurement is a parasitic signal dueto a disturbance The respective projection onto the Y -measurement can thus be lsquovetoedrsquoIn a more sophisticated approach the X-data might be used to model and then to elim-inate the disturbance as well as its projections on both quadrature measurements Theresult is a reduced spectral density of the actual phase quadrature measurement [Ast et al(2016)] Traces shown here are slightly sloped due to the decreasing transfer functions ofthe balanced homodyne detectors The resolution bandwidth was ∆Ω(2π) = 10 kHz thevideo bandwidth was 100 Hz All traces were averaged three times

signal to trigger its removal from the data streamThe question arose whether the additional information can be used to

reduce the actual noise spectral density of the first measurement ie to re-cover signals that were buried by parasitic interferences Very recently it wasshown that this is indeed possible Ref [Ast et al (2016)] reports a table-top proof-of-principle experiment in which the additional information in the

91

QDM approach could be used for improving the sensitivity of an interfero-meter The measurement sensitivity was improved from above-shot-noise tosub-shot-noise (sub-Poissonian) performance This result was possible notbecause the way the parasitic interference arose was known but because theadditional information provided by QDM allowed for fitting a model of theexcess noise to the readout data

Quantum dense metrology (QDM) as shown in Fig 32 improves a mea-surement by simultaneously reading out two conjugate observables Bothreadout observables show a squeezed quantum noise and act as estimatorsof independent physical quantities This situation was recently described aslsquoquantum-mechanics freersquo [Tsang and Caves (2012)] QDM is based on anEinstein-Podolsky-Rosen (EPR) entangled [Einstein et al (1935)] bi-partitesystem as described in Subsec 34 EPR entanglement was previously con-sidered for the quantum-informational task of dense coding which doublesthe capacity of quantum communication channels [Bennett et al (1992)Braunstein and Kimble (2000)] The application of EPR entanglement inmetrology was first proposed by DrsquoAriano et al [DrsquoAriano et al (2001)]

72 Conclusions

A single beam that carries an optimized spectrum of squeezed vacuumstates and that is injected into the interferometerrsquos dark port provides themost efficient and practically optimal approach to reduce the quantum noisein laser interferometers by means of nonclassical states (see Section 5) Theconclusion of the section here is that two entangled beams provide a superiorapproach if the interferometerrsquos sensitivity is limited by classical noise thatis not exclusively restricted to the actual observable which is the phasequadrature amplitude Y Parasitic interferences due to laser light that isbackscattered from vibrating surfaces are an example Current gravitational-wave detectors use light fluxes of about 1024 photons per second [Abbott(2016)] Just a single photon per second and hertz which leaves the mainlight beam and is backscattered from a vibrating surface and in this way getsfrequency shifted into the detection band produces a significant disturbancesignal The lsquoquantum-dense metrologyrsquo approach might provide a powerfultechnique to tackle this problem

Very recently it turned out that QDM is not the only technique that mayexploit EPR entanglement to improve phase measurements Ref [Ma et al(2017)] proposes to use EPR entanglement to simultaneously suppress shotnoise and radiation pressure noise in a gravitational-wave detector without

92

the need for an additional filter cavity (confer subsection 55) In this caseEPR entanglement is exploited that is carried by one broadband squeezedbeam and that is present between quadrature amplitudes defined with re-spect to different optical frequencies ω and ωprime as investigated in Ref [Hageet al (2010)] Such lsquofrequency multiplexedrsquo EPR entanglement might resultin considerably lower costs of building a gravitational-wave detector with abroadband simultaneous squeezing of shot noise and radiation pressure noiseAlso this proposal does not lead to a fundamentally lower quantum noise butrather improves on classical aspects of an interferometer

8 Summary and Outlook

In many cases experiments that involve interference of quantum statescan be described in a semi-classical way This description uses the classicalwave picture for the interference part of the experiment and subsequently theclassical particle picture when the states transfer their energy to a detectoror more generally to a thermal bath This semi-classical description is notpossible when using the specific class of lsquononclassicalrsquo states Squeezed statesof light are a prominent example of these Squeezed states and other nonclas-sical states allow for observations that made Einstein Podolsky and Rosenformulate their critical and seminal paper on quantum theory [Einstein et al(1935)]

In the review here it is argued that after many successful proof-of-principle experiments with nonclassical states in the past decades the routineuse of squeezed-light in observational runs of the gravitational-wave detectorGEO 600 goes beyond proof-of-principle and is a true application of nonclassi-cal light Since 2010 the squeezed-light source has improved the measurementsensitivity of GEO 600 in basically every observational run [Abadie (2011)Grote et al (2013) Dooley et al (2016)] At quantum noise limited frequen-cies ie above a few hundreds of hertz the sensitivity has been improvedcorresponding to a squeezing strength in the noise spectral density of up to37 dB which corresponds to an increase of the average gravitational-wavedetection rate by a factor of 043minus32 = 36 This success is a strong motiva-tion to also equip the Advanced LIGO Virgo and Kagra gravitational-wavedetectors with squeezed light Similar improvement factors even down tolower signal frequencies are expected [LSC (2013)] The achievable improve-ment factors are mainly limited by the optical loss on the squeezed states

93

and much higher factors are achievable in principleUp to now squeezed states have not been used to reduce the radiation-

pressure noise in gravitational-wave detectors The reason is that so far othernoise sources are larger than radiation pressure noise and such an effect can-not be observed It is expected however that future gravitational-wavedetectors will eventually be partly limited by radiation pressure noise Fromthis point on squeezed light will be used to simultaneously reduce shot noiseand radiation pressure noiseSqueezed states are the optimum nonclassical states for gravitational-wavedetectors or more generally for all laser interferometers operating with largeaverage photon numbers per measuring interval [Demkowicz-Dobrzanski et al(2013)] In addition to using higher light powers and heavier test mass mir-rors higher squeeze factors will thus contribute to mitigate the lightrsquos quan-tum noise in laser interferometers

Two-mode (bi-partite) squeezed light has not been used in gravitational-wave detectors so far They are not capable of further reducing the quantumnoise in laser interferometers but they can be used to mitigate classicalnoise that originates from fluctuating phase space displacements A well-known such noise source is back-scattered light Proof-of-principle experi-ments were performed recently [Steinlechner et al (2013) Ast et al (2016)]This new technique could turn out to be valuable in next generations ofgravitational-wave detectors in particular in those targeting high sensitivi-ties at low sub-audio signal frequencies and using high light powers Suchan implementation in gravitational-wave detectors does not require any newtechnology Compared to a squeezed-light enhanced interferometer just asecond squeezed-light source is required

It is certainly remarkable that those quantum states that made EinsteinPodolsky and Rosen falsely think quantum theory incomplete are now ex-ploited as new technologies in measurement devices targeting new observa-tions in nature

Acknowledgements

RS thanks M Ast J Bauchrowitz C Baune S Chelkowski J DiGugliel-mo A Franzen B Hage J Harms A Khalaidovski L Kleybolte NLastzka M Mehmet S Steinlechner and H Vahlbruch for their contri-butions many fruitful discussions and their support with the figures and JFiurasek for many valuable comments on the manuscript Thanks are also

94

due to Y Chen F Khalili and H Miao for fruitful discussions within thequantum noise working group of the LIGO Scientific Collaboration (LSC)Special thanks are due to H Vahlbruch and H Grote together with theGEO 600 team for their pioneering work on the squeezed-light implementa-tion in GEO 600 RS is supported by the Deutsche Forschungsgemeinschaft(Grant No SCHN 757-6) and by the European Research Council (ERC)project lsquoMassQrsquo (Grant No 339897)

References

References

Aasi J et al Jul 2013 Enhanced sensitivity of the LIGO gravitationalwave detector by using squeezed states of light Nature Photonics 7 (8)613ndash619URL httpwwwnaturecomdoifinder101038nphoton2013177

Aasi J et al Apr 2015 Advanced LIGO Classical and Quantum Gravity32 (7) 074001URL httparxivorgabs14114547httpstacks

ioporg0264-938132i=7a=074001key=crossref

20895763c84bce3f8929251031b2475c

Abadie J et al Sep 2011 A gravitational wave observatory operatingbeyond the quantum shot-noise limit Nature Physics 7 (12) 962ndash965URL httparxivorgabs11092295httpwwwnaturecom

doifinder101038nphys2083

Abbott B P et al Feb 2016 Observation of gravitational waves from abinary black hole merger Phys Rev Lett 116 061102URL httplinkapsorgdoi101103PhysRevLett116061102

Acernese F et al 2015 Advanced virgo a second-generation interferome-tric gravitational wave detector Classical and Quantum Gravity 32 (2)024001URL httpstacksioporg0264-938132i=2a=024001

95

But the question remains what property of coherent states justifies thename lsquoclassicalrsquo even though coherent states are quantum states and showquantum uncertainties My answer to this question is the following Allexperiments which only involve coherent states and mixtures of them allowfor a description that uses a combination of classical pictures As we will seebelow this description swaps between two different classical pictures and isthus not truly classical but semi-classical (A more precise description of thenature of coherent states uses the term lsquosemi-classicalrsquo)

Let us consider a laser interferometer that uses light in a coherent stateFirstly the light beam is split in two halves by a beam splitter The twobeams travel along different paths and are subsequently overlapped on abeam splitter where they interfere exactly as classical waves would do Theelectric fields superimpose thereby producing the phenomenon of interfer-ence Up to this point there is no reason to argue light might be composedof particlesSecondly the new (still coherent) beams that result from the interference areabsorbed for instance by a photo-electric detector In the case of coherentstates the detection process can be perfectly described in the classical parti-cle picture in which the particles appear independently from each other in atruly random fashion yielding the aforementioned Poisson statistic Duringthe detection process no wave feature of the light is present Let us havea closer look A truly random (lsquospontaneousrsquo) event is an event that hasnot been triggered by anything in the past This allows us to make a clearcut between the first part of the experiment described by the classical wavepicture and the second part of the experiment described by the classicalparticle picture Both lsquoworldsrsquo are disconnected The subsequent applicationof two classical pictures is not truly classical but lsquosemi-classicalrsquo It is in-deed the observation that the photons occur individually with truly randomstatistics that allows this semi-classical description In the case of a mixtureof coherent states the photon statistics are super-Poissonian which can beunderstood as a mixture of different Poissonian distributions In the caseof a slowly changing coherent state the mean value n depends on time Inall these cases the semi-classical description is appropriate Let me pointout that in this very reasonable description photons do not exist before theyare detected eg absorbed Further note that the famous double-slit exper-iment with coherent states also allows for the same semi-classical description

For squeezed states [Yuen (1976) Walls (1983)] the situation is different

6

As before the interference can be fully described by the classical wave pic-ture The result of the detection process however is different from that ofmutually independent random events It is also different from any super-Poissonian statistics that could be produced by mixing an arbitrary num-ber of different andor time-dependent Poissonian distributions Insteadthe squeezed probability distribution in Fig 1 suggests that the probabil-ity of detecting a photon decreases with the more photons that are alreadydetected in the same time interval over which a single measurement is in-tegrated From this observation one must conclude that the photons donot individually appear in a random fashion upon detection There mustbe lsquoquantumrsquo correlations between the photons These correlations mustexisted before detection since there is no interaction between the photonsduring their detection Pre-existing correlations between detected photonsseem to imply that the photons themselves existed before detection ie attimes when interference occurred In a semi-classical description howeverphotons are classical particles and cannot interfere for instance on a beamsplitter At this point the semi-classical picture breaks down Squeezedstates are therefor lsquononclassicalrsquoThe failure of the semi-classical model described above generally certifiesnonclassicality

Squeezed states are usually not characterized by counting their pho-tons but by measuring canonical continuous-variable phase-space observ-ables Measurements are performed as usual on an ensemble of identicalstates and quasi-probability density functions are calculated from the dataThe Glauber-Sudarshan P -function is the quasi-probability density distribu-tion over coherent states If the P -function of a state is entirely positivethe state is a coherent state or a (classical) mixture of coherent states Thestate is considered as semi-classical If the P -function is not a positive-valuedfunction the state cannot be expressed as a (classical) mixture of coherentstates and is thus nonclassical [Gerry and Knight (2005) Vogel and Welsch(2006)] A non-positive-valued P -function is the sufficient and necessary con-dition for the failure of the semi-classical model The Wigner function is thequasi-probability phase-space representation over the canonical continuous-variable phase-space observables themselves [Gerry and Knight (2005)] TheWigner functions of squeezed states are entirely positive Although subject todiscussion this fact does not mean that squeezed states are less nonclassicalthan Fock states or cat states which not only have a nonclassical P -function

7

but also a partially negative Wigner function (A cat state is a quantum su-perposition of two macroscopically distinct states [Monroe (2002)] referringto Schrodingerrsquos-cat gedanken experiment [Schrodinger (1935)]) In practicesqueezed states can even be regarded as superior nonclassical states becausethey represent the only nonclassical state that has been produced in a steadystate fashionIn almost all experiments so far the generation of Fock states and cat statesinvolves a probabilistic event such as the detection of a photon in anotherbeam path to herald these states In fact squeezed states provide the non-classical resource for the probabilistic preparation of Fock states as well as catstates But only the squeezed states themselves show a nonclassical effect in astationary way Limited only by the time duration and the frequency span ofthe mode that is in a squeezed state the squeezing effect can be continuouslyobserved independently of the time when the measurement is performed andalso independently of the measurement integration time This fact is of greatimportance for applications of squeezed states in measurement devices sincea squeezed-light-enhanced measurement remains unconditional and the ef-fective measurement time is not reduced

In past decades squeezed states of light were used in many proof-of-principle experiments to research their potential for improving the sensitivityof laser interferometers [Grangier et al (1987) Xiao et al (1987) McKen-zie et al (2002) Vahlbruch et al (2005) Goda et al (2008) Taylor et al(2013)] or the performance of imaging beyond the shot-noise limit [Lugiatoet al (2002) Treps et al (2003)] both accompanied by a huge number oftheoretical works Potential applications in secure optical communication(quantum key distribution) were also proposed and proof-of-principle ex-periments demonstrated [Ralph (1999) Furrer et al (2012) Gehring et al(2015)] This review restricts itself to the improvement of laser interfero-meters since only here has the application of squeezed light gone beyondproof-of-principle The gravitational-wave detector (GWD) GEO 600 hasoperated with squeezed light now for more than seven years starting in 2010[Abadie (2011) Grote et al (2013)] GEO 600 is a 600 m long Michelsonlaser interferometer built for the detection of gravitational waves Thesewaves are audio-band and sub-audio-band changes of space-time curvatureoriginating from cosmic events such as the merger of neutron stars or blackholes as detected recently [Abbott (2016)] In GWDs such as GEO 600 [Doo-ley et al (2016)] Advanced LIGO [Aasi (2015)] Advanced Virgo [Acernese

8

(2015)] and KAGRA [Aso et al (2013)] conventional laser technology hasbeen pushed to extremes over the past decades Noise spectral densities nor-malized to space-time strain of less than 10minus23 Hzminus12 have been measured[Abbott (2016)] Progress will continue and based on the successful appli-cation in GEO 600 squeezed light is now widely accepted to provide a newadditional technology to contribute to the new field of gravitational-waveastronomy It was also successfully tested in one of the LIGO detectors in2013 [LSC (2013)] and is an integral part of the European design study forthe 10 km Einstein-Telescope [Punturo et al (2010)]

GEO 600 has already taken several years of lsquosqueezedrsquo observational datawhich has increased its sensitivity at signal frequencies above 500 Hz Withthe implementation of a squeezed light source in GEO 600 the applicationof nonclassical states in metrology has been pushed beyond merely proof-of-principle

lsquoTwo-mode squeezed statesrsquo show a squeezed uncertainty in at least onejoint continuous variable of two subsystems lsquoArsquo and lsquoBrsquo Examples of jointvariables are differences and sums of phase-space observables of A and BTwo-mode squeezed states not only belong to the class of nonclassical statesbut due to their bi-partite character also to the class of lsquoinseparablersquo orlsquoentangledrsquo states They are the ideal states to demonstrate the Einstein-Podolsky-Rosen paradox [Einstein et al (1935)] as first achieved in [Ouet al (1992)] Apart from fundamental research on quantum mechanicsrecent proof-of-principle experiments demonstrated their usefulness in inter-ferometric measurements that go beyond the application of simple squeezedstates [Steinlechner et al (2013) Ast et al (2016)] This experiment is thefinal topic of this review

2 Observations on light fields in squeezed states

Generally there are two different kinds of observables that can be subjectof a measurement performed on a quantum system The first kind is associ-ated with the systemrsquos wave property In optics it corresponds to the electricfield strength at a given phase angle ϑ The according (dimensionless) opera-tors are called the quadrature amplitudes Xϑ and have a continuous spectrumof eigenvalues Quadrature amplitudes are measured in very good approx-imation with a balanced homodyne detector using the interference with a

9

bright local oscillator beam see Fig 3 (a) In practice any measurement ofXϑ integrates over some sideband (Fourier) spectrum within the angular fre-quencies Ω plusmn ∆Ω2 The sideband information always needs to be quotedA straight forward but rather untypical way is by adding subscripts whichleads to Xϑ

Ω∆Ω The classical analogue of the quadrature amplitude operator

XϑΩ∆Ω is the modulation depth of the optical field at modulation phase angle

ϑ and at angular modulation frequency Ω measured over the band ∆Ω lt 2ΩThe uncertainties of the statersquos quadrature amplitudes at different phases ϑare limited by a Heisenberg uncertainty relation see section 3 The secondkind of measurement is associated with the systemrsquos particle property andis given by the photon number operator n associated with a measuring timeinterval ∆t Its precise measurement requires a photon counter ideally withsingle photon resolution The measurement result obviously has a discretespectrum Continuous as well as discrete observables are usually subject toquantum uncertainties and thus quantum noiseUsually the measurementrsquos integration time and frequency band actuallydefine the physical system that is characterized In quantum optics experi-ments the interrogated physical system is called a lsquomodersquo

21 Definition of a lsquosingle modersquo

Let us define a light field or generally any quantum system to be asingle mode if it corresponds to the lsquosmallest entity of a waversquo In this caseits spectral and temporal distributions as well as waist size and divergenceare at their Fourier limits and all other properties such as optical axis waistposition and polarization are well defined For instance a linearly polarizedlongitudinal resonance of an optical standing-wave cavity defines such a singlemode if the cavity finesse is high and transversal modes are non-degenerateThe complete photo-electrical detection of a cavity mode however is notstraight forward Most quantum optical experiments are instead performedon propagating light In this case single modes are defined by spatial filtersand by temporal-spectral measurement windows both being at the Fourierlimit Examples for single modes are a laser pulse and a spectraltemporalcutout from a continuous observation of a quasi-monochromatic continuous-wave light beam in the spatial TEM00 mode both at the Fourier limits

In classical physics the only remaining free parameter of a given singlemode is its excitation energy In quantum physics the situation is differentFor a given energy a single mode can be in many different quantum states

10

which differ in their quantum statistics Examples are coherent states num-ber (Fock) states and squeezed states

22 Observations on squeezed states using a single PIN photo-diode

An ideal PIN photo-diode absorbs the full energy of a light mode andproduces one photo electron for every absorbed photon energy It uses theinternal photo-electric effect inside a semiconductor such as silicon or In-GaAs In contrast to avalanche photo-diodes PIN photo-diodes operatewith unity gain lsquoPINrsquo stands for lsquopositiversquo lsquointrinsicrsquo and lsquonegativersquo and isdescribing the doping of the semiconductor layers A PIN photo-diode is op-timally suited for the continuous monitoring of a rather bright light field ofup to several tens of milliwatts An example is the photo-diode in the outputport of a gravitational-wave detector as shown in Fig 2 (a) The prominentwavelength of 1064 nm which is emitted by NdYAG lasers has an opticalfrequency of ν = ω(2π) = 282 middot 1014 Hz The period of the field oscillationis a few femtoseconds and cannot be directly resolved with photo-electric de-tectors However variations of the electric field around the averaged opticalfield oscillation on longer time-scales can be resolved Applying an electronicbandpass filter at the sideband angular frequency Ω plusmn ∆Ω2 to the photovoltage provides information about the lsquodepth of the lightrsquos amplitude mod-ulationrsquo which is also called the lsquoamplitude of the amplitude quadraturersquo Itcan also slowly vary in time and reads

Xϑ=0

Ω∆Ω(t) equiv XΩ∆Ω(t) equiv X (1)

The subscript is usually skipped as it is done with the time dependenceas indicated on the right Applying the electronic bandpass filter in factdefines the mode of the light being detected The structure of the defini-tion in Eq (1) forms the basis of interferometric signals and quantum noisealso in the semi-classical case of coherent states Lets take an example Inthe recent observation of gravitational waves [Fig 1 bottom row in Abbott(2016)] the time-frequency representation of the gravitational-wave signalcorresponded to the amplitude quadrature amplitude XΩ∆Ω(t) of the inter-ferometer output light Note that a larger value of ∆Ω allows for changes ofthe quadrature amplitude on shorter time scales

If the light fieldrsquos lsquomodulation modersquo does not contain any quanta simplybecause there are no photons that have a frequency difference of plusmnΩ with

11

respect to the carrier it is in its ground state In this case lsquovacuum noisersquois observed which originates from the ground state uncertainty Since thevacuum noise only becomes measurable as a beat with a bright light field itcan also be seen as the carrierrsquos band-path filtered shot noise A modulationmode in a displaced vacuum state (a coherent state) corresponds to nonzerocoherent modulationThe measured level of the vacuum noise generally depends on the power ofthe bright carrier light and on the electronic amplification In any case itprovides the reference for certifying lsquosqueezingrsquo Observations using a singlePIN photo-diode require an independent measurement to quantify vacuumnoise A necessary condition is that attenuating the total fieldrsquos light powerresults in the same attenuation of the measured XΩ∆Ω values If they showa stronger attenuation a coherent modulation or thermal noise might bepresent If they show a weaker attenuation the photo-diode and its electron-ics might be saturated

Fig 2 (b) illustrates how a broadband squeezed field improves the mea-surement of an amplitude modulation in time domain based on a PIN photo-diode Shown is a simulated time sequence of XΩ∆Ω-data sampled from thephotoelectric voltage In this simulation all sideband frequencies from zero(DC) to the cutoff frequency of the detector electronics (Ωcut) are included(Ω = ∆Ω2 = Ωcut2) No additional band pass filter is applied making it amaximally broadband detection Although the data in Fig 2 (bi) contains aclassical amplitude modulation of the detected light this signal is not visibledue to random noise here representing shot noise Fig 2 (bii) shows thesame situation but with shot noise that is squeezed over the full detectionband The quantum uncertainty of the modulation depth is squeezed andthe classical signal becomes visible

It needs to be noted that a single PIN photo-diode can only measure theamplitude of the amplitude quadrature XΩ∆Ω(t) but not the non-commutingobservable the lsquoamplitude of the phase quadraturersquo

Xϑ=90

Ω∆Ω (t) equiv YΩ∆Ω(t) equiv Y (2)

For values that are small compared to the field strength of the bright fieldthe quantity Y approximately describes the bright fieldrsquos lsquophase modulationdepthrsquo

12

23 Observations on squeezed states using a balanced homodyne detector

-72

-69

-66

-63(i)

(ii)

(iii)

(a)

LO

(Squeezed)signal input

Phaseshifter

PD1

PD2

5050

01 0200 03

-75

-60

-57Balanced homodyne detector

(b)

Time [s]

Noi

se p

ower

[dB

m]

Figure 3 Balanced homodyne detection (BHD) ndash (a) Setup The quadrature atchoosable angle ϑ of the signal field is measured by overlapping the latter with a localoscillator (LO) field of the same mode parameters on a balanced beam splitter and record-ing the difference voltage from two PIN photo-diodes as shown In order to meet theBHD approximation the LO needs to be much more intense than the signal field A closeto perfect mode overlap between LO and signal input field is crucial For a non-perfectoverlap the detector measures the input state with unwanted contributions of the vacuumstate (b) Noise power measurements (i) on an electronically amplified and band-passfiltered quadrature amplitude of the vacuum field (Xvac

Ω∆Ω) (signal input blocked) (ii) on

a squeezed quadrature (XsqzΩ∆Ω) of a squeezed vacuum state (ϑ = 0) and (iii) on re-

spective quadratures of the same squeezed state where the phase angle ϑ was continuouslyshifted by changing the optical path length of the LO The measurement data shows about5 dB of squeezing and was first published in [Chelkowski et al (2007)] Ω2π = 5 MHz∆Ω2π = 100 kHz

In contrast to a single PIN photo diode a balanced homodyne detector(BHD) is suitable to measure the quantum statistic of all types of modu-lations ie for all angles ϑ Such a detector consists of two identical PINphoto-diodes a balanced beam splitter and an external homodyne local os-cillator field that is much brighter than the signal beam and that has anadjustable phase The signal beam corresponds to the squeezed field whichin many experiments is in a squeezed vacuum field having an optical powerthat usually corresponds to just a few photons per mode The two beams areoverlapped on the balanced beam splitter with close to perfect mode match-ing and the two interference outputs are focussed onto the photo diodes seeFig 3 (left) The electric output signal of the BHD is the difference of the

13

photo diode voltages The LO takes over the role of the carrier light fieldbut with the possibility to choose the phase shift ϑ This way eigenvaluesof X Y or Xϑ can be measured where the latter is given by the followinglinear combination of the first two

Xϑ(t) = cos(ϑ) X(t) + sin(ϑ) Y(t) (3)

If the modulation depths of signal and local oscillator beams are weak com-pared to their coherent amplitudes |α| and |αLO| the output voltage of aBHD corresponds to eigenvalues of the following operator

V (t) prop 2cos(ϑ) |αLO||α|+ |αLO| Xϑ(t) + |α| XϑLO(t) (4)

The lsquohomodyne approximationrsquo further involves |αLO| |α| such that theterm on the right can be neglected even if the local oscillator shows someclassical quadrature excitation The output voltage of a BHD is usually spec-trally analysed or at least spectrally filtered which removes the DC part infull analogy to a single photo diode (see previous subsection) Sampling thefiltered voltage provides eigenvalues proportional to the generalized quadra-ture amplitude in Eq (3)

V BHDΩ∆Ω(t) prop |αLO| Xϑ

Ω∆Ω(t) (5)

Fig 3 (a) shows the setup of a balanced homodyne detector for the char-acterization of squeezed states Setting ϑ = 0 eigenvalues of the ampli-tude modulation depths can be sampled from the photo voltage accordingto Eq (5) Setting ϑ = 90 eigenvalues of the phase modulation depths aremeasured The datarsquos expectation values 〈Xϑ〉 provide the coherent displace-ment of the squeezed state The datarsquos variances

∆2Xϑ equiv 〈(Xϑ)2〉 minus 〈Xϑ〉2 (6)

provide the statersquos (quantum) noise A pure squeezed state as well as asqueezed state that experienced photon loss have Gaussian quantum statisticsand are thus fully described by the expectation values and variances (first andsecond moments) of two orthogonal quadratures but only if one quadraturereflects the lowest quadrature variance

14

In most experiments with squeezed light the photo electric voltage ac-cording to Eq (5) is not sampled with a data aquisition system but the signalis directly fed into a spectrum analyser measuring the noise power of the volt-age If the expectation value 〈Xϑ〉 is zero the noise power is proportionalto the variance ∆2Xϑ in Eq (6) The reference for quantifying the squeezefactor is measured by blocking the (squeezed) signal field in Fig 3 (a) Themeasured vacuum noise level corresponds to the LOrsquos (electronically ampli-fied) shot noise level

Traces (ii) and (iii) in Fig 3 (b) show measured noise powers of the mod-ulation mode (Ω2π = 5 MHz ∆Ω2π = 100 kHz) being in a squeezed vac-uum state (i) is proportional to the variance of the ground state uncertainty∆2Xvac

Ω∆Ω (ii) is proportional to the quantum noise variance of the squeezed

quadrature amplitude ∆2XsqzΩ∆Ω (iii) is proportional to the quantum noise

variance of the quadrature amplitude with scanned phase ∆2XΩ∆Ω(ϑ(t))

To fully characterize a quantum state ie to do quantum state tomogra-phy [Vogel and Risken (1989)] a BHD is a prerequisite But also interfero-metric measurements with balanced homodyne detectors instead of singlePIN photo-diodes have several advantages A correctly implemented BHDreadily provides the vacuum noise level when the signal beam is blockedWith a BHD the optimum operating point of the interferometer is preciselyat a dark fringe If a perfect dark fringe can practically be achieved ampli-tude noise of the laser does not couple into the signal port If the interfero-meter has balanced arm length also frequency noise of the laser then does notcouple into the signal port Some quantum non-demolition schemes with theprospect of evading quantum radiation pressure noise require the detection ofa non-canonical quadrature angle [Jaekel and Reynaud (1990) Kimble et al(2001)] Here the adjustable phase of a BHD provides a straight forwardapproach The experimental exploration of BHDs for gravitational-wave de-tectors only has started recently [Steinlechner et al (2015)]

A light field can be analysed with respect to many different modulationfrequencies Ω The result constitutes a spectrum [Breitenbach et al (1998)]where in principle every modulation mode can be in a different quantumstate Fig 4 shows spectra of squeezed states from 5 MHz to 100 MHz with∆Ω2π = 1 MHz The lower curve shows the spectrum of the most strongly

15

-14-12-10-8-6- 4-20 2 4 6 8

10 12 14 16 18

6 8 10 20 40 60 80 1005

Squeezed noise

Anti-squeezed noise

Vacuum noise

Frequency [MHz]

Nor

mal

ized

noi

se v

aria

nce

[dB

]

Figure 4 Spectrum of quadrature amplitude variances ndash Shown are the quantumnoise properties of a large number of modulation modes having a resolution bandwidth of∆Ω(2π) = 1 MHz For all traces the balanced homodyne detector output was analysedwith a spectrum analyser Squeezing of XΩ∆Ω (bottom trace) and anti-squeezing of

YΩ∆Ω (top trace) versus f = Ω(2π) are shown relative to the vacuum noise variance Thespectrum below 5 MHz is not shown since it contained less squeezing due to laser relaxationoscillation in the carrier field as well as disturbances from back-scattered light [Vahlbruchet al (2007)] Disturbances at frequencies above 70 MHz originated from relatively largedetector dark noise which was subtracted from all traces shown here The thin linerepresents a theoretical model that takes into account for the linewidth of the squeezingcavity The data was first presented in in Ref [Mehmet et al (2010)]

squeezed variances in this case the variances of XΩ∆Ω The upper spectrum

shows the variance in the orthogonal quadrature amplitude (YΩ∆Ω) Allvariances are normalized to those of the corresponding vacuum state Thesqueeze factor reduces towards higher frequencies due to the linewidth ofthe squeezing cavity The anti-squeezing is always higher than the absolutevalue of the squeezing due to Heisenbergrsquos uncertainty relation and due to thepresence of optical loss The curves do not represent pure squeezed states butmixed squeezed states with a significant contribution from vacuum statesdue to optical loss Pure squeezed states can only be produced by making

16

the influence of all decoherence processes negligibleThe choice of the resolution bandwidth (RBW ∆Ω) during data taking

and processing defines the spectral-temporal modulation modes includingtheir number within the detected spectrum For any setting of the RBWthe quantum mechanical properties of the quadrature amplitudes XΩ∆Ω and

YΩ∆Ω [Caves (1985)] fully correspond to those introduced for quadratures instandard text books and which are reviewed in Sec 3

24 Observations on two-mode squeezed states using balanced homodyne de-tectors

Two-mode squeezed states are composed of two subsystems lsquoArsquo and lsquoBrsquoand are bi-partite entangled states with a Gaussian quantum statistic Toavoid conflicts with different usage of the term lsquomodersquo they can synony-mously be named lsquobipartite Gaussian entangled statesrsquo or lsquobipartite squeezedstatesrsquo which will be mainly used in this Review In the same way multi-partite Gaussian entangled states correspond to multi-partite squeezed states

The measurement observables that prove or disprove the bi-partite squeez-ing property are XA

Ω∆Ω minus XBΩ∆Ω and Y A

Ω∆Ω + Y BΩ∆Ω where the minus and

plus signs may be swapped Bi-partite squeezed states are precisely thosestates that were discussed by Einstein Podolsky and Rosen (EPR) in theirseminal paper [Einstein et al (1935)] Fig 5 shows a measurement resulton bi-partite squeezed light [Eberle et al (2013)] The variances of bothjoined observables are squeezed as shown in the two lower traces They wererecorded consecutively by adding or subtracting the outputs of two balancedhomodyne detectors But by interfering the subsystems on a beam splitterone could even measure both joined observables simultaneously This pos-sibility is correctly described in quantum theory since their commutator iszero

The so-called EPR paradox arises as follows If we either measure XAΩ∆Ω

and XBΩ∆Ω or Y A

Ω∆Ω and Y BΩ∆Ω it is obvious from the data in Fig 5 that

we can always predict the measurement result at subsystem lsquoBrsquo when know-ing the result at subsystem lsquoArsquo This seems to suggest that both quantitiesat lsquoBrsquo are precisely defined simultaneously before the measurement on lsquoArsquowhich contradicts the rigorous (and correct) interpretation of their non-zerocommutator that they are not precisely defined simultaneously

To solve this paradox EPR conjectured that the wavefunction as definedby quantum theory does not provide the full information This led to a

17

discussion of whether hidden variables existed that needed to be included ina complete theory of quantum mechanics (see also Bell [Bell (1966)]) Theexperimentally observed violation of Bellrsquos inequality [Bell (1964) Aspectet al (1981) Giustina et al (2013) Hensen et al (2015)] however ruled outthe existence of (local) hidden variables

Based on that the EPR paradox needs to be solved in a different wayContrary to what EPR assumed it is in fact possible to predict the valueof an arbitrary observable of a physical system A with certainty via a mea-surement on system B although this observable was not defined before themeasurement Without any interaction a measurement on subsystem lsquoArsquo notonly creates lsquorealityrsquo of eg XA

Ω∆Ω simultaneously lsquorealityrsquo is also created

regarding the observable XBΩ∆Ω describing subsystem lsquoBrsquo Here the term

lsquorealityrsquo has the meaning as defined by EPR [Einstein et al (1935)] Simi-larly the detection of one photon of a two photon entangled number statenot only produces the reality of this photon but also that of a second oneA discussion of Einstein-Podolsky-Rosen entanglement can also be found in[Schnabel (2015)] Note that the EPR paradox can also be described aslsquoquantum steeringrsquo [Schrodinger (1935) Cavalcanti et al (2009) Handchenet al (2012)] It should also be mentioned that two-mode squeezing beingdetected with BHDs and not with photon counters cannot be used to violatea Bell inequality The latter topic is outside the scope of this Review

Bi-partite squeezed states were first characterized with balanced homo-dyne detectors by the group of J Kimble in 1992 [Ou et al (1992)] Gener-ally the EPR paradox becomes more pronounced the stronger the bi-partitesqueezing is A measure of the strength of EPR entanglement was introducedby M Reid [Reid and Walls (1985)] According to this measure the resultin Fig 5 can be quantified to ε2 = 00309 where the critical value is one Itcorresponds to the strongest Gaussian EPR entangled state generated so far

For a long time it looked like that two-mode squeezed states are notuseful for laser interferometers The reason for that belief was that a laserinterferometer as any other measurement device too is built to measureone observable It seems to be ideal already if the quantum noise in thissingle observable is squeezed The increased quantum noise in the orthog-onal observable is not harmful in this case and squeezing in two differentobservables useless Only recently realistic scenarios were discussed in whichtwo-mode squeezing in fact does improve the performance of a laser inter-

18

-12

-10

-8

-6

-4

-2

0

0 1 2 3 4 5 6 7 8 9 10

100

dB

109

dB

Δ (XAvac + X B

vac ) = Δ (Y

Δ (XA + X B)

Avacminus Y B

vac )2 2

2

Δ (YAminus Y B)2

Nor

mal

ized

noi

se v

aria

nce

[dB

]

Times [s]

Figure 5 Two-mode squeezing measurement ndash For this measurement the outputs oftwo balanced homodyne detectors are added or subtracted and the variances (noise pow-ers) of the results recorded The upper trace was measured with modes lsquoArsquo and lsquoBrsquo beingin their ground states This measurement served as a reference level Strong two-modesqueezing was observed as shown by the lower two traces The sideband frequency wasΩ(2π) = 8 MHz and the resolution bandwidth was ∆Ω(2π) = 200 kHz The measure-ment results were first published in Ref [Eberle et al (2013)]

ferometer [Steinlechner et al (2013)] The proof-of-principle experiment isreviewed in Sec 7

25 Observations using photon counters

Alternatively to field quadratures an optical mode in a squeezed statecan also be characterized at least partly by detecting its photon numberdistribution For a pure squeezed vacuum state such a measurement wouldreveal the existence of solely even photon numbers including a large prob-ability for zero photons The average photon numbers of squeezed vacuumstates with feasible squeeze factors are very small of the order of one persecond and bandwidth in hertz see Fig 13 (a) ndash (c) A distribution with closeto zero probability of odd photon numbers however has not been measuredso far The reason is the lack of ideal photon counters First of all theefficiency of these detectors ie their probability of converting one photoninto one click and no photon into no click must be almost perfect lsquoLostrsquophotons as well as dark counts wash out the oddeven oscillations Further-more most detectors available can only distinguish between zero and one

19

photon This problem can be solved by distributing the squeezed mode ontoa large number of single photon detectors using an array of beam splitterssuch that all paths have a low probability of carrying more than one pho-ton Photon number measurements on squeezed vacuum states neverthelessplay an extremely important role in quantum optics When the squeezingstrength is very low the probability of detecting more than 2 photons can beneglected and the detection of a photon heralds the existence of a second one

0

300

600

900

1200

-40 -20 0 20 400

5

10

15

20

25

Two-

fold

coi

ncid

ence

s [1

(4s

)](a) (b)

Non-degeneratetwo-mode squeezing

5050

APDA1

APDA2

APDHerald

Photon coincidencedetection

Delay [ns]

Thr

ee-f

old

coin

cide

nces

[1

(4s)

]1500

Figure 6 Coincidence clicks from non-degenerate photon pairs ndash The first suchexperiment was reported in Ref [Hong and Mandel (1986)] (a) shows a setup with threeavalanche photo-diodes (APDs) for proving the successful heralding of a single photonnumber state (b) Histograms of the two-fold coincidence detections at APDHerald andAPDA1 (red) and at APDHerald and APDA2 (yellow) with theoretical models (solid lines)If the two-mode squeezing just carried one photon in each spatial subsystem the three-foldcoincidence detection should be zero Indeed the according histogram (grey points righty axis) shows only a few events These are produced by false (dark) counts of the APDsThe delay for the three-fold coincidences is defined as the time between counts at lsquoA1rsquo andlsquoA2rsquo given that the trigger APDHerald detected a photon (within a 100 ns time window)The data was taken on photons that were up-converted from 1550 nm to 532 nm and itwas first published in Ref [Baune et al (2014)]

If a mode of light is always excited by either zero or two photons lsquocondi-tionalrsquo or lsquoheraldedrsquo one-photon Fock states can be realized (Measurementson an ensemble of the n-photon Fock state would always produce the mea-surement result n ie Fock states have a zero photon number uncertaintyThey are also called lsquonumber statesrsquo) The above concept of producing aone-photon Fock state obviously requires the deterministic and balanced dis-tribution of the down-converted signal and idler fields into two different paths

20

In order to achieve this the signal and idler fields need to be non-degenerateUsually a mode in a squeezed state is composed of degenerate signal andidler fields and this degeneracy thus needs to be removed Possible waysare producing the down-converted fields at well separated wavelengths [Vil-lar et al (2005) Su et al (2006) Li et al (2010) Samblowski et al (2011)]separating the upper and lower sidebands belonging to an ordinary squeezedmode by frequency filters [Schori et al (2002) Hage et al (2010)] and us-ing spatial filters [Hong et al (1987)] A frequently used approach is usingtype II parametric down-conversion where the photons within a pair are al-ways orthogonally polarized [Ou et al (1992) Kiess et al (1993) Kwiat et al(1995)]The list of experiments with conditional or heralded photon number states islong They showed for instance nonclassical g(2)-functions [Hong et al (1987)]and violations of Bell inequalities [Weihs et al (1998)] Fig (6) shows a re-sult from a more recent experiment in which a bipartite-squeezed state withsubsystems at 1550 nm and 810 nm was produced the subsystem at 1550 nmsubsequently up-converted to 532 nm and the lsquoquantum non-Gaussianityrsquo ofheralded up-converted single photons demonstrated [Baune et al (2014)]Squeezed states are also the resource for the conditional generation of super-positions of coherent states [Ourjoumtsev et al (2006) Neergaard-Nielsenet al (2006)] and so-called N00N-states [Afek et al (2010)]

The generation of nonclassical states mentioned in the paragraph aboveis not stationary but relies on a probabilistic trigger event The produc-tion of squeezed states themselves usually happens in a stationary fashionThis distinction has an important consequence for applications of nonclassi-cal states in measurement devices Only (stationary) squeezed states allowfor a continuous improvement of a measurement Avoiding any loss of mea-suring time is generally of high relevance for the detection of short-livedsignals with unknown arrival time as well as for the detection of long-livedquasi-monochromatic signals since the signal-to-noise-ratio (SN) improveswith measuring time

26 Conclusions

The detection of squeezed light produces measurement results that canbe considered as remarkable Let us focus on experiments where a mode in abright coherent state is overlapped with a mode in a squeezed vacuum state

21

as shown in Figs (1) and (3) In both setups the squeezed vacuum field caneasily be blocked which allows us to compare the measurement results ona bright coherent state with and without the interference with the squeezedvacuum state Without squeezing the photo-electric detectors measure alarge number of photon events with a large quantization noise (shot noise)The large noise reflects the fact that all photon events were independent fromeach other as shown in Fig 2 (bi) With squeezing the photo-electric de-tectors again measure a large number of photon events with an expectationvalue that is even slightly higher but nevertheless the quantization noise ofall detected photons is significantly reduced Fig 2 (bii)Based on the discussion of EPR entanglement in Subsec 24 the photo-electric detection of the output light of a squeezing-enhanced laser inter-ferometer (with αlowastα1) produces the reality of photons This way we cankeep the lsquowave picturersquo in which no photons exist when light travels alongthe interferometer arms and when it interferes at the beam splitter Whenthe energy of the beam is elevating electrons to the conductance band ofthe photo-diodersquos semi-conductor n photon events simultaneously appearwithin the measuring interval with probability P (n) What conclusion hasto be drawn if the probabilities resemble a sub-poissonian statistic ndash Theoccurrence of photon events is still truly random but in this case not forindividual photons The occurrence of photons is correlated in such a waythat the probability of detecting an additional photon in the same time in-terval reduces the larger the number of already detected photons is Whatfollows from the discussion of EPR entanglement for a photon counting ex-periment with pure squeezed vacuum and ideal photon counters Here theprobabilistic detection of one photon entails the detection of a second onewith certainty With some smaller probability a third photon is detectedwhich entails the detection of a fourth photon with certainty and so on

If a photon of a mode that was not interrogated by the environment beforeis absorbed its reality is created in this very moment If the photon belongsto a squeezed state this process instantaneously influences the probability ofother photons becoming reality

Of course a more general statement can be made based on the insightthat interaction with the environment creates the reality of any kind ofquanta including electrons atoms and molecules

22

3 Theoretical description of squeezed states

31 The quadrature amplitude operators

Consider a single mode of light at optical frequency ω Its Hamiltonoperator reads

Hω = ~ω(n+

1

2

)= ~ω

(adaggerωaω +

1

2

)= ~ω

(X2ω + Y 2

ω

) (7)

where n is the photon number operator and aω and adaggerω are the annihilationand creation operators which obey the commutation rule

[aω a

daggerω

]= 1 The

operator aω has a complex-valued dimensionless eigenvalue spectrum andcorresponds to the complex amplitude αω in classical optics Xω and Yω arethe hermitian amplitude and phase quadrature operators The eigenvaluesof the quadrature operators are also dimensionless and proportional to theelectric fields at the oscillationrsquos antinode and at the oscillationrsquos node Inthe above equation they are defined such that their variances are ∆2Xω =∆2Yω = 14 if the oscillator is in its ground state ie if 〈n〉 = 0

Although Eq (7) simply describes the energy of an harmonic oscillator itis the essence of quantum theory since it mathematically describes the wave-particle dualism Whereas the eigenvalues of n have a discrete spectrum theeigenvalues of Xω and Yω have a continuous spectrum In classical opticsthe phase quadrature is zero In quantum optics its expectation value is alsozero but its uncertainty contributes to the overall energy

Eq (7) describes a cavity mode as well as a section that is cut from apropagating quasi-monochromatic light beam The latter example is of highrelevance in actual experiments By setting the sectionrsquos time window ie themeasuring time interval the time-frequency (lsquomodulationrsquo) mode is defined

The quadrature operators introduced in Eq (7) and displayed in Fig 7 donot correspond to lsquoXrsquo and lsquoY rsquo that are of relevance in laser interferometry andin optical communication and which were already discussed in Subsec 22and 23 The optical frequency of visible and near-infrared light is far too highto be transferred to an oscillation of photoelectric voltage Quite general alaser interferometer targets signals at audio or radio band frequencies Ωi ωSuch a measurement is achieved as stated before by decomposing the photo-electric voltage from the photo diode at the interferometer output into asingle-sided spectrum (positive frequencies only) of intervals of Ωplusmn∆Ω2

23

(a)

(b)

(c)

(d)

2π 4π

2π 4π

2π 4π

2π 4π

Figure 7 Phase spaces and electric field oscillations of monochromatic lightndash Top Left Monochromatic light in a coherent state is represented by a phasor (whitearrow) including its quantum uncertainty (white dashed circle and fuzzy area) located inthe phase-space spanned by the quadratures Xω and Yω When the phase space rotateswith optical frequency ω2π the projection of the quantum phasor onto a fixed (vertical)axis corresponds to the electric field E(t) as shown on the right side (a) Weakly displacedcoherent state (b) Corresponding amplitude squeezed state The electric field uncertaintyaround the zero average field region is anti-squeezed (c) Vacuum state at the same opticalfrequency (d) Corresponding squeezed vacuum state The meaning of the uncertaintycould be carved out by supplementing them with monochromatic waves all having theoptical frequency ω2π Changing amplitudes then display amplitude quadrature noiseChanging shifts along the time axis model the electric field uncertainty at the expectedzero crossing They are not implemented in the graphics here however since any of thosewaves does not exist due to Heisenbergrsquos uncertainty relation

24

The signals as well as the quantum uncertainties carried by a beam oflight are thus described by a spectrum of pairs of non-commuting quadratureoperators Mathematically every such operator is defined by an integral overthe Fourier components within the bandwidth The spectral weighting of theFourier components is called the lsquowindow functionrsquo By going to sidebandintervals a spectrum of a new type of optical mode is defined which describesthe modulation of the electric field in the respective frequency interval Ω plusmn∆Ω2 In this Review we call it a lsquomodulation modersquo

The quadrature operators that are defined around a modulation frequencyΩ with a bandwidth of ∆Ω are the quadrature amplitude operators thatare relevant in laser interferometry Whenever they are not related to aspecific band we use the short form XΩ∆Ω(t) equiv X and YΩ∆Ω(t) equiv Y cfEqs (1) and (2) These operators can slowly vary with time where thetime dependence is limited by ∆Ω (The time dependence is not due toquantum uncertainty which usually is time independent but for instancedue to the time dependence of the signal eg a passing gravitational wave)Let us consider now a pair of quadrature operators for a particular sidebandΩ plusmn ∆Ω2 The Hamilton operator of the corresponding modulation modeis found by switching to the frame rotating at optical frequency ω Thetransition is done by applying the unitary transformation U = exp(iωadaggerat)generating a new Hamiltonian H = U daggerHωU minus i~UpartU daggerpartt The Hamiltonianof the modulation mode reads

H = ~Ω

(nΩ +

1

2

)= ~Ω

(adaggera+

1

2

)= ~Ω

(X2 + Y 2

) (8)

where nΩ is the (occupation) number operator for the modulation modeand a and adagger its annihilation and creation operators The commutation rule[a adagger

]= 1 is unchanged X and Y are the amplitude and phase quadra-

ture amplitude operators respectively They correspond to the depth of theamplitude modulation and for weak excitations to the depth of the phasemodulation respectively They are the conventional hermitian field operatorsin experimental quantum optics Note that modulation modes at angularfrequency Ω can be described by a superposition of three optical frequenciesa carrier at ω an upper sideband at ω + Ω and a lower sideband at ω minus ΩThe quantum mechanical description of modulation states in connection tooptical carrier and upper and lower sidebands is known as the lsquoTwo-PhotonFormalismrsquo [Caves and Schumaker (1985) Schumaker and Caves (1985)]

The quadrature amplitude operators in Eq (8) are again defined such

25

that the variances of the uncertainty of a modulation field in its ground stateor in a coherent state are

∆2Xvac = ∆2Yvac = 14 (9)

Generally quadrature operators X and Y as defined in Eqs (7) and (8) arethe real and imaginary parts of the annihilation operator

a = X + iY hArr adagger = X minus iY (10)

hArr X =1

2

(a+ adagger

) Y =

1

2i

(aminus adagger

) (11)

They satisfy the commutation relation[X Y

]=i

2 (12)

and their variances are limited by a Heisenberg uncertainty relation of thefollowing form

∆2X∆2Y ge 1

16 (13)

A quantum state is called a ldquosqueezed staterdquo [Bachor and Ralph (2004)]if ∆2Xϑ lt 14 for an arbitrary field quadrature Xϑ = X cosϑ + Y sinϑ seeEq (3) The angle of the lowest variance below 14 is called the squeeze angleθ The largest factor by which the variance is below 14 is called the squeezefactor often given on a decibel (dB) scale using the following transformation

minus 10 middot log10

(∆2Xθ

∆2Xvac

) (14)

The squeeze factor can also be described by the squeeze parameter r

eminus2r =∆2Xθ

∆2Xvac

(15)

When a squeezed state experiences optical loss it remains squeezed butthe squeeze factor is reduced Also the statersquos purity is reduced ie theproduct of the quadrature uncertainties increases above the minimum valueOptical loss corresponds to mixing the state with the vacuum state Let∆2Xϑ be the variance of a quadrature amplitude ∆2Xvac the variance of the

26

(quadrature angle independent) ground state uncertainty and (1 minus η2) therelative energy loss Then the resulting quadrature variance reads

∆2Xprime

ϑ = η2∆2Xϑ + (1minus η2)∆2Xvac (16)

To maximize the benefit from squeezed states in applications stronglysqueezed states need to be generated and optical loss minimized Opticalloss occurs due to absorption and scattering in the optical components inthe path of the squeezed beam including the squeezing resonator itself anddue to non-perfect matching to the interferometer mode non-perfect inter-ference contrast of the interferometer and non-perfect quantum efficiency ofthe photo diodes The sum of all losses including those outside the inter-ferometer need to be less then 10 to allow a nonclassical quantum noisesuppression of a factor of 10 in power ie 10 dB

32 Phase space representations of squeezed states

The Wigner function ndash The properties of squeezed states are nicely dis-played by the Wigner function W (X Y ) [Wigner (1932)] An example interms of a squeezed vacuum state is shown in Fig 8 It is a quasi-probabilitydistribution which contains the statersquos full information including its quan-tum statistic There are two ways how a Wigner function provides a sufficientcriterion for nonclassicality First by containing negative values second byfeatures that have a smaller (squeezed) width compared with the Wignerfunction of the ground state Integrating the Wigner function over Y pro-vides the probability density of measurement results ie of the eigenvaluesof the observable X and vice versa

infinintminusinfin

W (X Y )dY = p(X)

infinintminusinfin

W (X Y )dX = p(Y ) (17)

where p(X) and p(Y ) are the observed probability distributions also exem-plarily shown in Fig 8

The ground state coherent states as well as (quadrature) squeezed stateshave quadrature eigenvalue probability densities that are Gaussian TheirWigner functions are also Gaussian and thus entirely positive Wigner func-tions of other nonclassical states for instance Fock states exhibit negativevalues For this reason the Wigner function is called a quasi -probabilityfunction

27

X Y XY

Figure 8 Wigner function and its projections ndash Displayed is the full information ofa squeezed vacuum state The quasi probability density of the Wigner function (centre)is plotted along a third dimension and is color-coded Also shown are the actual mea-surement results from which the Wigner function is reconstructed They are representedby the squeezed and anti-squeezed Gaussian projections onto the X and Y axes Theirprobability densities p(X) and p(Y ) are given on the vertical axes The Gaussian measure-ment statistic of the first clearly shows squeezing compared to the ground state statistic(dashed) The squeeze factor is 116 dB and the squeeze parameter r = 1335 [Mehmetet al (2010)]

Fig 9 shows the Wigner functions for (a) a coherent state (b) the ground(vacuum) state (c) a displaced squeezed state and (d) a squeezed vacuumstate All Wigner functions describe a modulation of the carrier light atsideband frequency Ω integrated over the frequency interval ∆Ω The car-rier light is not part of these Wigner functions The displacement in (a)represents a classical amplitude modulation (b) corresponds to the absenceof any photons with a frequency offset of plusmnΩ from the local oscillator field(c) and (d) represent states whose amplitude modulation depth is more pre-cisely defined than that of the ground state Fig 10 shows Wigner functionspectrum for a broadband squeezed vacuum field Every Wigner functiondescribes the modulation field at some modulation frequency Ωi integrated

28

X

Y

X

(a) (b)

X X

(c) (d)

Y

Y Y

Figure 9 Simplified representation of Wigner functions ndash The darker the shadedareas the larger is the phase-space quasi-probability Shown are four different (time-independent) states of a modulation mode at frequency Ω for a specific resolution bandwidth ∆Ω Panel (a) represents a coherent state the displacement (α) corresponds to aclassical amplitude modulation Panel (b) represents the ground (vacuum) state (c) adisplaced squeezed state and (d) a squeezed vacuum state both with squeeze angle θ = 0The latter is in analogy to Fig 8 Again the light field that carries the modulation is notpart of the pictures

over the resolution bandwidth (RBW) of ∆Ω

The Glauber-Sudarshan P -function ndash The P -function [Glauber (1963)Sudarshan (1963)] is calculated by de-convoluting the Wigner function fromthe ground state uncertainty [Gerry and Knight (2005)] For displaced vac-uum states (coherent states) the P -function corresponds to a displaced δ-function The mathematical expression of the P -function of a squeezed statecontains infinitely high orders of derivatives of the δ-function [Vogel andWelsch (2006)] Such a function contains negativities but cannot be dis-played It is possible however to define a phase-space quasi probabilityfunction for squeezed states that can be displayed and that does show neg-

29

0

Ω1

Ω2

Ω

Single-sided spectrum

Y

X

Figure 10 Hint of a Wigner function spectrum ndash A single-sided spectrum (positivefrequencies only) with respect to the carrier field can be used to visualize a broadbandsqueezed field Shown are two examples displaying a squeezed vacuum state at Ω1 anda displaced amplitude squeezed state at Ω2 The individual Wigner functions cover theresolution bandwidth ∆Ω gt 0 (not shown) In general the squeezing strength as well asthe squeeze angle and the displacement are a function of sideband frequency

ativities as a sufficient and necessary condition for certifying the squeezingeffect This lsquononclassicality functionrsquo is calculated by de-convoluting theWigner function from an uncertainty distribution that is steeper than theGaussian distribution A pronounced negativity of a squeezed vacuum stateof up to 69 standard deviations was found [Kiesel et al (2011)]

The double-sided phasor picture ndash This phasor picture links quantumstates of modulations with the quantum states of the contributing opticalfields [Bachor and Ralph (2004)] and is mathematically described by thetwo-photon-formalism [Caves and Schumaker (1985) Schumaker and Caves(1985)] Generally a weak amplitude or phase modulation at frequency Ω ofa carrier field at optical frequency ω can be understood as the carrierrsquos beatwith two optical frequencies at ωplusmnΩ The double-sided phasor picture is ableto display a spectrum of different and independent modulation frequenciesin the rotating frame of the carrier field The carrier light field is time-independent but the upper and lower sidebands are not They rotate withplusmnΩi(2π) respectively around the frequency axis

30

ω0

Yω

Xω

Upper sideband

Lower sideband

ω0+Ω1

ω0+Ω2

ω0-Ω2

ω0-Ω1

ω

Upper sideband

Lower sideband

Yω

Xω

ω0+Ω1

ω0+Ω2

ω0-Ω2

ω0-Ω1

ω

ω0

Figure 11 Double-sided phasor pictures ndash Phase spaces at optical frequency ω0 plusmnΩirotate around the frequency axis with sideband frequency plusmnΩi Its sign determines thedirection of rotation with respect to the local oscillator in the rotating frame at ω0 Apair of phase spaces need to be superposed to provide a description of a modulation fieldat |Ωi| Top Amplitude quadrature squeezed field with a coherent displacement at |Ω2|The displacement corresponds to a classical amplitude modulation The uncertaintiesof all optical frequencies are circular and larger than that of the ground state (dashed)Quantum correlations are indicated by additional symbols More details are given in themain text Bottom Corresponding spectrum of (displaced) vacuum states which do nothave any quantum correlations

31

Fig 11 shows such a double-sided phase space picture where the carrierrsquosmodulation at Ω1 is in a squeezed vacuum state and where the modulationat Ω2 is in a displaced squeezed state The picture shows how a classicalamplitude modulation as well as the quantum statistic of a modulation fieldis decomposed into contributions from upper and lower sidebands For asqueezed modulation field the upper and lower sidebands show no squeezedbut circular thermally excited quantum uncertainties The uncertainties of apair of sidebands however show correlations as well as anti-correlations InFig 11 these (anti-) correlations are marked with times and + for the modulationfrequency Ω1 and with N and bull for the modulation frequency Ω2

33 Covariance matrix representation of (single-party) squeezed states

Since squeezed states have a Gaussian quantum statistic four numbersare sufficient for their full description These numbers are the second mo-ment of the quadrature amplitude showing the strongest squeezing and thesecond moment of its orthogonal quadrature amplitude as well as their firstmoments describing the displacement These four numbers are sufficientto calculate the Wigner function shown in Fig 8 In general the quadra-ture of strongest squeezing is not perfectly aligned with one of the axesof the measurementrsquos coordinate system The so-called covariance matrix(VXXVXY VY XVY Y ) [Simon et al (1994)] accounts for phase space rotationsand enables the calculation of how these states evolve within an interfero-metric arrangement Their components are normalized to the vacuum noisevariance ∆2Xvac = ∆2Yvac and read

VXY =

langXY + Y X

rangminus 2

langXranglang

Yrang

2∆2Xvac

(18)

The following examples represent the ground state a pure 10 dB am-plitude quadrature squeezed state and a pure 10 dB squeezed state with asqueeze angle of 45

Vvac =

(1 00 1

) V0

01 =

(01 00 10

) V45

01 =

(505 495495 505

) (19)

with V4501 =RT

45V001R45 where Rα = (cosα minussinα sinα cosα) is the rota-

tion matrix

32

34 Phase space representation of two-mode (bi-partite) squeezed states

XA

YA

XB

YB

Figure 12 Bi-partite squeezed vacuum state ndash Shown is a Wigner-function-basedphase space representation in close analogy to the single party version in Fig 9 (d) Thepicture describes a single modulation at frequency Ω with bandwidth ∆Ω Measurementsat party A and B reveal local Wigner functions that correspond to thermal states sincethe uncertainties (indicated by the color and by the large circles) are symmetric and largerthan that of the ground state (indicated by the small dashed circles) The uncertaintieshowever show correlations and anti-correlations here indicated by + and times respectivelyThe strength of these (anti-)correlations are indicated by ellipses Bi-partite squeezingie entanglement is present if the short axes of the ellipses are shorter than the diameterof the ground state uncertainty The picture in fact represents Einstein-Podolsky-Rosenentanglement [Einstein et al (1935)] From a measurement of XA or YA the correspondingmeasurement result at party B can always be inferred with an uncertainty that is smallerthan the ground state uncertainty

A bi-partite state enables a measurement on subsystem A and simulta-neous a measurement on subsystem B For a large number of simultaneousensemble measurements of the same quadrature amplitude Xϑ the followingtwo joint quadrature variance can be calculated

∆2(XAϑ plusmn XB

ϑ ) (20)

A state that is symmetrically shared between two parties (A and B) is calleda two-mode squeezed state if the variances of joint quadrature measurementsfulfill the following inequality [Duan et al (2000)] ie

∆2(XA minus XB)

∆2(XAvac minus XB

vac)+

∆2(Y A + Y B)

∆2(Y Avac + Y B

vac)lt 2 (21)

33

with ∆2(XAvac plusmn XB

vac) = ∆2(Y Avac plusmn Y B

vac) = 2∆2Xvac A lsquotwo-mode squeezedstatersquo reveals entanglement in the second moments of the measurement statis-tics It is thus a lsquobi-partite Gaussian entangled statersquo

Fig 12 displays a (pure) bi-partite squeezed vacuum state distributedbetween A and B The state shows full symmetry regarding its subsystemsat the two sites The large circles and the colored area represent Wignerfunctions of the subsystems Measurements of the quadrature amplitudesXA YA XB and YB show identical variances and the correlations and anti-correlations have identical strength since ∆2(XA minus XB) = ∆2(YA + YB) lt12 for our normalization of quadrature amplitudes having a ground statevariance of 14

Generally a symmetric bi-partite squeezed state fulfills another quantita-tive (Gaussian) entanglement criterion if less than 50 of the vacuum state issymmetrically mixed into the initially pure state Bi-partite squeezed statesare always entangled but in this case they are even Einstein-Podolsky-Rosen(EPR) entangled [Reid (1989)] allowing the demonstration of the quantumsteering effect [Einstein et al (1935) Schrodinger (1935) Reid (1989) Cav-alcanti et al (2009)] The first such experiment was performed by Ou etal [Ou et al (1992)] using type II parametric down-conversion (PDC) Laterexperiments produced bi-partite squeezed vacuum states by overlapping twosqueezed vacuum states each produced with type I PDC on a balanced beamsplitter and used the entangled output for the demonstration of quantumteleportation [Furusawa et al (1998) Bowen et al (2003ca)] The criterionin Eq (21) and the EPR criterion from [Reid (1989)] was experimentallycompared in Ref [Bowen et al (2003b)] The steering effect in asymmet-ric bi-partite squeezed states were recently experimentally characterized inRef [Handchen et al (2012)]

Fig 12 shows features similar to those in the top part of Fig 11 Thisis not a coincidence and shows that a bi-partite squeezed state can also begenerated by spatially splitting the upper and lower sideband of a (single-party) squeezed state This was first experimentally demonstrated by thegroup of E Polzik [Schori et al (2002)] and later used for EPR multiplex-ing of a single longitudinal mode of a squeezing resonator [Hage et al (2010)]

35 Covariance matrix representation of bi-partite squeezed states

Also the full information of bi-partite states including the entanglementcan be cast by the covariance matrix [Simon et al (1994)] which can be used

34

to calculate the propagation of these states in laser interferometers Againall variances are normalized to the vacuum noise variance in full analogy toEq (18) The generic bi-partite covariance matrix has dimension 4times4 andreads

Vbp =

VXAXA

VXAYA VXAXBVXAYB

VYAXAVYAYA VYAXB

VYAYBVXBXA

VXBYA VXBXBVXBYB

VYBXAVYBYA VYBXB

VYBYB

with

VXAYB =

langXAYB + YBXA

rangminus 2

langXA

ranglangYB

rang2∆2Xvac

ϑ

(22)

Due to the symmetry in Eq (22) the 4times4 covariance matrix is fully spec-ified by just ten independent coefficients If the phase spaces at A and Bare aligned along the strongest correlations and anti-correlations the ma-trix components referring to different quadrature amplitudes eg VXAYA arezero Such entangled states can be produced by overlapping two squeezedfields with a squeeze angle difference of 90 on a balanced beam splitter

A symmetric bi-partite squeezed vacuum state which is also called anlsquoS-classrsquo [DiGuglielmo et al (2007)] bi-partite squeezed vacuum state shows(anti-)correlations in two joint quadratures as defined in Eq (21) For a puresuch state of 10 dB squeezing the covariance matrix reads

Vbp10|10 =

505 0 495 0

0 505 0 minus495495 0 505 0

0 minus495 0 505

The following covariance matrix describes a so-called lsquoV-classrsquo 10 dB bi-partite squeezed vacuum state Here only one joint quadrature shows 10 dBsqueezing whereas the orthogonal joint quadrature shows vacuum noise Thestate is obtained by overlapping one 10 dB squeezed state with a vacuumstate on a balanced beam splitter

35

Vbp0|10 =

055 0 045 0

0 55 0 minus45045 0 055 0

0 minus45 0 55

The first measurement of all elements of such a covariance matrix was achievedin [DiGuglielmo et al (2007)]

36 Photon numbers of squeezed states

In contrast to the ground state squeezed vacuum states do have photonexcitations As said earlier quantum theory links the wave and the particlepictures Indeed the squeeze factor of a modulation mode is directly con-nected to a certain photon number excitation Squeezed states of light areproduced via spontaneous photon pair generation eg by parametric down-conversion The following operator S is called the lsquosqueeze operatorrsquo [Gerryand Knight (2005)] It creates and annihilates photon pairs

|r θ〉 = S(r θ) |0〉 (23)

where |r θ〉 is a squeezed vacuum state with squeeze parameter r and squeezeangle θ and |0〉 is the vacuum state The definition of the squeeze operatoris

S(r θ) = exp

[1

2

(reminusiθa2 minus reiθadagger 2

)] (24)

The following shows that this definition indeed results in a state with squeezedquadrature amplitude variances Lets set θ = 0

〈0| Sdagger(r 0) X S(r 0) |0〉 =1

2〈0| Sdagger(r 0)

(a+ adagger

)S(r 0) |0〉 (25)

〈0| Sdagger(r 0) Y S(r 0) |0〉 =i

2〈0| Sdagger(r 0)

(aminus adagger

)S(r 0) |0〉 (26)

Using the Baker-Hausdorff formula we get

Sdagger(r θ) a S(r θ) = a cosh r minus adaggereiθsinh r (27)

Sdagger(r θ) adagger S(r θ) = adaggercosh r minus aeminusiθsinh r (28)

36

Since 〈0| X |0〉 = 〈0| Y |0〉 = 0 also Eqs (25) and (26) are zero To finallycalculate the variances we need

〈0| Sdagger(r 0) X2 S(r 0) |0〉 =1

4〈0| Sdagger(r 0)

(a2 + adaggera+ aadagger + adagger2

)S(r 0) |0〉

〈0| Sdagger(r 0) Y 2 S(r 0) |0〉 = minus1

4〈0| Sdagger(r 0)

(a2 minus adaggeraminus aadagger + adagger2

)S(r 0) |0〉

Given that SSdagger is the identity and using again Eqs (27) and (28) we get theexpected variances

∆2X =1

4

[cosh2 r minus 2cosh r sinh r + sinh2 r

]=

1

4eminus2r

∆2Y =1

4

[cosh2 r + 2cosh r sinh r + sinh2 r

]=

1

4e2r

Since the squeeze operator can only create and annihilate photon pairsa squeezed vacuum state without photon loss must correspond to an evennumber of photons But not only photon loss also a coherent displacementleads to flattening out the odd-even oscillations The probability of detectingN photons in a pure displaced squeezed state are derived for instance in[Gerry and Knight (2005)] and read

P (N) = |〈N |α r θ〉|2 =(05 tanh r)N

N cosh rexp

[minus|α|2 minus 1

2(αlowast2eiθ + α2eminusiθ)tanh r

]times∣∣∣HN

[(αcosh r + αlowasteiθsinh r)

radiceiθsinh(2r)

]∣∣∣2 (29)

where HN is the N th Hermite polynomialFig 13 shows the photon number distributions for 5 different pure squeezed

states according to Eq (29) Panels (a) to (c) show squeezed vacuum stateswith 43 dB 86 dB and 172 dB of squeezing Panel (d) shows the moregeneral case of a squeezed state with a coherent displacement α 6= 0 Dueto θ = 0 the state is amplitude quadrature squeezed Panel (e) refers to thecorresponding phase quadrature squeezed state For comparison panel (f)shows the photon number distribution of the coherent state with the samedisplacement

The panels in Fig 13 represent the diagonal elements of the statersquos den-sity matrix in number basis Only the latter also contains the coherences

37

002

004

006

000 000

002

004

006

008

403020100 Photon number n

403020100Photon number n

Pro

babi

lity

403020100000

005

010

015

100 2 4 6 8100 2 4 6 800

04

06

02

100 2 4 6 800

03

02

01Pro

babi

lity

Pro

babi

lity

08

00

04

06

02

08

(a) (b)

(c) (d)

(e) (f)

Figure 13 Photon number distributions ndash All panels represent pure states (a) 43 dBsqueezed vacuum state (r = 05 α = 0) (b) 86 dB squeezed vacuum state (r = 1 α = 0)(c) 172 dB squeezed vacuum state (r = 2 α = 0) (d) displaced 86 dB squeezed state(r = 1 θ = 0 α = 4) (e) displaced 86 dB squeezed state (r = 1 θ = π2 α = 4) (f)Coherent state (r = 0 α = 4) The average photon numbers are about 027 14 131174 174 and exactly 16 see Eq (30)

between photon numbers [Gerry and Knight (2005)] Figures as shown heregenerally do not give full descriptions of the states

38

A squeezed vacuum state (r 6= 0) always has a non-zero photon numberand can not be the ground state The average photon number of a puresqueezed vacuum state can be calculated using Eq (8) With the maximallysqueezed quadrature variance ∆2Xsqz the average photon number is given by

n = 〈n〉|α=0θr〉 = ∆2Xsqz +(∆2Xsqz)

minus1

16minus 1

2=eminus2r

4+e2r

4minus 1

2 (30)

with the vacuum noise variance normalized to one quarter A coherent dis-placement further adds |α|2 photons on average

4 Squeezed-light generation

41 Overview

Squeezed light was first produced in 1985 by Slusher et al using four-wave-mixing in sodium atoms in an optical cavity [Slusher et al (1985)]Shortly after squeezed light also was generated by four-wave-mixing in an op-tical fibre [Shelby et al (1986)] and by degenerate parametric down-conversion(PDC) in a 2nd-order nonlinear crystal placed in an optical cavity [Wu et al(1986)] The pumped cavity was operated below its oscillation thresholdie the parametric gain did not fully compensate the round trip losses whichis also called lsquocavity-enhanced optical-parametric amplification (OPA)rsquo

The early day experiments achieved squeeze factors of a few percent up toabout 3 dB Today squeeze factors of more than 10 dB are directly observedin several experiments [Vahlbruch et al (2008) Eberle et al (2010) Stefszkyet al (2012) Vahlbruch et al (2016)] All of them are based on cavity-enhanced OPA (below threshold) The parametrically amplified mode isdegenerate ie signal and idler modes are identical In particular the down-conversion process is of lsquotype Irsquo which means that the amplified mode has awell-defined polarization Squeezed states can also be generated above oscil-lation threshold In Refs [Villar et al (2006) Jing et al (2006)] bi-partitesqueezing was generated with above-threshold PDC Both experiments usedtype II PDC which provides orthogonally polarized signal and idler fieldsType II PDC below threshold was also used to generate squeezed and bi-partite squeezed fields [Grangier et al (1987) Ou et al (1992)] All these

39

experiments were performed in the continuous-wave regime which is also thefocus of this Review Squeezed states of modulations of trains of laser pulseshowever have been also generated since the 1980s using either PDC or theoptical Kerr effect [Slusher et al (1987) Bergman and Haus (1991) Our-joumtsev et al (2006) Dong et al (2008)] For an overview of the develop-ments in squeezed-light generation in the continuous-wave as well as pulsedregime see Ref [Bachor and Ralph (2004)] Squeezed-light generation inopto-mechanical setups [Aspelmeyer et al (2014)] which use the intensitydependent phase shift from radiation pressure was discussed in Refs [Paceet al (1993) Rehbein et al (2005) Corbitt et al (2006)] and recently ex-perimentally achieved by several groups [Brooks et al (2012) Safavi-Naeiniet al (2013) Purdy et al (2013)]

42 Degenerate type I optical-parametric amplification (OPA)

This section provides a graphical description of how degenerate type IOPAPDC turns a vacuum state into a squeezed vacuum state and a coher-ent state into a displaced squeezed state The process requires a bright pumpfield and a 2nd-order nonlinear crystal For simplicity we set all nonlinearitiesabove 2nd-order to zero

Let us consider a short segment of the second-order nonlinear crystalpumped with light of optical frequency 2ν All other modes that enter thecrystal shall not contain any photons ie are in their vacuum states Ofthese the only mode of interest is that at optical frequency ν which spatiallyoverlaps with the pump mode Fig 14 shows the total electric field of theoptical input E in and the 2nd-order nonlinear dielectric polarisation of thecrystal P The latter is proportional to the total electric field of the outputEout The pump field at 2ν periodically drives the vacuum field at ν betweenregions of low and high polarisation This process transforms the vacuumstate into a squeezed vacuum state in the output [Bauchrowitz et al (2013)]The output further contains the hardly depleted pump field and frequencydoubled parts of the pump field at 4ν It is again emphasized that Fig 14displays OPA in a small segment of the crystal In reality the nonlinear effectaccumulates over the crystal length or even over several passages since thecrystal is usually put into an optical resonator A noticeable effect is achievedif all infinitesimal contributions constructively interfere This is achieved incase of phase matching ie if the wave fronts of the modes at 2ν and νpropagate with the same speed and thus do not run out of phase Note that

40

t

t

Squeezing

Figure 14 Optical parametric amplification of a vacuum state ndash The upper leftcorner shows the crystal polarization P(E) = ε0

(χ(1)E + χ(2)E2

) ie the separation of

charge carriers by the electric component of an optical field E The graph illustrates howan input quantum field (from below) is projected into an output quantum field (towardsthe right) In the example shown here the input field is composed of a classical pumpfield E in

2ν at frequency 2ν and zero-point fluctuations E inν of a field at frequency ν cf

Fig 7(c) The superposition E in of these two fields is transferred into a time-dependentdielectric polarization that is the source of (and thus directly proportional to) the elec-tric component of the output field Eout The quantum uncertainty of the output fieldshows a phase-dependent (parametric) amplification at frequency 2ν Spectral decompo-sition of the output field Eout reveals coherent amplitudes at frequencies 2ν and 4ν and asqueezed vacuum state Eout

sqzν cf to Fig 7(d) The concept of this figure was published inRef [Bauchrowitz et al (2013)]

41

t

t Amplitudequadraturesqueezing

Figure 15 Optical parametric amplification of a coherent state ndash The pictureshows how a displaced vacuum state is transformed into a displaced squeezed state Thepumprsquos electric field is maximal when the input field at fundamental frequency ν shows azero crossing The phase relation described results in an output state that is amplitudequadrature squeezed If the pump field at the input in phase was shifted by half of itswavelength the squeezed output field were a phase quadrature squeezed The squeezinggeneration displayed here corresponds to the transition from Fig 7(a) to Fig 7(b) but withan additional reduction of the displacement The concept of this figure was published inRef [Bauchrowitz et al (2013)]

in actual squeezing experiments the 4ν component is usually suppressed byphase miss-matching

Fig 15 shows the same process but now for an input field at frequencyν in a coherent state In this case the relative phase between the two inputstates ∆ϕ = ϕ2ν minus 2ϕν is relevant In Fig 15 the relative phase is set suchthat the expectation value of the field at frequency ν is zero when the pump

42

(a) (b)

(c) (d)

X

Y

X

YΔϕ = ndash50deg

X

Y

Δϕ = 0deg

X

Y

Δϕ = 90deg

Figure 16 Phase-space illustration of degenerate OPA ndash The (displaced) dashedcircle in each diagram represents the uncertainty of the initial state at optical frequencyν The (displaced) shaded area represents the state after degenerate optical parametricamplification All quantum uncertainties shown correspond to pure states The boldgreen arrow describes the bright second-harmonic pump field whose uncertainty can beneglected The phase between the 2nd harmonic pump and the initial state (∆ϕ = ϕ2ν minus2ϕν) determines the result of the parametric amplification

field reaches its maximum (∆ϕ = 90) The output at the fundamentalfrequency is then an amplitude squeezed state with a deamplified coherentamplitude

Fig 16 summarizes the squeezing operation on the vacuum state as wellas on displaced vacuum states for different phase relations ∆ϕ between thetwo input fields

43 Cavity-enhanced OPA

Placing the nonlinear crystal inside a cavity can greatly enhance thedown-conversion efficiency but not only that A cavity introduces a thresholdfor the pump power above which the parametric gain is infinite just limited

43

by the finite pump power In this case the vacuum uncertainty of the inputfield at frequency ν is amplified to a bright laser field at frequency ν Thedevice is then called an optical-parametric oscillator (OPO) For the gener-ation of squeezed states however the pump power is usually kept (slightly)below threshold Due to nonzero optical loss there exists a pump powersmaller than the threshold above which the tiny improvement of squeezingis not noticeable anymore Getting the pump power closer to the thresholdcould even reduce the observed squeeze factor if a fluctuating squeeze an-gle projects anti-squeezing into the observed quadrature amplitude [Franzenet al (2006) Suzuki et al (2006) Dwyer et al (2013)] The cavity has an-other important purpose It confines the transverse spatial mode usuallyto TEM00 This mode confinement is crucial for any efficient application ofthe squeezed state in laser interferometry since it allows the suppression ofanti-squeezing from other transversal modes The squeezing process requiresa nonlinear material that should show negligible absorption at both opticalfrequencies involved in particular at the wavelength of the squeezed modeIn Refs [Vahlbruch et al (2008) Mehmet et al (2009)] 10 dB and 116 dBof squeezing were achieved using MgOLiNbO3 The highest squeeze factorstoday are produced in (quasi phase matched) periodically poled KTP [Eberleet al (2010) Mehmet et al (2011) Stefszky et al (2012) Vahlbruch et al(2016)]

The optical cavity that is built around the nonlinear crystal is vital forsqueezed-light generation and it deserves a detailed consideration Gener-ally the mode propagating away from a cavity is the result of interferenceat the cavity coupling mirror One contribution is given by the intra-cavityfield attenuated by the amplitude transmission coefficient t of the outcouplingmirror The second contribution is given by the outside field that is reflectedby the same mirror with amplitude reflectivity r =

radic1minus t2 and spatially

overlapped with the first Also the mode from a squeezing resonator is suchan interference product

The impedance matched resonator

Let us consider first an empty optically stable and loss-less Fabry-Perotresonator built from two identical mirrors each with amplitude reflectivityr = r1 = r2 lt 1 A propagating field be perfectly mode-matched to one ofthe cavity resonances In this setup the resonator shows zero reflection andthe resonator is said to be impedance matched (for all such input fields)

44

|α〉

r1α

r12

= +

|α〉 r22 = r1

2

minusr1α equiv t1 αcav = 1minus r12 minusα sdot r1

1minus r12

0

XY

Figure 17 Empty impedance-matched resonator ndash Mode-matched and resonant lightthat enters the cavity from the left here displayed by a quantum phasor for a coherentstate is fully transmitted including its quantum uncertainty The back-reflected statedestructively interferes with itself for all frequencies well within the cavity linewidth The180 phase shift of the transmitted cavity field amplitude minusr1α is a necessary conditionin order to fulfill energy conservation on cavity resonance Impedance matching is forinstance achieved for a resonant cavity with matched mirror reflectivities (r2

2 = r21) and

zero optical loss The complex amplitude of the field traveling towards left inside the cavityis readily deduced from the figure and reads αcav = minusαr1(1minus r2

1)minus12 (highlighted by thedashed circle) In the displayed setup zero field uncertainties are reflected however alsothe vacuum state that enters the cavity from the right is fully transmitted (not shown)

r1α

XY

r12 r2

2 =1

= + |α〉

minus 1+ r1( )α equiv t1 αcav = 1minus r12 minus 1+ r1( )α

1minus r12

|α〉

Figure 18 Empty maximally overcoupled resonator ndash Maximal overcoupling isachieved for a resonant cavity with a perfect end-mirror reflectivity (r2

2 = 1) and zerooptical loss For a given input-mirror reflectivity r2

1 the intracavity light power is maximalMode-matched and resonating light entering the resonator from the left is fully reflectedThe complex amplitude of the field traveling towards left inside the cavity is readilydeduced from energy conservation to αcav = minusα(1 + r1)(1 minus r2

1)minus12 (dashed circle) Inthis setup no uncertainty from the right couples to the left side of the cavity

45

Obviously the interference described in the previous paragraph is fullydestructive The same resonator also shows zero reflection of the input fieldrsquosquantum uncertainty since the interference happens between parts of thesame quantum state The mode propagating away from such a resonatorhowever is not in a nonclassical but in a vacuum state because the vacuumstate that enters the cavity through the opposite site is also fully transmittedThe interference at the coupling mirror of an impedance matched resonatoris displayed in Fig 17

r1α

r12

+

=

OPA PDC

Xcavg r1(1+r1) Xcav

Δ2Xg 0 Δ2Yg

XY

r22 =1

|α〉

Figure 19 Squeezing resonator ndash Shown is the interference at the zero-loss squeezingresonator operated at threshold The lower line represents the perfectly squeezed modepropagating away from the cavity towards the left The parametric gain medium inside thecavity deamplifies the X quadrature of the cavity mode (Xcav) by the factor r1(1 + r1)which is the ratio of the intra-cavity field amplitudes of the two previous figures The Xquadrature of the field that is back-reflected towards the left destructively interferes withitself similar to the situation of the impedance matched cavity in Fig 17 The parametricpower gain per resonator round-trip (G) needs to mimic the effect of an end mirror withreflectivity r2

2 = r21 For this reason the deamplification of Xcav corresponds to a round-

trip deamplification factor of r1 equivradic

1G The round-trip amplification factor for Ycav

then is 1r1 equivradicG which exactly compensates for the outcoupling and thus determines

the parametric oscillation threshold (threshold for bright lasing) The variances of thequantum uncertainties ∆2Ycav and ∆2Y are thus infinite In this setup no field uncertaintyfrom the right couples to the left of the cavity and a perfectly X-quadrature-squeezed fieldoutside the squeezing resonator is produced

46

The perfectly over-coupled single-ended resonator

We now increase the reflectivity of the far mirror lsquo2rsquo to being perfect (r2 = 1)This way the counter-propagating vacuum state can not enter the cavityAgain a propagating field be perfectly mode-matched through mirror lsquo1rsquo toone of the cavity resonances For frequencies well inside the cavity linewidththe situation is displayed in Fig 18 The setup protects the left side of thecavity against vacuum fluctuations entering through mirror lsquo2rsquo but of coursedoes not squeeze quantum noise The intra-cavity built-up factor is too highfor achieving destructive interference below the vacuum uncertainty on theleft side of the resonator

The impedanced-matched single-ended squeezing resonator

Building on the two previous concepts the straight forward approach now isto start from the perfectly over-coupled single-ended resonator and insert anattenuator into the cavity that does not couple the cavity mode to any bathbut still results in a roundtrip efficiency of precisely r1(lt 1) in amplitudeOptical loss is not appropriate since it increases the coupling of the cavitymode to a thermal bath neither would any phase-insensitive atenuator beappropriate It is easy to show that a phase-insensitive attenuator adds ad-ditional uncertainty since otherwise the commutation relation [a adagger] = 1 isviolated The amplification process that matches our requirement is OPA Toachieve infinite squeezing in X on cavity resonance a second-order nonlinearcrystal needs to be put into the cavity and pumped such that the intra-cavityamplitude quadrature is attenuated by the factor (1 + r1)r1 (on cavity reso-nance) with respect to the empty cavity This factor is readily deduced fromFigs 17 and 18 Due to the symmetry in parametric amplification the intra-cavity phase quadrature is then amplified by (1 + r1)r1 and the round-tripgain has a value of 1r1 in amplitude In this situation not only infinitesqueezing but also the (laser) threshold of the resonator is achieved sincethe round-trip gain of the intra-cavity phase quadrature equals its roundtriploss here fully given by the incoupling mirror

The physical descriptions in Figs 17 to 19 are fully consistent with ob-servations in squeezing experiments The consideration above in particularshows that the intra-cavity field shows a finite squeezing strength while theexternal field shows infinite squeezing The strongest intra-cavity squeezefactor possible is (1 + r1)2r2

1 In the high reflectivity limit this factor corre-

47

sponds to 6 dB Averaged over the full cavity mode the squeeze factor of thecavity mode is in this limit even limited to 3 dB [Walls and Milburn (2008)]Higher intra-cavity squeeze factors are possible for lower mirror reflectivities

44 The generation of squeezed light for laser interferometry

With the insights gained in the previous subsection we now turn to ac-tual experiments The application of squeezed states in laser interferometrycertainly requires large squeeze factors (idealy accompanied with the highestpossible purity) to maximize the impact in terms of sensitivity improvementIn cavity-enhanced OPA the highest parametric gain is achieved on cavityresonance ie at zero sideband frequency But this is not the main reasonwhy this Subsection focusses on the generation of squeezed states at lowsideband frequencies The application of squeezed states in a laser inter-ferometer requires that their sideband frequencies cover the devicersquos signalband Ground-based gravitational wave (GW) detectors have a detectionband from about 10 Hz to 10 kHz frequencies which can be considered aslsquolowrsquo compared to typical frequencies in quantum optics experiments

Squeezing at MHz sideband frequencies is easier to observe than at acous-tic frequencies because the latter are often polluted with excess noise fromlight beams that serve as control beams [Bowen et al (2002) McKenzie et al(2004)] and parasitic interferences from back-scattered light [Vahlbruch et al(2007)] Furthermore the observation of squeezing at low sideband frequen-cies requires a more stable setup since larger measuring times are necessaryThe observation of strong squeezing at MHz frequencies however alreadysets an upper limit to the optical loss of the setup At least the same squeezefactor can be observed at lower frequencies

There are two different main topologies for squeezing resonators TheFabry-Perot-type standing-wave resonator consists of a minimum number ofmirror surfaces and has the advantage of being compact and thus robustagainst mechanical vibrations Usually one or even two mirror coatings aredirectly placed on the spherical and polished surfaces of the nonlinear crys-tal itself [Wu et al (1986) Grangier et al (1987) Breitenbach et al (1998)Vahlbruch et al (2008) Eberle et al (2010) Vahlbruch et al (2016)] TheBowtie traveling-wave resonator has the advantage of providing a separa-tely accessible counter propagating mode for cavity length control [Ou et al(1992) Takeno et al (2007)] It shows no direct back-reflection of incoupledlight which helps reducing parasitic interferences [Stefszky et al (2012)]

48

(a)

(b)

(c)

(d)Squeezingresonator

To inter-ferometer

OPA

SHG

BHD

Mode cleaner

LO

Laser

DBS

DBS

Figure 20 Generation of squeezed light ndash (a) Example of a 2nd-order nonlinearcrystal for the squeezed-light generation at 1064 nm Shown is a bi-convex 65 mm long7MgOLiNbO3 crystal whose polished surfaces also carry the mirror coatings of the res-onator The crystal thus realizes a monolithic squeezing resonator as it was used forthe first demonstration of 10 dB squeezing [Vahlbruch et al (2008)] (b) Optical con-figuration of a half-monolithic (hemilithic) standing-wave squeezing resonator Here thecavity length can be adjusted by displacing the coupling mirror The crystal surface insidethe cavity is anti-reflection coated The photograph shows a 10 mm long PPKTP crystalsqueezing resonator as used for the GEO 600 squeezed-light source [Abadie (2011)] (c)Mechanically stable housing of a standing-wave squeezing resonator The crystalrsquos temper-ature is stabilized at its phase matching condition using Peltier elements (d) Schematicfor the squeezed-light generation After spatial filtering of continuous-wave laser lighttwo hemilithic standing-wave resonators are employed The first generates second har-monic pump light (SHG) The second (OPA) generates a squeezed vacuum field at theinitial wavelength The squeezed states are observed by a balanced homodyne detector(BHD) or alternatively sent and mode-matched to the optical mode of an interferometerbeforehand LO local oscillator DBS dichroic beam splitter

49

Fig 20 (a) and (b) show photographs of typical nonlinear crystals used forsqueezed-light generation at near infra-red wavelengths The crystals shownhere form a monolithic standing-wave squeezing resonator (a) or are partof a half-monolithic standing-wave squeezing cavity (c) shows a tempera-ture stabilized and mechanically stable housing of the squeezing resonator(d) shows a schematic of a full setup for the generation of squeezed vacuumstates of light for an application in a laser interferometer The only brightinput required for the squeezing resonator (OPA) is the second-harmonicpump field The resonator mode at fundamental frequency is thus initiallynot excited by photons ie it is in its ground state characterized by vacuumfluctuations due to the zero point energy see Fig 7 (c) [Gerry and Knight(2005)] The pump field spontaneously decays in the degenerate pair of sig-nal and idler fields The combined down-converted field leaving the resonatorexhibits quantum correlations which give rise to a squeezed photon countingnoise when overlapped with a bright coherent local oscillator beam Thedetection is done either in a balanced homodyne detector (BHD) or with asingle photo diode The squeeze factor increases the closer the pump powerof the squeezing resonator gets to the oscillation threshold and the lower theoptical loss on down-converted photon pairs is

441 High squeeze factors ndash minimizing decoherence

Squeezed states of light have significant impact on the sensitivity of laserinterferometers if large squeeze factors can be produced Squeezing of 3 dBimproves the signal-normalized quantum-noise spectral density by a factorof 2 This factor corresponds to doubling the (coherent state) light powercirculating inside the interferometer Squeezing of 10 dB corresponds to aten-fold power increase The experimentally demonstrated squeeze factorswere considerably improved in recent years [Takeno et al (2007) Vahlbruchet al (2008) Polzik (2008) Eberle et al (2010) Stefszky et al (2012)]culminating in a value of as large as 150 dB [Vahlbruch et al (2016)] Thisvalue corresponds to the same reduction of signal-normalized quantum noisethat is achieved by increasing the light power by a factor of 32 (At this pointit is already noted that squeezing the quantum noise can simultaneouslyreduce quantum measurement noise (shot noise) as well as quantum backaction noise (radiation pressure noise) This is not possible with scaling thelight power of coherent states see Subsec 55)

50

Ideally a parametric squeezed-light source can produce an infinite squeez-ing level see Fig 19 fundamentally just limited by the energy provided bythe pump field In practice the limit is set by decoherence mechanismsThe by far most important one is optical loss Optical loss occurs duringsqueezed-light generation its propagation through the interferometric setupincluding imperfect mode matchings and finally the photo-electric detec-tion Also detector dark noise [Schneider et al (1998)] phase noise [Takenoet al (2007)] and excess noise [Bowen et al (2002)] impair the observablesqueezing strength

Optical loss is usually understood as coupling the squeezed mode to a zerotemperature bath ie overlapping it with a vacuum mode For any amountof loss the resulting state is still squeezed But to be able to directly observesay 10 dB of squeezing the total loss on the state needs to be less than 10in this example cf Eq (16) To minimize optical loss the nonlinear crys-tal as well as lenses and beam splitters in the interferometric path need toshow very low absorption and scattering at the wavelength of the squeezedlight PPKTP shows absorption of about 10minus4cm and below at near-infraredwavelengths Low OH content fused silica is a suitable material for all otheroptics Absorptions of less than 10minus6cm were measured [Hild (2007)] Coat-ings on crystal surfaces and on all other optical components should also showlowest optical loss Total loss of the 10minus6 level are available today Superpol-ished surfaces which show roughnesses with less than 1 A root mean square(integrated over spatial scales from approximately 1 micron to 100 microns)and thus very low scattering are necessary to achieve these low numbersMinimizing the total number of optical components is essential From thisperspective a monolithic squeezing resonator as shown in Fig 20 (a) is theoptimum choice The squeezed mode needs to be matched to the mode ofthe laser interferometer or to the mode of the balanced homodyne detectorVisibilities of up to 998 have been achieved [Eberle et al (2010)] whichcorresponds to a loss of about 04 Of great importance also is the quantumefficiency of the photo-diodes used for detecting the squeezed field (togetherwith the interferometric signal) Recently a quantum efficiency of photo-diodes in a squeezing experiment of (995plusmn 05) was measured [Vahlbruchet al (2016)] To minimize photon loss the photo-diodes had no protectionwindow an anti-reflection coating on the semi-conductor material and theremaining reflection was re-focussed with an external mirror

Also the dark-noise spectral density of the detection electronics reducesthe observable squeezing and needs to be as low as possible Similar to optical

51

noise it also provides a contribution to the observed variance The dark noiseof the detection electronics needs to be much lower than the detected photoncounting noise In [Vahlbruch et al (2016)] it was 28 dB below shot noisebut still reduced the observable squeeze factor from 153 dB to 150 dB

Excess noise emerges if the squeezed mode couples to a nonzero tem-perature bath or to a mode whose excitation is strongly fluctuating (Thecoupling process can always be understood as a beam splitter coupling andis physically described by overlapping electric fields Coupling to a zero tem-perature bath leads to Eq (16)) The captured excess noise variance thenneeds to be added to the initial squeezing variance which deteriorates theobserved squeezing stronger than just mixing in the vacuum mode Excessnoise is less likely to occur at MHz frequencies but can be significant ataudio-band sideband frequencies and below and is thus a serious issue ingravitational-wave detectors [Chua et al (2014)] The reason for that isthat acoustically or thermally excited motions of surfaces produce frequencyshifts of back-scattered light mainly at these low frequencies [Vahlbruch et al(2007)]

Phase noise corresponds to stochastic phase fluctuations between thesqueezed field and the local oscillator within the measuring time It cor-responds to mixing the squeezed mode with itself with a fluctuating squeezeangle [Suzuki et al (2006) Franzen et al (2006)] Phase noise in squeez-ing experiments typically is less of an issue than optical loss [Dwyer et al(2013) Oelker et al (2016) Vahlbruch et al (2016)] The setuprsquos phasenoise can be reduced by making the squeezing resonator more compact andthus mechanically more stable against acoustic and thermal fluctuations ofthe environment and by improving the quality of the servo loops for cavitylength and propagation length controls Operating a squeezed-light resonatorin vacuum might also be beneficial The ability to run a high performancesqueezed-light generator in vacuum was demonstrated in [Wade et al (2015)]

442 Squeezing in the gravitational-wave (GW) detection band

High squeeze factors have been first demonstrated at sideband frequen-cies of a few MHz and above where excess noise is generally negligible whenworking with visible or near-infra-red light Today we know that extendingthe squeezing spectrum towards the audio-band and even below is technicallynot always easy but straight forward once a high squeeze factor is achieved

52

at MHz frequencies In most squeezing experiments the main laser light

Figure 21 Photograph of the GEO 600 squeezed-light source ndash The breadboarddimensions are 135 cm times 113 cm The squeezing resonator is high-lighted by the white ar-row and is set up as a standing-wave hemilithic cavity containing a plano-convex PPKTPcrystal of about 10 mm length (see also Fig 20 b) It is pumped with continuous-wave532 nm light that is produced by frequency doubling of light (at angular frequency ω)from a commercial NdYAG laser Two more laser fields at about 1064 nm having fre-quency offsets of more than 10 MHz with respect to ω(2π) serve as optical control fieldsBoth fields are mode-matched and injected into the squeezing resonator together with thesecond-harmonic pump field

at the squeezing wavelength is accompanied by significant noise up to thelaser relaxation oscillation For this reason laser control fields at the opticalcarrier-frequency in the optical path of the squeezed mode need to be avoided[Bowen et al (2002) Schnabel et al (2004) McKenzie et al (2004)] and thesqueezing resonator length and the optical path stabilized by other means[McKenzie et al (2005) Vahlbruch et al (2006)] Furthermore and mostimportantly excess noise due to back-scattering is an issue Back-scattering(also called lsquoparasitic interferencesrsquo) is produced if DC light scatters out ofthe optical path hits a vibrating surface and re-scatters back into the opticalpath [Vahlbruch et al (2007)] Significant back-scattering can be produced

53

in interferometers for the detection of gravitational waves since light powersof several hundreds of kilowatts are used Even back-scattering from the mil-liwatt local oscillator of balanced homodyne detectors is an issue at acousticsideband frequencies and below The recipe for avoiding parasitic interfer-ences turns out to be threefold (i) avoiding scattering by using ultra-cleansuperpolished optics with close to perfect anti-reflex coatings (ii) avoidingback-scattering by carefully blocking all residual (faint) light fields and (iii)reduce the vibrationally and thermally excited motion of all mechanical andoptical parts that could potentially act as a re-scattering surface with re-spect to the optical path [Vahlbruch et al (2007) McKenzie et al (2007)]The insights described above led to the first demonstration of audio-bandsqueezing at frequencies down to 200 Hz [McKenzie et al (2004)] and laterto the first demonstration of squeezing over the full gravitational-wave de-tection band even from as low as 1 Hz [Vahlbruch et al (2007)] Whilea standing-wave squeezing resonator [Ou et al (1992)] can be built in avery compact way that is rather insensitive against mechanical vibrations[Chelkowski et al (2007)] a traveling-wave bow-tie squeezing resonator [Wuet al (1986)] is more tolerant against back-scattered light [Chua et al (2011)]The strongest squeezing in the audio-band of up to 116 dB was reported inRef [Stefszky et al (2012)]

443 The first squeezed-light source for GW detection

The first squeezed-light source for the continuous operation in GW de-tectors had been designed and completed between 2008 and 2010 [Vahlbruch(2008) Vahlbruch et al (2010)] Since then this source has been producingsqueezed vacuum states in a fully phase controlled way using co-propagatingfrequency-shifted bright control beams [Vahlbruch et al (2006)] as an inte-gral part of the GW detector GEO 600 The source is a turn-key device witha fully automated re-lock system [Vahlbruch et al (2010) Khalaidovski et al(2012)] Re-locking is required if the temperature of the environment changessignificantly which drives the actuators outside their dynamic ranges

Up to 9 dB of squeezing over the entire GW detection band was observedusing a balanced homodyne detector (BHD) located close to the squeezingresonator The squeeze factor has been limited by optical loss due to ab-sorption in the PPKTP crystal transmission of the back-surface and thenon-perfect AR-coating of the crystalsrsquos intra-cavity surface The adjustableair gap has been necessary to allow for an easy way to apply length control

54

-12

-8

-4

0

4

8

12

16

20

10 100 1k 10k

Rel

ativ

e no

ise

pow

er [

dB]

Frequency [Hz]

shot noise (a)

squeezed noise (b)

anti-squeezed noise (c)

~ 9dB

Figure 22 Broadband squeezing spectrum ndash Noise power spectra measured on theoutput of the GEO 600 squeezed-light source shown in Fig 21 with a balanced homodynedetector The traces correspond to the spectra of quadrature amplitude variances normal-ized to vacuum noise The resolution bandwidth used increases towards higher frequenciesto reduce the measurement time (a) Shot noise normalized to unity which serves as thereference level (0 dB) (b) Squeezed noise covering the complete detection band of ground-based GW detectors (c) Anti-squeezed noise Peaks at 50 Hz and 100 Hz are the electricmains frequency and its first harmonic The data was published in Ref [Vahlbruch et al(2010)]

Additional optical loss in the path to the balanced homodyne detector mainlyarose due to a Faraday isolator that eliminated parasitic interferences Fi-nally the mode missmatch to the BHD as well as its non-perfect quantumefficiency provided additional loss Inferring the squeeze factor without theBHD detection loss more than 10 dB of squeezing are provided by the sourceSince 2010 it has been used in basically all observational runs of the GEO 600GW detector see Section 6

444 Generation of two-mode (bi-partite) squeezing

lsquoTwo-mode squeezed lightrsquo or lsquobi-partite squeezed lightrsquo is light that allowsfor joint measurements at two locations A and B These joint quadraturemeasurements reveal correlations and anti-correlations with a remaining un-certainty smaller than the ground-state uncertainty which certifies the pres-

55

ence of entanglement cf Subsec 34 Bi-partite squeezed light has been gen-erated by type I and by type II parametric down-conversion In case of type Ithe squeezed fields from two squeezing resonators as described in Subsec 44are overlapped on a balanced beam splitter with a 90 phase shift The twooutput fields together represent the entangled mode [Furusawa et al (1998)Bowen et al (2003c) Eberle et al (2013)] In case of type II signal and idlerfields are non-degenerate regarding polarisation and a single cavity contain-ing an appropriate crystal and a polarising beam splitter are sufficient for theproduction of bi-partite squeezing Also in this case the measurements ofthe quadrature amplitudes of signal and idler fields show large uncertaintiestogether with bi-partite correlations and anti-correlations that are strongerthan the ground state uncertainty of individual subsystems [Ou et al (1992)Villar et al (2006) Jing et al (2006)]

To date the strongest entanglement of bi-partite squeezed light has beenproduced based on type I parametric down-conversion [Eberle et al (2013)]The requirements of producing strong entanglement are identical to those ofproducing strong squeezing outlined above The strength of bi-partite en-tanglement can be given in decibels in full analogy to the squeeze factorPractically the strength of bi-partite squeezing is always somewhat smallerthan that of single party squeezing since it requires an additional mode-matching that results in additional optical loss

45 Conclusions

The first observation of squeezed light was achieved in 1985 [Slusher et al(1985)] Shortly after cavity-enhanced optical parametric amplification forsqueezed-light generation was demonstrated [Wu et al (1986)] which todayenables the observation of up to 15 dB of squeezing [Vahlbruch et al (2016)]Quite generally the maximum squeezing level that is observed does not de-pend on the strength of the optical nonlinearity Squeezing cavities can easilybe operated at their oscillation threshold where they should provide infinitesqueezing if decoherence is zero The main limiting factor is optical lossincluding that of the photo-electric detection

Dedicated experimental research and development towards a squeezed-light source for applications in gravitational-wave detectors can be tracedback to 2002 [McKenzie et al (2002) Bowen et al (2002)] Since then a sur-prising amount of progress has been made culminating in the first squeezed-light source specifically built for the integration into a gravitational-wave

56

detector For the future squeeze factors above 15 dB will certainly be possi-ble by further reducing optical loss This statement is supported by the highdegree of matching between experimental data and a theoretical loss modelas presented in Fig 3 of [Vahlbruch et al (2016)]

5 Quantum noise in laser interferometers

51 Interferometric measurements

The purpose of a laser interferometer is the precise measurement of smallchanges of an optical path length with respect to a reference path Forthis the interferometer transfers the change of the phase difference betweentwo light fields into an amplitude quadrature change of the interferometerrsquosoutput light The latter can easily be detected by a single photo diode Ofgeneral interest are differential changes of the optical path length that aremuch smaller than the laser wavelength ie differential phase changes thatare much smaller than 2π In this case the differential phase change can bedescribed in very good approximation as a differential change of the phasequadrature instead

In order to transfer the phase quadrature signal with minimum loss ahigh interference contrast at the interferometerrsquos beam splitter is essentialAdditionally instrumental noise in terms of unwanted excitations of the out-putrsquos amplitude quadrature needs to be reduced to a minimum Noise arisesdue to power fluctuations of the input laser light back-scattered laser lightinside the interferometer thermally driven displacements of mirror surfacesand in many more ways The important measure of the sensitivity of aninterferometer obviously is its signal-to-noise-ratio The most useful measureis given in terms of the noise spectral density S(f=Ω2π) that is normalizedto the physical unit of the signal S(f) is in fact a lsquonoise-to-signal-ratiorsquoand can be seen as the signal-normalized variance of the photo diode out-put decomposed into spectral components versus sideband frequency f withthe resolution bandwidth of 1 Hz As an example S(100 Hz) = 10minus39 m2Hzmeans that the instrumental noise in the one hertz band around 100 Hz equalsa signal that would be produced if the mirror of one interferometer arm os-cillates with an amplitude of just

radic10 middot 10minus20 m in the very same band Such

small spectral densities are achieved by gravitational-wave detectors [Abbott(2016)]

57

52 Quantum measurement noise and shot noise

The most fundamental noise source in laser interferometers is due to thequantum noise of light which is in fact two-fold [Caves et al (1980)] Firstof all there is lsquoquantum measurement noisersquo which arises in the processof photo-electric detection For coherent states the quantum measurementnoise is the lsquophoton counting noisersquo of mutually independent photons andusually simply called lsquoshot-noisersquo Fig 2 (b i) shows a time series of suchnoise hiding the actual signal The frequency components of the shot noiseare well described by the quantum uncertainty of the output fieldrsquos amplitudequadratures XΩ∆Ω see Subsec 22 (Recall this quantity corresponds to the

differential phase quadrature YΩ∆Ω of the light beams in the interferometerarms) The photon counting noise has a white Fourier spectrum howeverthe lsquoshot noisersquo of an interferometer is usually normalized to the signal whosetransfer function is usually not white for instance due to the presence of armcavities or a signal-recycling cavity

All current and planned gravitational-wave detectors are Michelson-typelaser interferometers with operating points very close to a dark fringe Thelight power in the output port is just a couple of tens of mW which canbe handled by a single photo diode In this configuration the signal-to-shot-noise-ratio is actually maximized which can be shown in three steps [Bachorand Ralph (2004)] For the first step we use plane waves to describe theelectric field in the output port of a Michelson interferometer For perfectinterference contrast at the balanced beam splitter ie for perfect modematching and for defining φ = 0 as the dark port condition we get

Eout(t φ) =1

2E0 sin(ωt+ φ)minus 1

2E0 sin(ωt) (31)

where E0 is the amplitude of the total internal field whose two parts hasaccumulated a differential phase It directly follows for the squared fields

E2out(t φ) =

(sin

φ

2

)2

(E0 cos(ωt+ φ2))2 (32)

We now turn to a light beam with a localized transversal mode that can befocussed onto a photo-electric detector The photo diode has perfect quantumefficiency ie the rate of photo electrons is not only proportional to the rateof output field photons but also has a unity slope efficiency Since the optical

58

frequency is too high to be resolved we consider the averaged light power

P out(φ) =

(sin

φ

2

)2

P (33)

The next step is a formulation of the signal being the derivative of detectedphoton number versus phase Let n be the average value of the photonnumber per measuring time interval Eq (33) can then be rewritten as

nout(φ) =

(sin

φ

2

)2

n (34)

rArr dnout(φ) = n sinφ

2cos

φ

2dφ (35)

The final step is the calculation of the signal-to-shot-noise-ratio Shot noiserefers to coherent states which have a standard deviation of the photonnumber of σ(n) =

radicn

dnout(φ)

σ(nout)=n sinφ

2cosφ

2dφ

radicn sinφ

2

(36)

and find for a signal-to-noise ratio of unity for coherent states and for anon-zero but still small phase difference ∆φCoh 2π

1 =radicn cos

φ

2∆φCoh with φ 6= 0 (37)

In this equation the smallest measurable phase difference is given for φrarr 0

∆φCohmin =

1radicn (38)

This is the well-known shot-noise limit of high-precision phase sensing ∆φCohmin

is the smallest phase shift that can be measured with a signal-to-noise ra-tio of one when using n mutually independent photons per measuring time(those of a coherent state) when the loss of photons is assumed to be zeroThe typical purpose of a laser interferometer is the continuous sensing (moni-toring) of a continuously changing phase An illustrative example is the phasesignal produced by the black hole merger measured by Advanced LIGO onSept 14 in 2015 (Fig 1 in [Abbott (2016)]) The measuring interval should

59

be short to be able to resolve the time-evolution of the signal Generally themeasurement of an arbitrary signal that lasts for a finite time thus needs tobe understood as l subsequent measurement intervals using n photons eachIt can be shown that Eq (38) is also valid for interferometers operated athalf fringe ie when each output port contains the same light power In thiscase photo diodes need to be placed in both output ports and the actualsignal is provided by their difference voltage

Due to its importance the shot-noise limit deserves some remarksThe phase φ in Eq (31) is the phase difference of two mode-matched fieldsand might be accumulated by a single pass along the length L such as ina Mach-Zehnder interferometer or in a double pass such as in a (simple)Michelson interferometer or in four passes as realized in a Michelson inter-ferometer with folded arms [Grote (2005)] The shot-noise limit in Eq (38)and its scaling therefore holds independent of the number of passes Theclaim in Ref [Higgins et al (2007)] that the scaling according to Eq (38) canbe surpassed by multiple passes is not justifiedThe fact that Eq (38) is derived by approaching φ rarr 0 correctly describesthe actual operation point of gravitational-wave detectors which is close tobut not exactly at a dark port In practice a tiny offset from dark port ischosen at which the shot noise is well above the photo diodersquos electronic darknoiseEq (38) solely depends on the number of quanta but not on the lightrsquos wave-length λ Of course the shot-noise limit for the change of an optical pathlength ∆L does depend on the wavelength and ∆φCoh

min needs to be replacedby ∆φCoh

min = 2π∆LCohminλ

Finally an essential result of the shot-noise limit is that the ideal precisemeasurement should use lsquoas much quanta as possible per measuring inter-valrsquo which translates to lsquoas much light power in the interferometer armsas possiblersquo Eq (38) is indeed the one and only reason why gravitational-wave detectors use high power lasers power-recycling and arm resonatorsExtending the measuring time for a given light power can also improve thesensitivity but only if the signal repeats ie is periodic Let us assume thatone period of the signal is resolved by l intervals using n photons each Inthis case repeating the overall measurement k times improves Eq (38) by1radick The fundamental statement of Eq (38) however does not change

since the actual photon number n may then simply incorporate the factor k

60

For a given average photon number the shot-noise limit in Eq (38) canonly be surpassed by using photons that are quantum correlated ie by usingnonclassical states of light How is the shot-noise limit surpassed with thehelp of squeezed states A Michelson interferometer that is operated closeto a dark fringe acts like an almost perfect mirror for both input ports Allthe input light is back-reflected towards the laser source This also accountsfor the quantum uncertainty of the input light The quantum uncertaintythat impinges onto the photo diode thus (mainly) enters the interferometerthrough its (almost) dark port An interferometer that uses displaced co-herent states entering from one port can thus be improved by replacing theordinary vacuum entering the signal output port by a squeezed vacuum stateThis was the proposal by CM Caves in 1981 [Caves (1981)] which is labeledhere with lsquoCSVrsquo If the differential phase quadrature of the interferometer issqueezed Eq (38) then within the limit of large coherent state displacementα sinh2r improves to

∆φCSVmin asymp

eminusrradicn (39)

(The above expression is an approximation since the squeezing operationproduces a small number of photons that are not accounted for here) Ofcourse the mode of the squeezed vacuum needs to be precisely matchedto the mode of the interferometer The first experimental demonstrations ofsqueezed phase measurements used a Mach-Zehnder [Xiao et al (1987)] and apolarization interferometer [Grangier et al (1987)] Fig 2 shows how spatialdegeneracy between an externally generated squeezed mode and the signalmode in a Michelson interferometer is achieved using a polarizing beam split-ter and a Faraday rotator Again the limit in Eq (39) can only be achievedif optical loss is zero Optical loss not only reduces the signal but here alsoreduces the squeeze parameter see Eqs (15) and (16)

Let us consider an example The sensitivity of a laser interferometer thatuses coherent states with an excitation of 1023 photons per second can beimproved by a factor of

radic10 by either adding 09 middot 1024 photons per second

or by adding about just 2 photons per second and bandwidth in hertz thatbelong to the 10 dB squeezed vacuum confer Eq (30) Since the full sig-nal band of ground-based GW detectors covers sideband frequencies up to10 kHz just 2 middot 104 photons per second are necessary At a wavelength ofλ = 1064 nm these values correspond to a power increase by 168 kW and

61

37 fW respectively

The question arises whether a scaling of the sensitivity better than propradic1n is possible It was theoretically shown that in principle the scaling

can indeed considerably be improved yielding the so-called Heisenberg limitor Heisenberg scaling [Bondurant and Shapiro (1984) Yurke et al (1986)Braunstein (1992) Holland and Burnett (1993)]

∆φHLmin prop

1

n (40)

The Heisenberg scaling requires nonclassical states that have a certain num-ber of quanta similar to Fock states ie n = n The theoretically optimalstates describe a superposition of n (N) indistinguishable photons in oneinterferometer arm while having zero (0) photons in the second arm and viceversa and were named lsquoN00Nrsquo-states [Dowling (2008)] A specific propertyof these states is lsquosuper-resolutionrsquo The output ports of the interferometershow an n-times faster oscillation of the interference fringes when changingthe phase between the two interferometer arms Super-resolution correspondsto an n-times improved signal transfer function and was demonstrated forinstance in Refs [Rarity et al (1990) Kuzmich and Mandel (1998) Mitchellet al (2004) Afek et al (2010)] The presence of this nonclassical phe-nomenon however does not prove a sensitivity better than the semi-classicalbound according to Eq (38) Sensitivity is rather related to the signal-to-noise-ratio and needs to take into account all imperfections as well as theprobability of a successful detection of the sensing state [Thomas-Peter et al(2011)] All experiments so far used post-selection on particular measure-ment outcomes and neglected the typically large probability that nothingwas detected

Super-resolution was demonstrated with up to n = 5 [Afek et al (2010)]In addition to the fact that super-resolution does not prove a sensitivity bet-ter than the semi-classical bound photon numbers in state-of-the-art super-resolution experiments are extremely small compared to the photon numberof about 1023 (within a measuring interval of one second) of coherent statesused in Ref [Abbott (2016)] and of about 1022 using coherent states plussqueezed vacuum states used in Ref [Abadie (2011)]( which did prove a sen-sitivity better than the semi-classical bound)

Another interesting and related question is what the smallest phase is

62

that can be estimated in a single measurement again using a given numberof quanta Taking into account that no prior information about the phaseshift exists still a scaling proportional to 1n is possible In the limit of largen however an additional factor of π is required in the nominator of Eq (40)[Sanders and Milburn (1995) Berry and Wiseman (2000)] yielding

∆φHLmin asymp πn (41)

The state that can actually achieve this bound is different from the N00Nstate and was found in [Summy and Pegg (1990) Luis and Perina (1996)Berry and Wiseman (2000)] A N00N state is not the optimum state forphase estimation (via a single measurement) since it only provides one bit ofinformation A recent review on generell aspects on phase measurements isgiven by Ref [Demkowicz-Dobrzanski et al (2015)]

It is important to note that Eqs (38-40) do not consider photon lossExperiments that demonstrated super-resolution and aimed for achievingthe scaling in Eq (40) were conditioned on zero photon loss Let η gt 0 bethe average efficiency of detecting (all) photons Eq (38) then reads

∆φηCohmin =

radic1

η n (42)

Eq (39) turns into

∆φηCSVmin asymp

radicηeminus2r + 1minus η

η n(43)

and Eq (40) turns into [Demkowicz-Dobrzanski et al (2012 2013)]

∆φηmin =

radic1minus ηη n

for 0 lt η lt 1 (44)

For non-zero photon loss most interestingly the ultimate sensitivity of aphase measurement for a given photon number also shows a 1

radicn -scaling

The difference between the CSV strategy of using bright coherent states incombination with squeezed vacuum states which is bounded by Eq (43) andthe strategy of using the optimal nonclassical state which is bounded byEq (44) is marginal in practice [Demkowicz-Dobrzanski et al (2013)] Forgravitational-wave detectors and for any other laser interferometer using in-tense light there is no need for an alternative to the CSV strategy

63

We now turn back to the shot noise according to Eq (38) Generally noisecan be decomposed into its spectral contributions For a simple Michelsoninterferometer without arm resonators and without a signal-recycling cavitythe square-root of the single-sided shot-noise spectral density normalized tothe differential arm length change x in units of m

radicHz is given by [Saulson

(1994)] radicSMI

SNx =

radic~c2

2ωPprop 1radic

P (45)

where ω is the optical angular frequency of the quasi-monochromatic carrierlight and P the total light power in both arms including the built-ups fromcavities In combination with a squeezed vacuum whose relative phase gen-erates squeezing of the output lightrsquos amplitude quadrature the right handside reduces according to the factor eminusr Note that the single-sided spectraldensity is only defined for positive sideband frequencies and thus twice aslarge as the double-sided spectral density

The spectral density of the measurement of a GW induced strain is givenby the same expression but normalized to h = xL (If the gravitational waveis oriented in an optimal way with respect to the Michelson interferometerone arm is squeezed while the other is expanded by the same amount of ∆L =x2 and h then corresponds to the actual gravitational-wave amplitude)The square-root of the single-sided shot-noise spectral density normalized tostrain in units 1

radicHz is given by

radicSMI

SNh =

radic~c2

2L2ωP (46)

Equations (45) and (46) show that the smallest measurable signal (corre-sponding to unity signal-to-shot-noisendashratio) is inversely proportional to thesquare root of the laser power and has a white spectrum for sideband frequen-cies much smaller than the carrier frequency see horizontal line in Fig 23

All first- and second-generation GW detectors use power-recycling andadditional cavities to improve their sensitivities Fabry-Perot arm resonatorsdo not only increase the light power but additionally also increase the signalfor signal frequencies inside the resonator linewidth For lossless Fabry-Perot

64

arm resonators the spectral densities in Equations (45) and (46) need to bemultiplied by the following factor [Kimble et al (2001)]

HFP =

radicL2(γ2

FP + Ω2)

c2 (47)

where γFP = cTFP(4L) is the Fabry-Perot arm resonatorrsquos half bandwidthand TFP is the light power transmission of the input mirror The end mirrorsare assumed to have perfect reflectivity A similar expression can be derivedfor describing the improvement due to signal-recycling [Buonanno and Chen(2001)]

In summary shorter laser wavelengths higher light powers and squeezingof the amplitude quadrature of the interferometer output reduce shot noise ina broadband way ie for all signal frequencies Fabry-Perot arm resonatorsas well as signal-recycling provide improvements mainly for frequencies insidethe resonator linewidths

53 Quantum back-action and quantum radiation pressure noise

In laser interferometers quantum back-action noise results from the un-certainty of the lightrsquos radiation pressure force on the interferometer mirrorsand is also called lsquo(quantum) radiation pressure noisersquo (RPN) Its origin isthe quantum uncertainty of the differential amplitude quadrature XΩ∆Ω ofthe fields in the interferometer arms It results in an uncertain momentumtransfer to the mirrors and thus in an position uncertainty of the mirrors atfuture times with respect to their differential mode of motion [Caves et al(1980)] The physical mechanism of radiation pressure corresponds to anintensity dependent phase shift [Pace et al (1993)]The higher the light power in the arms of a laser interferometer the loweris its shot-noise spectral density see Eq (45) Unfortunately the spectraldensity of quantum back-action noise increases with light power The single-sided force noise spectral density reads

radicSRPNF =

radic8~ωPc2

(48)

Whereas the force noise of the quantum radiation pressure has a white spec-trum the RPN does not since the mirrorrsquos reaction to external periodicforces depends on frequency The link between the Fourier component of an

65

external force F (Ω) and the Fourier component of the displacement x(Ω) isgiven be the mechanical susceptibility HM It reads for an harmonic oscillatorwith mass M

HM(Ω) =1

M | minus Ω2 + Ω2M + iΩΩMQ|

(49)

where ΩM is the oscillatorrsquos resonance frequency and Q its quality factorThe square root of the single-sided spectral density of the RPN normalized

to the displacement of an harmonic oscillator with mass M is then given by

radicSRPNx = HM(Ω)

radic8~ωPc2

(50)

In GW detectors the test mass mirrors are suspended as pendula with highmechanical Q-factors and their centre of mass motion corresponds to that ofa harmonic oscillator The resonance frequencies of the pendula are lowerthan the detection band of interest The mechanical susceptibility is thereforeoften approximated for the so-called free-mass regime as H fm

M (Ω) = (mΩ2)minus1The square root of the single-sided spectral density of the RPN normalizedto differential displacement of two mirrors with each of mass M in a simpleMichelson interferometer is given by [Saulson (1994)]radic

SfmMIRPNx =

radic2~ωPc2m2Ω4

propradicP (Ω ΩM) (51)

where m = M2 is the mirrorsrsquo reduced mass In case of a simple Michel-son interferometer that is enhanced with arm cavities the spectral density inEq (51) needs to be multiplied with the expression given in Eq (47) In com-bination with a squeezed vacuum whose relative phase generates squeezing ofthe output lightrsquos phase quadrature the right hand side reduces according tothe factor eminusr Note if the radiation pressure noise is squeezed the shot noisemust be anti-squeezed or vice versa The radiation pressure noise calibratedto strain of space time is given by the right side of Eq (51) divided by theinterferometer arm length LIn summary heavier masses longer laser wavelengths lower light powersand squeezing of the amplitude quadrature in the interferometer arms re-duce radiation pressure noise

66

1 10 100 1000Frequency [Hz]

Radiation pressure noise

Shot noise

SQL

10-21

10-18

10-15

10-12

radicSx

[mradic

Hz]

ndashndash

ndashndash

Figure 23 Displacement-normalized quantum noise spectral densities ndash Consid-ered is a simple Michelson interferometer with neither arm cavities nor signal recyclingThe two end mirrors (m = 100 g) of the interferometer arms are suspended as pendulahaving a resonance frequency of ΩM2π = 1 Hz and a Q-factor of 107 The interferometeruses quasi-monochromatic light (in coherent states) with a total power of 4 kW Opticalloss and the offset from a dark output fringe is assumed to be negligible Wavelengthλ = 1550 nm The standard quantum limit (SQL) corresponds to the lowest noise achiev-able at a given sideband frequency when varying the light power without using quantumcorrelations

54 Interferometer total quantum noise and the standard quantum limit

Both shot noise and radiation pressure noise contribute to the total quan-tum noise of a given interferometer If they are not quantum correlatedwhich is the case for a conventional Michelson interferometer when detectingthe output lightrsquos amplitude quadrature their variances add up (The resultis not shown in Fig 23) It can easily be deduced from the previous sectionsthat changing the laser power will shift the two quantum noise contributionsHowever the total quantum noise never goes below the standard quantumlimit (SQL) [Braginsky and Manukin (1967)]

Let us consider Fig 23 for sideband frequencies much greater than thependulum resonance Here the test mass mirrors react as free masses whenexerted to external forces The SQL in this free-mass regime is calculated

67

by minimizing the sum of the squares of Eqs (45) and (51) [Saulson (1994)]

Sfmtotx =

~c2

2ω

[1

P+

4ω2

c4m2Ω4P

] (52)

Its derivative reads

dSfmtotx

dP=minus1

P 2+

4ω2

c4m2Ω4 (53)

Setting the above equation to zero provides the optimum laser power versussideband frequency in order to achieve the lowest total quantum noise

P fmopt =

c2mΩ2

2ω (54)

Inserting the optimal light power into Eq (57) provides the square root ofthe single-sided noise spectral density of the free-mass SQL in m

radicHzradic

SfmSQLx =

radic2~mΩ2

(55)

Again m is the reduced mass and dividing by the interferometer arm lengthL yields normalization to the GW-induced strain h Eq (55) shows that theSQL falls off with sideband frequency The corresponding equation for aMichelson interferometer that uses arm cavities readsradic

SfmFPSQLx =

radic~

mΩ2

(1

HFP

+HFP

) (56)

with HFP according to Eq (47)Using the expression for the SQL the square root of the total quantum

noise spectral density of a Michelson interferometer in the free-mass approx-imation can be written asradic

SfmFPtotx =

radicSfmFP

SQLx

2

[1

k+ k

] (57)

with the radiation pressure coupling parameter

k(Ω) =2ωP

mc2Ω2 (58)

68

For a fixed light power and fixed reduced mass of the mirrors the quantumnoise limited interferometer reaches the SQL when k = 1 which is realizedat the angular sideband frequency ΩSQL =

radic2ωP(mc2)

Note that neither squeezing the phase quadrature nor squeezing the am-plitude quadrature of the interferometer light leads to sub-SQL performance[Caves (1981)] also confer Ref [Schnabel (2005)] Fig 3 (left) As we willsee in the next sections the standard quantum limit can be surpassed if shotand radiation pressure noise are correlated Then the total quantum noise isnot given by the sum of the variances ie the sum of the squares in Eqs (45)and (51)

55 Squeezed light for surpassing the standard quantum limit

A measurement with sensitivity better than the standard quantum limit(SQL) is also called a lsquoquantum non-demolition (QND)rsquo measurement [Bra-ginsky and Khalili (1995 1996) Kimble et al (2001)] Several QND tech-niques for laser interferometers were proposed in recent decades [Jaekel andReynaud (1990) Kimble et al (2001) Purdue and Chen (2002) Chen (2003)McClelland et al (2011) Danilishin and Khalili (2012) Graf et al (2014)]What they have all in common is they exploit quantum correlations betweenobservable uncertainties

Arguably the most extensive way of introducing quantum correlationsand surpassing the SQL is the injection of squeezed states of light [Jaekeland Reynaud (1990)] If the squeezed quadrature angle of the injected statesis neither 0 nor 90 the quantum uncertainties of the amplitude and phasequadrature amplitudes that describe the differential field in the two interfe-rometer arms become correlatedLet us consider a very simplified setup that just consists of a quasi-mono-chromatic light field that is back-reflected from a quasi-free mirror Thelight power and the mass be such that reflected light in a coherent stateresults in a measurement of the mirror position with a noise spectral den-sity at the SQL at sideband angular frequency ΩSQL At this frequencyquantum measurement noise and back-action noise are of the same sizeie the uncertainty in X produces an equally large additional uncertaintyin Y Upon reflection the quadrature amplitude variances change from∆2X = ∆2Y = 14 to 2∆2X = ∆2Y = 12 This result corresponds tothe situation in Fig 23 at the crossing frequency of shot noise and radiation

69

pressure noise The coupling of the uncertainty variances can be describedby the matrix K = (1 minusk 0 1) where k = 1 at the SQL If the modulationstate at ΩSQL is the ground state its variances are transferred according to

KT

(1 00 1

)K =

(1 0minus1 1

)(1 00 1

)(1 minus10 1

)=

(1 minus1minus1 2

) (59)

In accordance with Fig 23 the variance of YΩSQLis twice as large as the vac-

uum noise varianceNow let the quantum noise of the light field be 10 dB squeezed at 45

(Eq (19)) The projection of the quantum uncertainty onto the X-observableproduces the radiation pressure noise by being transferred with the couplingfactor k = 1 at the SQL into the Y -observable in fact with negative signsince a larger value of X produces a larger optical path length and thus aretardation of the phase Due to the squeezing at 45 the initial uncertaintyin Y cancels with the additional uncertainty that originates from the onein X The following calculation shows that the strength of the cancellationcorresponds to the initial squeezing strength Upon reflection the quantumuncertainties transform in the following way(

1 0minus1 1

)(505 495495 505

)(1 minus10 1

)=

(505 minus01minus01 02

) (60)

The state of light after reflection has a squeezed phase quadrature amplitudeThe improvement in comparison to Eq (59) is exactly 10 dB The quantumnoise improvement corresponds to the input squeeze factor and is also a mea-sure by what factor the SQL is surpassed Squeezed vacuum injection thusallows surpassing the SQL upon measuring the conventional Y -quadrature(which is realized by a single photo diode in the interferometerrsquos output port)as first realized by [Unruh (1983) Yuen (1983) Jaekel and Reynaud (1990)]

In the example above the input squeeze angle is optimized for a sin-gle sideband frequency Injecting a broadband squeezed vacuum field withfrequency-independent squeeze angle of 45 would result in a rather bad inter-ferometer quantum-noise performance at frequencies far smaller or largerthan ΩSQL Fig 24 shows the quantum-noise performance if the input fieldhas squeeze angles that are optimized for every k(Ω) as given in Eq (58)Injected squeezing can thus lead to a broadband sub-SQL performance ifthe quantum measurement noise (shot noise) and the quantum back-actionnoise (radiation pressure noise) are correlated in an optimal way Due to the

70

10 50 100 500 1000 5000Frequency [Hz]

Dis

plac

emen

t noi

se sp

ectra

l den

sity

[mradic

Hz]

10-20

10-19

10-18

Total quantum noise 0dB

Total quantum noise -10dB

SQL

Y

X

Y

X

Y

X

Figure 24 Surpassing the SQL with squeezed-light injection ndash At shot-noise lim-ited sideband frequencies squeezing of the Y -quadrature amplitude improves the noisespectral density of the interferometer At radiation-pressure-noise limited sideband fre-quencies squeezing of the X-quadrature amplitude improves the noise spectral density ofthe interferometer If both kinds of quantum noise contribute equally (at the SQL markedwith a dot) a squeeze angle of 45 results in surpassing the SQL by the full squeeze factorsee Eq (60) In the graph here the squeeze angle is optimized for all frequencies result-ing in a broadband quantum noise reduction [Jaekel and Reynaud (1990)] Measurementsensitivities beyond the SQL (shaded area) are in the so-called quantum non-demolition(QND) regime [Kimble et al (2001)] Dashed horizontal lines represent the (squeezed)shot noise Dashed straight lines with negative slope represent the (squeezed) radiationpressure noise The calculations use 10 dB of squeezing a conventional Michelson inter-ferometer with neither arm resonators nor signal recycling a light power at the beamsplitter of 1 MW at a wavelength of λ = 1550 nm and mirror masses of 1 kg

correlation shot noise and radiation pressure noise can be squeezed simulta-neously

71

Light with a frequency-dependent squeeze angle

The discovery that shot noise and radiation pressure noise can be squeezedsimultaneously and thus a broadband reduction of quantum noise beyond theSQL be achieved required the insight that the spectral analysis of light definesa spectrum of many lsquosideband modulation modesrsquo that all can be in differ-ent quantum states An ordinary squeezing resonator which is on resonancefor light at twice the pump wavelength produces a spectrum of modulationmodes that all have the same squeeze angle A frequency-dependent squeezeangle can be introduced by reflecting such a field from a detuned single-ended filter cavity which was suggested by Kimble and coworkers [Kimbleet al (2001)] They showed that the optimal frequency dependence thatleads to the broadband improvement shown in Fig 24 can be realized byusing altogether two filter cavities as shown in Fig 25 Motivated by this re-sult research and development on filter cavities for optimizing the frequencydependence of broadband squeezed fields has been very active in recent years[Corbitt et al (2004) Chelkowski et al (2005) Dwyer et al (2013) Kweeet al (2014) Straniero et al (2015) Oelker et al (2016)]

Photo diode Squeezed vacuum

Faraday Rotator

Coherent light

YΩΔΩ

Figure 25 Frequency dependent squeezing injection ndash A broadband squeezed fieldwith a frequency-dependent squeeze angle that is optimal for gravitational-wave detectorsis produced by reflecting off an ordinary broadband squeezed field from two detuned opticalfilters [Kimble et al (2001)]

A light field with a frequency-dependent squeeze angle was first demon-strated in Ref [Chelkowski et al (2005)] see Figs 26 and 27 The experi-

72

Detuned filter cavityLockingphoto diode

Isolator

EOM R asymp 1

SHG

EOM

DBSOPA

LaserIsolator

Mode cleaner

LO

Homodyne detector

Figure 26 Generation of a frequency-dependent squeezing ndash A frequency-dependent orientation of the squeeze ellipse was first demonstrated in Ref [Chelkowskiet al (2005)] Initially a conventional spectrum of squeezed vacuum states of light was gen-erated in a squeezing resonator (lsquoOPArsquo) The squeezed vacuum was transmitted throughan optical isolator to a detuned filter cavity After reflection the squeezed vacuum stateswere absorbed in a balanced homodyne detector (BHD) The phase of the BHDrsquos localoscillator (LO) was changed for quantum state tomography of the squeezed states in diffe-rent regions of the spectrum The result showed a frequency-dependent orientation of thesqueeze ellipse see Fig 27 SHG second harmonic generation EOM electro-optical mod-ulator for applying phase modulation sidebands for cavity length control DBS dichroicbeam splitter R mirror reflectivity λ4 quarter wave plate for turning linear polarizedlight into circular polarised light and vice versa

ment consisted of a standing-wave squeezing resonator which produced ans-polarized broadband amplitude quadrature squeezed field accompanied bya dim continuous-wave DC control field with a wavelength of λ = 2πcω =1064 nm The squeeze bandwidth covered sideband frequencies up to aboutΩ(2π) = 30 MHz which corresponded to the linewidth of the squeezing res-onator The optical cavity for producing the frequency dependence of thesqueeze angle was a standing-wave cavity composed of a plane incouplingmirror of reflectivity r1 =

radic097 and a concave end mirror of reflectivity

r2 =radic

09995 The cavity length was L = 50 cm resulting in a linewidthof 147 MHz The squeezed field first passed a Faraday isolator to preventinterference effects between the filter cavity and the squeezing resonator A

73

Phase quadrature

min

max

-2 -1 0 1 2

141 MHz

-2

-1

0

1

2

Am

plit

ud

e q

uad

ratu

re

Figure 27 Frequency-dependent squeezing ndash Picture top right Reconstructed con-tour plot of the Wigner function of the sideband modulation at Ω(2π) = 141 MHz afterreflection from a 1515 MHz detuned filter cavity The state shows quantum correlationsbetween phase and amplitude quadratures ie squeezing at an angle of here about 40The white circle visualizes the standard deviation of the vacuum state uncertainty Thewhite ellipse represents the standard deviation of the squeezed uncertainty Small picturesMeasurement results on the same continuous-wave laser beam at various sideband frequen-cies around 15 MHz For each tomographic picture noise histograms of 100 equidistantquadrature angles were measured In each case the laser beam was phase locked to a ref-erence beam and the quadrature angle stably controlled and stepwise rotated The phasereference was given by a phase modulation at 198 MHz [Chelkowski et al (2005)] Thepicture was first published in Ref [Schnabel (2005)] (copyright ccopy2007 by Imperial CollegePress)

λ4-waveplate turned the s-polarized field into a circularly polarized beamwhich was then mode matched into the detuned cavity The retro-reflectedfield was analyzed by a balanced homodyne detector (BHD) for quantumstate tomography The filter cavity was electro-optically controlled to be de-tuned by 1515 MHz with respect to the DC control field The cavity length

74

control was achieved by the Pound-Drever-Hall (PDH) locking techniqueutilizing a circularly polarized laser beam that carried 15 MHz phase mod-ulation sidebands and was coupled into the filter cavity from the back Theoutput voltage of the BHD was characterized by a spectrum analyser as wellas used to perform quantum state tomography In the latter case the BHDoutput voltage was mixed down with an electronic local oscillator at differentradio-frequencies around 15 MHz and low-pass filtered to set the resolutionbandwidth to ∆Ω(2π) = 100 kHz The final electric signal corresponds to atime series of quadrature amplitude measurements XθΩi∆Ω Quantum statetomography is a method to reconstruct the phase space quasi-probability dis-tribution (Wigner function) of quadrature amplitudes from sets of measuredXθΩ∆Ω distributions when varying the angle θ [U Leonhardt (1997)] Forevery sideband frequency Ωi 100000 quadrature values were measured di-vided up on 100 equidistant quadrature angles Each quadrature angle wasstably controlled with a precision of plusmn1 Fig 27 shows the reconstructedWigner functions which were all measured on the same laser beam but atdifferent sideband frequencies For these measurements the detuned filtercavity was locked to the lower sideband at minus1515 MHz The result clearlyshows the frequency-dependent orientation of the squeeze ellipse In a morerecent experiment a frequency-dependent squeeze angle was also realized inthe kHz regime [Oelker et al (2016)]

56 Optomechanically induced (ponderomotive) squeezing

The radiation pressure of light when acting on a movable mirror re-sults in an intensity dependent phase shift [Pace et al (1993)] The cou-pling produces a so-called lsquoponderomotive effectrsquo [Braginsky and Manukin(1967)] which is of third order optical nonlinearity and which transforms abright coherent state inside an interferometer into a squeezed state of light[Vyatchanin and Matsko (1993)] This type of squeezed-light generationis usually called lsquoponderomotive squeezingrsquo or lsquooptomechanical squeezingrsquoConsequently even if no squeezed field is injected into the interferometercorrelations between the quadrature amplitudes are generated that allow forsurpassing the SQLPonderomotive squeezing as produced by the interferometer itself can onlybe exploited for evading back-action (radiation pressure noise) It can notbe used to squeeze the interferometer shot noise This is why ponderomotivesqueezing is fundamentally less extensive than injecting externally producedsqueezed states of light [Corbitt et al (2006)] suggested an external pon-

75

deromotive squeezing source for gravitational-wave detectors In this casedue to its external generation also the interferometerrsquos shot noise can besqueezed Recently ponderomotive squeezing was observed for the first time[Brooks et al (2012) Purdy et al (2013)] The achieved squeeze factors aremuch smaller than those produced by optical-parametric down-conversion[Vahlbruch et al (2016)]

Let us have a look again at Eq (59) Rotating the covariance matrix onthe right by arctan(minus

radic54minus 12) asymp minus58 indeed reveals squeezing

(cos 58 minussin 58

sin 58 cos 58

)(1 minus1minus1 2

)(cos 58 sin 58

minussin 58 cos 58

)asymp(

262 00 038

) (61)

The vacuum-noise normalized variance of 038 corresponds to about 42 dBof ponderomotive squeezing This is the general value that is produced atthe angular sideband frequency ΩSQL At higher frequencies the squeezefactor gets smaller at lower frequencies higher The squeezing strength of42 dB can be observed if the photo diode in the interferometer output portis replaced by a balanced homodyne detector using a local oscillator phase ofabout minus58 It can be shown however that the optimal signal-to-quantum-noise-ratio at the SQL is achieved for a local oscillator phase of exactly 45At this angle back-action is fully evaded

Full evasion of radiation pressure noise at all frequencies requires an op-timized frequency dependence of the relative local oscillator phase This canbe achieved by reflecting off the interferometer output field from two detunedfilter cavities [Kimble et al (2001)] The scheme was called lsquovariational out-putrsquo In the case of zero optical loss this scheme can fully evade radiationpressure noise just leaving the shot noise as the only quantum noise contri-bution

The variational-output scheme can be used to enhance the frequency-dependent squeezed input scheme The right site of Eq (60) shows thatthe output statersquos squeezing is not optimally detected in the Y -quadratureRather than with a single photo diode the detection should be done witha balanced homodyne detector with optimized phase of its local oscillatorIn this case the output lightrsquos quantum noise is solely given by squeezedshot noise The total quantum noise in Fig 24 would then be given by thelowest (dashed) horizontal line This combined scheme was called lsquosqueezed

76

variationalrsquo [Kimble et al (2001)] It can be realized by reflecting off theinterferometer output light from in total two optical filter cavities placed infront of the balanced homodyne detector

57 Conclusions

The highest quantum-noise-limited sensitivities of high-precision laserinterferometers are achieved by employing a large number of quanta tomaximize the signal strength in combination with strongly squeezed statesto minimize the quantum noise From this perspective it is clear that thequantum-noise-limited sensitivity of future gravitational-wave detectors willbe further improved ndash by increasing the light power and the squeeze factorTo be able to do so the optical loss in these devices needs to be reduced

In principle the optical loss in laser interferometers can be made smallbut never zero Recent theoretical research has shown that for any non-zero loss the sensitivity scales proportional to 1

radicn at best where n is the

average photon number per measurement This scaling is efficiently achievedby combining strongly displaced coherent states with squeezed vacuum statesof light

If a repeated measurement is not only limited by quantum measurementnoise but also by quantum back-action noise squeezed states of light can beused to simultaneously reduce both ie in the case of an interferometer shotnoise and radiation pressure noise

6 The first application of squeezed light in an operating gravita-tional-wave detector

Squeezed states of light have been successfully used to improve the sensi-tivity of the gravitational-wave detector GEO 600 from 2010 up to the pointwhen this Review was written [Abadie (2011) Grote et al (2013)] Afterdecades of proof-of-principle experiments [Xiao et al (1987) Grangier et al(1987) McKenzie et al (2002 2004) Vahlbruch et al (2005 2006 20072008) Goda et al (2008)] the implementation of a squeezed-light source inGEO 600 has resulted in the first sensitivity improvement beyond shot noiseof a measurement device that targets new observations in nature The im-plementation of squeezed states in GEO 600 was not done to provide anotherproof-of-principle demonstration but was realized because it offered a rela-tively cheap way of further improving the measurement sensitivity Of course

77

the sensitivity of GEO 600 can also be further increased by purely classicalmeans however the implementation of arm resonators to enable higher lightpowers without increasing the thermal load on the beam splitter or eventhe realization of longer interferometer arms are much more expensive Inthis respect the sensitivity improvement of GEO 600 with squeezed light canarguably be regarded as the first lsquotruersquo application that developed out of thefield of lsquononclassical (quantum) metrologyrsquo (Note that the term rsquoquantummetrologyrsquo is currently defined in different ways [Giovannetti et al (2006)Gobel and Siegner (2015)] and the term rsquononclassicalrsquo referring to a non-classical P-function gives a distinct description)

61 Gravitational waves

Einsteinrsquos General Theory of Relativity [Einstein (1916)] or simply lsquoGen-eral Relativityrsquo (GR) predicts that accelerating mass distributions producegravitational radiation analogous to electromagnetic radiation from accel-erating charges Experimental evidence of their existence is given by theobservation of the slow spiraling together of two neutron stars caused bythe loss of orbital energy to gravitational waves The inspiral rate exactlymatches the predictions of Einsteinrsquos theory [Weisberg and Taylor (2005)]Recently Advanced LIGO observed gravitational waves for the first time[Abbott (2016)] thereby giving the go-ahead for gravitational-wave astron-omy The gravitational-wave source was the final inspiraling and the mergerof two black holes 13 billion light years away from earth

Gravitational-waves evolve in the far field of the source propagate withthe speed of light and are measurable on earth with laser interferometersFig 28 displays a gravitational wave propagating along a certain directionGravitational waves are dynamical changes of space-time They are transver-sal and quadrupolar in nature and have two polarization states

A variety of known astrophysical and cosmological sources are predictedto emit gravitational radiation that should reach the Earth with a measur-able strength [Sathyaprakash and Schutz (2009)] The first gravitational waveevent detected was produced by two black holes of 36 and 29 solar massesDuring the final 02 seconds of their inspiraling they produced a peak gravi-tational strain in our solar system of 10minus21 covering frequencies up to 250 Hz[Abbott (2016)] Other predicted sources are mergers of neutron stars supernovae and background signals from the Big Bang According to GR GWsfrom complex astrophysical sources carry a plethora of information that willhave a major impact on gravitational physics astrophysics and cosmology

78

L +Δ L

Binary system

fBS

Figure 28 Space-time oscillation ndash Gravitational waves are dynamical deformations ofspace-time that form in the plane perpendicular to the direction of wave propagation Asa result distances between free-falling test masses in a transverse plane will change witha strain h = ∆LL For black hole or neutron star binary systems with orbital frequencyfBS distances will oscillate at frequency fGW = 2fBS The wavelength of this oscillation isgiven by λGW = cfGW where c is the speed of light The wave of orthogonal polarizationwith respect to the one shown is rotated by 45 around the propagation axis

62 Interferometric detection of gravitational waves

Current gravitational wave detectors are kilometre-scale laser interfero-meters [Dooley et al (2016) Aasi (2015) Acernese (2015) Aso et al (2013)]Continuous-wave laser light is split into two beams traveling in orthogonaldirections Both beams are reflected back towards the central beam splitterwhere they interfere Gravitational waves change the optical path lengthdifference and thus the light power directed towards the photo-diode thatis positioned in the signal output port of the beam splitter A gravitationalwave at frequency fGW = ΩGW(2π) reveals itself as a light-power modula-tion at the same frequency The spectral decomposition of the output signalis described by a spectrum of the quadrature amplitude YΩ∆Ω introduced inSec 3 It corresponds to the amplitude quadrature amplitude of the outputlight and relates to the differential phase quadrature of the interferometerarms

79

Photo diode

Squeezed vacuum

Faraday Rotator

Signal

Quantum noise

Coherent light XΩΔΩ

YΩΔΩ

(a) (b)

YΩΔΩFigure 29 Squeezed-light-enhanced interferometric measurement ndash (a) Michelsoninterferometer with arm cavities power recycling and signal recycling (see main text forexplanation) The interferometer is operated close to a dark fringe such that the quantumnoise entering from the dark port is back-reflected The squeezed field is mode-matched tothe signal output field (b) Phase space diagram of the gravitational-wave signal outputat sideband frequency Ω(2π) The quantum noise is squeezed below the ground stateuncertainty and thus the signal to quantum noise ratio improved

The first key ingredient of an interferometric gravitational-wave detectorare suspended heavy mirrors that can be regarded as quasi-free in the di-rection of laser light propagation thereby acting as test masses that probespacetime Being on ground current detectors are located in rather noisy en-vironments that allow the realization of undisturbed quasi-free mirrors onlyabove a sideband frequency of the order of 10 Hz Since sufficiently strongGW signals are expected up to a frequency of 10 kHz todayrsquos gravitationalwave detectors target at signals in the acoustic band from 10 Hz to 10 kHzThe quasi-free motion of the test mass mirrors in this frequency regime isachieved by suspending the mirrors as sophisticated multiple-stage pendulain vacuum chambers [Aasi (2015)] Far above the pendularsquos resonant frequen-cies which are typically around 1 Hz the centre of masses of the mirrors areisolated from vibrations of the ground and they react on frequency compo-nents of small external forces approximately as free masses The mirrors andtheir suspensions are built from materials having exquisitely high mechani-cal quality factors This helps to concentrate the thermal energy that causes

80

displacements of the mirror surface into well-defined vibrational frequencymodes At these particular very sharp frequencies no gravitational wavescan be detected

The second key ingredient of an interferometric gravitational-wave de-tector is laser light with a power of up to hundreds of kilowatts or evenmegawatts The light is quasi-monochromatic and needs to show very lowamplitude (quadrature) noise and phase (quadrature) noise at sideband fre-quencies within the detection band Low amplitude noise is necessary toprovide a shot noise limited output field It is also necessary to avoid classi-cal radiation pressure noise which becomes an issue if the light power or themirror masses in the two arms are not identical Low phase noise is requiredif the storage time of the light in the two arms is not identical This mightaccidentally occur due to different linewidths of the arm cavities or mightbe part of the interferometer design to allow for the length control schemeproposed by Schnupp [Heinzel et al (1998)] To maximize the light power in-side the interferometerrsquos cavities it should be produced in an almost perfecttransversal spatial distribution of a Gaussian TEM00 mode

Light sources of gravitational-wave detectors are ultra-stable NdYAGmaster-slave systems that provide up to 200 W of light at 1064 nm [Winkel-mann et al (2011) Kwee et al (2012)] The high power in the interferometerarms is achieved by cavity built-ups in the so-called power-recycling cavityand in the arm cavities Power recycling uses a partially reflective mirror thatis located between the light source and the interferometer beam splitter Itssurface is matched to the lightrsquos wave front and forms an optical cavity to-gether with the rest of the interferometer Since gravitational-wave detectorsare operated close to a dark fringe large power built-ups can be achievedThe highest power built-up is achieved for a mirror transmission equal to the(given) interferometer round trip loss In this case an impedance-matchedcavity is achieved The power-recycling cavity as well as the arm cavitiesare stabilized on resonance for the input light The difference between theirfunctionality is that the power-recycling cavity does not limit the detectionbandwidth of the interferometer GEO 600 as well as Advanced LIGO em-ploy a third type of cavity the so-called signal-recycling cavity Similarlyto power recycling a partially reflecting mirror that is placed between theoutput port of the beam splitter and the photodiode is used to resonantly en-hance the GW signal [Meers (1988)] The signal-recycling cavity resonantlyenhances the signal modulation fields within its linewidth without furtherenhancing the carrier light power In combination with low linewidth Fabry-

81

Perot arm resonators it can also be used to extract the signal by reducing theeffective finesse of the arm resonators for the signal sidebands This schemeis called resonant sideband extraction [Heinzel et al (1996)] The signal-recycling cavity has also been tested in a detuned setting in which just theupper or lower sideband is extracted or resonantly enhanced respectively[Heinzel et al (2002)] Current gravitational-wave detectors however usecarrier-tuned signal recycling

All these techniques are lsquoclassicalrsquo approaches for maximizing the signal-to-shot-noise ratio At frequencies above a few hundred Hertz howevershot-noise is still the limiting noise source in gravitational-wave detectorsFuture gravitational-wave detectors will therefore use even higher light pow-ers but further increasing the light power becomes more and more challeng-ing Optical materials with less light absorption need to be found to coun-teract an increasing thermal load inside the mirrors Mirror masses need tobe further increased to counteract the increasing radiation pressure noiseNonclassical approaches are superior and become more and more attractivethe farther classical approaches are pushed to the extremes Nonclassicalapproaches allow for simultaneously increasing the signal-to-shot-noise ratioand the signal-to-radiation-pressure-noise ratio without changing light poweror mirror masses see Fig 24 They also allow for a complete evasion of radia-tion pressure noise [Braginsky and Khalili (1995 1996) Kimble et al (2001)]see Subsec 56

63 Squeezed-light enhancement of the gravitational-wave detector GEO 600

In 2010 GEO 600 was equipped with the squeezed-light source shownin Fig 18 The location of the squeezed-light source close to the outputport is shown in Fig 30 It was known that GEO 600 was shot-noise limitedat sideband frequencies above about 700 Hz In this frequency regime thereplacement of the ordinary vacuum states that entered the interferometerfrom the output port by a spectrum of squeezed vacuum states was expectedto reduce the noise spectral density into the nonclassical regime It was notprecisely clear what squeezing factors could be expected since the opticalloss upon mode-matching an external field into the output port propagationalong the arms and the final photo-electric detection was not determined

Critical components were the quantum efficiency of the photo-diode aswell as the optical loss of Faraday rotator for overlapping the squeezed fieldwith the output mode Also the transversal output mode of GEO 600 was an

82

issue since it contained significant contributions from higher order modesto which a good mode matching of the squeezed mode was not possible Theimplementation of the squeezed-light source thus was accompanied with anew custom made InGaAs photo-diode with 3 mm diameter The goal was aquantum efficiency of greater 99 [Vahlbruch et al (2016)] Also the Fara-day rotator was custom-made and optimized for lowest optical loss which

Figure 30 GEO 600 ndash View into the central building of the British-German GW detectorlocated close to Hannover Germany The vacuum chambers contain the suspended beamsplitter power- and signal recycling mirrors additional input and output optics as well asmirrors to realize a double pass of the laser light through the 600 m long interferometerarms By courtesy of the AEI

83

involved a rather precise rotation of the polarisation of 45 plusmn 05 over anaperture of more than 15 mm Finally a ring cavity (output mode cleaner)was placed in front of the photo-diode which acted as a passive filter forhigher transversal modes Since GEO 600 was not limited by radiation pres-sure noise and since it used a carrier-tuned signal-recycling cavity a frequencyindependent orientation of the squeezing angle was optimum After several

100 200 300 500 700 1000 2000 3000 500010-22

10-21

10-20

Obs

erva

tory

noi

se c

alib

rate

d to

str

ain

[1radic

Hz]

10-19

10-18

10-17

Noise calibrated to test m

ass displacement [m

radicHz]

Sideband frequency f

Figure 31 Nonclassical reduction of the GEO 600 instrumental noise ndash Shownare the square roots of the noise spectral densities without squeezed vacuum states (top)and with squeezed vacuum states (bottom) [Abadie (2011)] Their calibrations [Affeldtet al (2014)] to gravitational-wave strain and differential arm length change are shownon the left and right y-axes respectively Note that both traces increase towards higherfrequencies due to the frequency-dependent signal enhancement of the signal-recyclingcavity The injection of squeezed vacuum states leads to a broadband noise reductionof up to 35 dB at shot-noise limited frequencies The spectral features are for instancecaused by excited violin modes of the mirror suspensions (600ndash700 Hz and harmonics)Data by courtesy of the AEI

months the combination of the squeezed-light source and the gravitational-wave detector succeeded The main laser of the squeezed-light source wasphase locked to the main laser of GEO 600 and a stable mode matching be-

84

tween the squeezed field and the interferometer output field was achieved[Abadie (2011)] The effect on GEO 600rsquos (strain normalized) noise spectraldensity is shown in Fig 31 At frequencies above about 700 Hz the squareroot of noise spectral density was reduced by up to one third This value cor-responds to a quantum noise variance of 045 (minus35 dB) with the shot noisevariance normalized to unity For isotropically distributed gravitational-wavesources this factor produces a detection rate increase by a factor of 153 asymp 34After its integration into GEO 600 the squeezed-light source was used in allscientific runs seeking for gravitational waves for instance in the observa-tional run S6eVSR4 that was undertaken from June 3rd to September 5th

in 2011 [Grote et al (2013)]Towards the end of 2011 right before the start of the detector upgrade

to Advanced LIGO a nonclassical sensitivity improvement was also demon-strated in one of the LIGO detectors [Aasi (2013)] In this experiment anonclassical sensitivity improvement corresponding to up to 215 dB abovefrequencies of about 150 Hz was achieved The successful test is a strongmotivation for a squeezed-light upgrade of Advanced LIGO Note that thedesign of Advanced LIGO was completed in 1999 and squeezed-light sourceswere not mature at those times

In the past years the squeezing enhanced GEO 600 detector was not onlyused for observations but also was the control of the injected squeezed modefurther improved Stabilizing the overlap between squeezed mode and brightmode of the interferometer to close to perfect is necessary to reduce theeffective optical loss and to maximize the measurable squeezing factor Re-cently superior methods for stabilizing the longitudinal phase of squeezedvacuum mode were found [Dooley et al (2015)] and the first automatic align-ment system for stabilizing and optimizing the transversal mode overlap wasdemonstrated [Schreiber et al (2016)]

64 Are squeezed states the optimal nonclassical resource ingravitational-wave detectors

For a given number of photons Eq (41) quotes the ultimately smallestphase change that can be measured with a signal-to-noise-ratio of one Thescaling with number of photons per measuring time of this Heisenberg limitseems appealing compared to the scaling achievable with coherent states orsqueezed states according to Eqs (38) and (39) respectively The Heisenberglimit however is only valid for precisely zero photon loss Since the non-classical states required to achieve Eqs (40) and (41) show an exponentially

85

increasing sensitiveness to loss when increasing the photon number the ac-tual scaling can not be deduced from Eq (40) Proposals to use Fock statesand the so-called N00N states for optimizing interferometer sensitivities [Hol-land and Burnett (1993) Dowling (1998) Mitchell et al (2004) Afek et al(2010)] are thus only applicable when the experiment is conditioned on zerophoton loss As discussed in recent publications the correct expression forthe fundamental sensitivity limit needs to consider not only the total photonnumber inside the interferometer but also the total photon loss [Dorner et al(2009) Ko lodynski and Demkowicz-Dobrzanski (2010) Knysh et al (2011)Escher et al (2011) Demkowicz-Dobrzanski et al (2012)]

Based on these earlier works Ref [Demkowicz-Dobrzanski et al (2013)]proved that the nonclassical sensitivity enhancement of GEO 600 reportedin Ref [Abadie (2011)] has been exceedingly close to fundamental quantuminterferometry bound under given energy constraints and photon loss lev-els More than that it was generally proven that the approach of com-bining displaced coherent states and squeezed vacuum states is optimal forgravitational-wave detectors

In Ref [Abadie (2011)] the gravitational-wave detector GEO 600 used aneffective number of photons per second of approximately n = 2 middot 1022 whichcorresponded to a total optical power inside the interferometer arms of P asymp37 kW at a wavelength of 1064 nm The total optical loss was 1minus η asymp 038The injected squeezing factor was eminus2r asymp 01 For these numbers the ratioof Eqs (43) and (44) is calculated to

∆φCSVmin

∆φgenmin

asymp

radicηeminus2r + 1minus η

1minus ηasymp 108 (62)

which is a good approximation within the limit of large coherent state dis-placements α sinh2r The quantum noise of GEO 600 including thesqueezed-light source was just 8 above the fundamental quantum inter-ferometry bound An increased squeezing strength of 16 dB (eminus2r asymp 0025)which is in reach would bring the approach based on coherent states andsqueezed vacuum states to within just 2 above the fundamental bound

Future GW detectors will have significantly reduced optical loss values(1 minus η) lsquoLossrsquo includes scattering and absorption at mirrors non-perfectfringe contrasts and the non-perfect quantum efficiency of the photo detec-tor Optical loss reduction is important for at least four reasons First itleads to an increased signal second it leads to a reduced quantum noise when

86

employing squeezed states third less absorption reduces the thermal load onthe test mass mirrors and fourth less scattering reduces the probability ofback-scattered light which produces disturbance signals [Billing et al (1979)Vahlbruch et al (2007) Punturo et al (2014)] The higher the finesse valuesof the arm and signal-recycling cavities are the more significant is opticalloss at mirror test masses the beam splitter and the signal-recycling mirrorThe finesse value of the power-recycling cavity and the loss of mirrors andlenses that guide the output field to the photo-diode are less critical Suitablephoto detectors of 995 quantum efficiency are available today [Vahlbruchet al (2016)] but achieving a total optical loss of 10 is still challengingThe reason for that is that first of all a measurement device aiming for bestabsolute sensitivity should use as much quanta (photons) as possible Highfinesse values for the enhancement cavities are thus essential but results inan unavoidable scaling-up of the effect of mirror losses A realistic exampleof future gravitational wave detectors thus considers η = 09 with a squeezingfactor of 20 dB (eminus2r = 001) In this case the quantum noise will be about4 above the ultimate fundamental bound for a given photon number

From Eq (62) it can be concluded that there is no need for any moresophisticated nonclassical states than squeezed states In particular non-classical states with a defined photon number such as N00N states are notrequired Within the approximation quoted this result is independent ofthe photon number This result is also independent of the physical systemused for interferometric phase estimation and can also be made for quantum-enhanced atomic clock calibration in the presence of dephasing Here the-oretical results indicate that the precision of Ramsey interferometry withspin-squeezed states is close to the optimal one in the asymptotic regime ofa large number of atoms [Huelga et al (1997) Ulam-Orgikh and Kitagawa(2001) Escher et al (2011)] as already stated in Ref [Demkowicz-Dobrzanskiet al (2013)] More sophisticated nonclassical states with fixed number ofn quanta might still be useful for the exceptional case when the absorptionof one quantum already results in zero measurement sensitivity anyway Anexample is an ensemble measurement where the absorption of a single photondemolishes the source of the phase change to be characterized A typicallyused approach of conditioning the measurement result on n clicks of n singlephoton counters conditions on precisely zero loss and is thus able to use theadvantage of Eq (40) over Eq (39)

87

65 Conclusions

Squeezed states of light will contribute to realizing gravitational-wave ob-servatories with much higher sensitivities than existing or planned ones Tobenefit from squeezed states in a most efficient way optical loss in terms ofabsorption and scattering must be minimized In particular the optical lossof mirror coatings and mirror substrates need to be minimized The rele-vant mirrors include the test masses the balanced beam splitter the signalrecyclingextraction mirror and all optical components between the latterand the photo diode Excellent spatial mode matching between the brightinterferometer field and the squeezed vacuum field is also of great impor-tance Achieving this requires further improvement of the surface figures ofall reflective optical components of the interferometer as well as improvedhomogeneity of all optical components that the light passes through

The quantum noise reduction achieved in a gravitational-wave detectoris of course always smaller than the highest squeeze factor provided by thesqueezed-light source As an example let us consider the observation of 15 dBof nonclassical noise suppression directly at the source If the squeezed fieldsenses an additional loss of 5 when propagating through the interferometerwhich is a very challenging number from todayrsquos point of view the remainingsqueezing level is about 11 dB see Eq (16)

7 The application of 2-mode-squeezed light in laser interferome-ters

71 Quantum Dense Metrology

At first glance the application of bi-partite (two-mode) squeezed statesto a device whose goal is measuring a single observable seems meaninglessSqueezing the uncertainty of that observable should be the optimum one cando This is indeed true when concerning just quantum noise but recentlyit was discovered that in the presence of classical disturbances bi-partitesqueezing can improve such measuring devices [Steinlechner et al (2013)]The concept was named quantum dense metrology (QDM) The potentialimprovement of a gravitational-wave detector with bi-partite squeezed statesis shown in Fig 32 (a) A description is given in the caption The pre-condition for a potential improvement can be best understood within a phasespace diagram Fig 32 (b) contains two different kinds of lsquosignalsrsquo The firstis the actual signal which always shows up as a phase space displacement

88

along the Y axis The second is a disturbance signal that can produce adisplacement in arbitrary direction in phase space A prominent example

Faraday Rotator

Coherent light

Photo detectors Balanced homodyne detectors

Squeezed vacuum

Squeezed vacuum

Quantum noise in

Y (A) and X (B)

XΩΔΩ(B)

YΩΔΩ(A)

(a) (b)

Disturbance projected

onto Y

and X

XΩΔΩ(B)

YΩΔΩ(A)

Example of (unknown) disturbance

Signal

Figure 32 Bi-partite-squeezed-light-enhanced measurement ndash (a) Setup for theapplication of bi-partite (two-mode) squeezed light in a laser interferometer on the basisof QDM Two squeezed vacuum fields are overlapped on a balanced beam splitter withprogrammable squeeze angles for instance with a relative angle of 90 which produces abi-partite state as shown in Fig 12 The beam splitter outputs are entangled for any rela-tive angle greater than zero One part is matched to the interferometer mode The secondpart is kept outside as a reference beam The interference of the interferometer outputand the reference beam is arranged with such a phase difference that it reproduces the twosqueezed inputs on the photo detectors The two squeezed beams are photo-electricallydetected measuring the respective squeezed quadrature (using balanced homodyne detec-tors) Both beams carry half of all interferometer induced modulations which includesignals as well as disturbances A single readout as shown in Fig 21 cannot distinguishbetween the two kinds The double readout shown here provides additional informationand allows for recognition of the disturbance [Steinlechner et al (2013)] as well as in prin-ciple a modeling of the disturbance and with a correct model an improvement of thenoise spectral density of the interferometer [Ast et al (2016)] (b) Phase space diagramdescribing phase quadrature readout A as well as amplitude quadrature readout B Bothshow squeezed quantum noise The amplitude quadrature readout does not contain anygravitational-wave signal ie any feature in this channel must be due to disturbancesThis information can be used to improve the interferometer

89

for such a disturbance is parasitic interference due to back-scattered laserlight [Vahlbruch et al (2007)] Back-scattering is a limiting noise at lowsignal frequencies of gravitational-wave detectors [Billing et al (1979) Vinetet al (1997) Hild (2007) Ottaway et al (2012) Punturo et al (2014)] Notethat all noise that couples in via unwanted motions of the test mass mirrorsso-called lsquodisplacement noisersquo always produces a phase space displacementalong the Y axis and cannot be tackled with QDM

Fig 33 shows measurement results obtained in Ref [Steinlechner et al(2013)] In a table-top experiment one part of a bi-partite squeezed state ofa continuous-wave mode at 1064 nm was mode-matched into the output portof a Michelson laser interferometer operated at its dark fringe in full analogyto Fig 32(a) A lsquosignalrsquo was produced by driving the piezo behind one of theend mirrors at a frequency of 555 MHz The lsquodisturbancersquo was introducedby re-injecting a small amount of light that leaked through the second endmirror with an additional piezo-mounted mirror The piezo was driven ata frequency of 517 MHz to produce a phase modulation An additionalDC voltage defined an arbitrary and unknown optical path length of thelight before being re-injected and as such the phase space orientation of thedisturbance signal This mechanism of a parasitic interference is realizednaturally in any interferometric device due to back-scattering of quanta frommoving surfaces in the environment

The interferometer output consisted of the signal as well as the distur-bance with a quantum uncertainty given by one subsystem of the bi-partiteentanglement It was overlapped with the second subsystem of the entan-gled state on a balanced beam splitter and the two outputs were analysedwith balanced homodyne detectors The phases of the bi-partite entangle-ment and the BHD local oscillators were controlled to resemble Fig 32(b)ie both BHDs measured a squeezed uncertainty regardless of the phase ofthe (generally unknown) disturbance

The beam splitter that combines interferometer output and the entangledreference beam unavoidably splits the signal as well as the disturbance intotwo paths For a balanced beam splitter this generally reduces the signaland disturbance power by 3 dB for both quadrature measurements Fig 33shows however that both BHDs performed about 6 dB below shot noisewhich demonstrates the usefulness of the scheme The squeeze factor can inprinciple be infinite which thus qualifies the lsquo3 dB penaltyrsquo In the abovefigure the additional information from the second BHD output was used torecognize the parasitic interference in the first BHD output providing a lsquovetorsquo

90

50 51 52 53 54 55 56 57

50 51 52 53 54 55 56 57-70

-75

-80

-85

-90

-70

-75

-80

-85

-90

Y (A) (Ω2π)Ω∆Ω

Frequency ( f )

Noi

se p

ower

(dB

m)

Noi

se p

ower

(dB

m)

X (B) (Ω2π)Ω∆Ω

~ ndash6 dB

~ ndash6 dB

Shot noise reference

Shot noise reference

Signal Projected disturbance

Projected disturbance

Vet

o

No signal

Figure 33 Bi-partite-squeezed-light-enhanced measurement ndash The result wasachieved in a table-top setup [Steinlechner et al (2013)] In the two panels the lower

(blue) traces show the squeezed quadrature noise-power spectra ∆2Y(A)Ω∆Ω(Ω2π) (top) and

∆2X(B)Ω∆Ω(Ω2π) (bottom) as simultaneously measured with balanced homodyne detectors

lsquoArsquo and lsquoBrsquo respectively The conventional Y -measurement (top) cannot distinguish be-tween signal and disturbances The additional X-measurement (bottom) does not detectany phase quadrature signal thus any feature in this measurement is a parasitic signal dueto a disturbance The respective projection onto the Y -measurement can thus be lsquovetoedrsquoIn a more sophisticated approach the X-data might be used to model and then to elim-inate the disturbance as well as its projections on both quadrature measurements Theresult is a reduced spectral density of the actual phase quadrature measurement [Ast et al(2016)] Traces shown here are slightly sloped due to the decreasing transfer functions ofthe balanced homodyne detectors The resolution bandwidth was ∆Ω(2π) = 10 kHz thevideo bandwidth was 100 Hz All traces were averaged three times

signal to trigger its removal from the data streamThe question arose whether the additional information can be used to

reduce the actual noise spectral density of the first measurement ie to re-cover signals that were buried by parasitic interferences Very recently it wasshown that this is indeed possible Ref [Ast et al (2016)] reports a table-top proof-of-principle experiment in which the additional information in the

91

QDM approach could be used for improving the sensitivity of an interfero-meter The measurement sensitivity was improved from above-shot-noise tosub-shot-noise (sub-Poissonian) performance This result was possible notbecause the way the parasitic interference arose was known but because theadditional information provided by QDM allowed for fitting a model of theexcess noise to the readout data

Quantum dense metrology (QDM) as shown in Fig 32 improves a mea-surement by simultaneously reading out two conjugate observables Bothreadout observables show a squeezed quantum noise and act as estimatorsof independent physical quantities This situation was recently described aslsquoquantum-mechanics freersquo [Tsang and Caves (2012)] QDM is based on anEinstein-Podolsky-Rosen (EPR) entangled [Einstein et al (1935)] bi-partitesystem as described in Subsec 34 EPR entanglement was previously con-sidered for the quantum-informational task of dense coding which doublesthe capacity of quantum communication channels [Bennett et al (1992)Braunstein and Kimble (2000)] The application of EPR entanglement inmetrology was first proposed by DrsquoAriano et al [DrsquoAriano et al (2001)]

72 Conclusions

A single beam that carries an optimized spectrum of squeezed vacuumstates and that is injected into the interferometerrsquos dark port provides themost efficient and practically optimal approach to reduce the quantum noisein laser interferometers by means of nonclassical states (see Section 5) Theconclusion of the section here is that two entangled beams provide a superiorapproach if the interferometerrsquos sensitivity is limited by classical noise thatis not exclusively restricted to the actual observable which is the phasequadrature amplitude Y Parasitic interferences due to laser light that isbackscattered from vibrating surfaces are an example Current gravitational-wave detectors use light fluxes of about 1024 photons per second [Abbott(2016)] Just a single photon per second and hertz which leaves the mainlight beam and is backscattered from a vibrating surface and in this way getsfrequency shifted into the detection band produces a significant disturbancesignal The lsquoquantum-dense metrologyrsquo approach might provide a powerfultechnique to tackle this problem

Very recently it turned out that QDM is not the only technique that mayexploit EPR entanglement to improve phase measurements Ref [Ma et al(2017)] proposes to use EPR entanglement to simultaneously suppress shotnoise and radiation pressure noise in a gravitational-wave detector without

92

the need for an additional filter cavity (confer subsection 55) In this caseEPR entanglement is exploited that is carried by one broadband squeezedbeam and that is present between quadrature amplitudes defined with re-spect to different optical frequencies ω and ωprime as investigated in Ref [Hageet al (2010)] Such lsquofrequency multiplexedrsquo EPR entanglement might resultin considerably lower costs of building a gravitational-wave detector with abroadband simultaneous squeezing of shot noise and radiation pressure noiseAlso this proposal does not lead to a fundamentally lower quantum noise butrather improves on classical aspects of an interferometer

8 Summary and Outlook

In many cases experiments that involve interference of quantum statescan be described in a semi-classical way This description uses the classicalwave picture for the interference part of the experiment and subsequently theclassical particle picture when the states transfer their energy to a detectoror more generally to a thermal bath This semi-classical description is notpossible when using the specific class of lsquononclassicalrsquo states Squeezed statesof light are a prominent example of these Squeezed states and other nonclas-sical states allow for observations that made Einstein Podolsky and Rosenformulate their critical and seminal paper on quantum theory [Einstein et al(1935)]

In the review here it is argued that after many successful proof-of-principle experiments with nonclassical states in the past decades the routineuse of squeezed-light in observational runs of the gravitational-wave detectorGEO 600 goes beyond proof-of-principle and is a true application of nonclassi-cal light Since 2010 the squeezed-light source has improved the measurementsensitivity of GEO 600 in basically every observational run [Abadie (2011)Grote et al (2013) Dooley et al (2016)] At quantum noise limited frequen-cies ie above a few hundreds of hertz the sensitivity has been improvedcorresponding to a squeezing strength in the noise spectral density of up to37 dB which corresponds to an increase of the average gravitational-wavedetection rate by a factor of 043minus32 = 36 This success is a strong motiva-tion to also equip the Advanced LIGO Virgo and Kagra gravitational-wavedetectors with squeezed light Similar improvement factors even down tolower signal frequencies are expected [LSC (2013)] The achievable improve-ment factors are mainly limited by the optical loss on the squeezed states

93

and much higher factors are achievable in principleUp to now squeezed states have not been used to reduce the radiation-

pressure noise in gravitational-wave detectors The reason is that so far othernoise sources are larger than radiation pressure noise and such an effect can-not be observed It is expected however that future gravitational-wavedetectors will eventually be partly limited by radiation pressure noise Fromthis point on squeezed light will be used to simultaneously reduce shot noiseand radiation pressure noiseSqueezed states are the optimum nonclassical states for gravitational-wavedetectors or more generally for all laser interferometers operating with largeaverage photon numbers per measuring interval [Demkowicz-Dobrzanski et al(2013)] In addition to using higher light powers and heavier test mass mir-rors higher squeeze factors will thus contribute to mitigate the lightrsquos quan-tum noise in laser interferometers

Two-mode (bi-partite) squeezed light has not been used in gravitational-wave detectors so far They are not capable of further reducing the quantumnoise in laser interferometers but they can be used to mitigate classicalnoise that originates from fluctuating phase space displacements A well-known such noise source is back-scattered light Proof-of-principle experi-ments were performed recently [Steinlechner et al (2013) Ast et al (2016)]This new technique could turn out to be valuable in next generations ofgravitational-wave detectors in particular in those targeting high sensitivi-ties at low sub-audio signal frequencies and using high light powers Suchan implementation in gravitational-wave detectors does not require any newtechnology Compared to a squeezed-light enhanced interferometer just asecond squeezed-light source is required

It is certainly remarkable that those quantum states that made EinsteinPodolsky and Rosen falsely think quantum theory incomplete are now ex-ploited as new technologies in measurement devices targeting new observa-tions in nature

Acknowledgements

RS thanks M Ast J Bauchrowitz C Baune S Chelkowski J DiGugliel-mo A Franzen B Hage J Harms A Khalaidovski L Kleybolte NLastzka M Mehmet S Steinlechner and H Vahlbruch for their contri-butions many fruitful discussions and their support with the figures and JFiurasek for many valuable comments on the manuscript Thanks are also

94

due to Y Chen F Khalili and H Miao for fruitful discussions within thequantum noise working group of the LIGO Scientific Collaboration (LSC)Special thanks are due to H Vahlbruch and H Grote together with theGEO 600 team for their pioneering work on the squeezed-light implementa-tion in GEO 600 RS is supported by the Deutsche Forschungsgemeinschaft(Grant No SCHN 757-6) and by the European Research Council (ERC)project lsquoMassQrsquo (Grant No 339897)

References

References

Aasi J et al Jul 2013 Enhanced sensitivity of the LIGO gravitationalwave detector by using squeezed states of light Nature Photonics 7 (8)613ndash619URL httpwwwnaturecomdoifinder101038nphoton2013177

Aasi J et al Apr 2015 Advanced LIGO Classical and Quantum Gravity32 (7) 074001URL httparxivorgabs14114547httpstacks

ioporg0264-938132i=7a=074001key=crossref

20895763c84bce3f8929251031b2475c

Abadie J et al Sep 2011 A gravitational wave observatory operatingbeyond the quantum shot-noise limit Nature Physics 7 (12) 962ndash965URL httparxivorgabs11092295httpwwwnaturecom

doifinder101038nphys2083

Abbott B P et al Feb 2016 Observation of gravitational waves from abinary black hole merger Phys Rev Lett 116 061102URL httplinkapsorgdoi101103PhysRevLett116061102

Acernese F et al 2015 Advanced virgo a second-generation interferome-tric gravitational wave detector Classical and Quantum Gravity 32 (2)024001URL httpstacksioporg0264-938132i=2a=024001

95

As before the interference can be fully described by the classical wave pic-ture The result of the detection process however is different from that ofmutually independent random events It is also different from any super-Poissonian statistics that could be produced by mixing an arbitrary num-ber of different andor time-dependent Poissonian distributions Insteadthe squeezed probability distribution in Fig 1 suggests that the probabil-ity of detecting a photon decreases with the more photons that are alreadydetected in the same time interval over which a single measurement is in-tegrated From this observation one must conclude that the photons donot individually appear in a random fashion upon detection There mustbe lsquoquantumrsquo correlations between the photons These correlations mustexisted before detection since there is no interaction between the photonsduring their detection Pre-existing correlations between detected photonsseem to imply that the photons themselves existed before detection ie attimes when interference occurred In a semi-classical description howeverphotons are classical particles and cannot interfere for instance on a beamsplitter At this point the semi-classical picture breaks down Squeezedstates are therefor lsquononclassicalrsquoThe failure of the semi-classical model described above generally certifiesnonclassicality

Squeezed states are usually not characterized by counting their pho-tons but by measuring canonical continuous-variable phase-space observ-ables Measurements are performed as usual on an ensemble of identicalstates and quasi-probability density functions are calculated from the dataThe Glauber-Sudarshan P -function is the quasi-probability density distribu-tion over coherent states If the P -function of a state is entirely positivethe state is a coherent state or a (classical) mixture of coherent states Thestate is considered as semi-classical If the P -function is not a positive-valuedfunction the state cannot be expressed as a (classical) mixture of coherentstates and is thus nonclassical [Gerry and Knight (2005) Vogel and Welsch(2006)] A non-positive-valued P -function is the sufficient and necessary con-dition for the failure of the semi-classical model The Wigner function is thequasi-probability phase-space representation over the canonical continuous-variable phase-space observables themselves [Gerry and Knight (2005)] TheWigner functions of squeezed states are entirely positive Although subject todiscussion this fact does not mean that squeezed states are less nonclassicalthan Fock states or cat states which not only have a nonclassical P -function

7

but also a partially negative Wigner function (A cat state is a quantum su-perposition of two macroscopically distinct states [Monroe (2002)] referringto Schrodingerrsquos-cat gedanken experiment [Schrodinger (1935)]) In practicesqueezed states can even be regarded as superior nonclassical states becausethey represent the only nonclassical state that has been produced in a steadystate fashionIn almost all experiments so far the generation of Fock states and cat statesinvolves a probabilistic event such as the detection of a photon in anotherbeam path to herald these states In fact squeezed states provide the non-classical resource for the probabilistic preparation of Fock states as well as catstates But only the squeezed states themselves show a nonclassical effect in astationary way Limited only by the time duration and the frequency span ofthe mode that is in a squeezed state the squeezing effect can be continuouslyobserved independently of the time when the measurement is performed andalso independently of the measurement integration time This fact is of greatimportance for applications of squeezed states in measurement devices sincea squeezed-light-enhanced measurement remains unconditional and the ef-fective measurement time is not reduced

In past decades squeezed states of light were used in many proof-of-principle experiments to research their potential for improving the sensitivityof laser interferometers [Grangier et al (1987) Xiao et al (1987) McKen-zie et al (2002) Vahlbruch et al (2005) Goda et al (2008) Taylor et al(2013)] or the performance of imaging beyond the shot-noise limit [Lugiatoet al (2002) Treps et al (2003)] both accompanied by a huge number oftheoretical works Potential applications in secure optical communication(quantum key distribution) were also proposed and proof-of-principle ex-periments demonstrated [Ralph (1999) Furrer et al (2012) Gehring et al(2015)] This review restricts itself to the improvement of laser interfero-meters since only here has the application of squeezed light gone beyondproof-of-principle The gravitational-wave detector (GWD) GEO 600 hasoperated with squeezed light now for more than seven years starting in 2010[Abadie (2011) Grote et al (2013)] GEO 600 is a 600 m long Michelsonlaser interferometer built for the detection of gravitational waves Thesewaves are audio-band and sub-audio-band changes of space-time curvatureoriginating from cosmic events such as the merger of neutron stars or blackholes as detected recently [Abbott (2016)] In GWDs such as GEO 600 [Doo-ley et al (2016)] Advanced LIGO [Aasi (2015)] Advanced Virgo [Acernese

8

(2015)] and KAGRA [Aso et al (2013)] conventional laser technology hasbeen pushed to extremes over the past decades Noise spectral densities nor-malized to space-time strain of less than 10minus23 Hzminus12 have been measured[Abbott (2016)] Progress will continue and based on the successful appli-cation in GEO 600 squeezed light is now widely accepted to provide a newadditional technology to contribute to the new field of gravitational-waveastronomy It was also successfully tested in one of the LIGO detectors in2013 [LSC (2013)] and is an integral part of the European design study forthe 10 km Einstein-Telescope [Punturo et al (2010)]

GEO 600 has already taken several years of lsquosqueezedrsquo observational datawhich has increased its sensitivity at signal frequencies above 500 Hz Withthe implementation of a squeezed light source in GEO 600 the applicationof nonclassical states in metrology has been pushed beyond merely proof-of-principle

lsquoTwo-mode squeezed statesrsquo show a squeezed uncertainty in at least onejoint continuous variable of two subsystems lsquoArsquo and lsquoBrsquo Examples of jointvariables are differences and sums of phase-space observables of A and BTwo-mode squeezed states not only belong to the class of nonclassical statesbut due to their bi-partite character also to the class of lsquoinseparablersquo orlsquoentangledrsquo states They are the ideal states to demonstrate the Einstein-Podolsky-Rosen paradox [Einstein et al (1935)] as first achieved in [Ouet al (1992)] Apart from fundamental research on quantum mechanicsrecent proof-of-principle experiments demonstrated their usefulness in inter-ferometric measurements that go beyond the application of simple squeezedstates [Steinlechner et al (2013) Ast et al (2016)] This experiment is thefinal topic of this review

2 Observations on light fields in squeezed states

Generally there are two different kinds of observables that can be subjectof a measurement performed on a quantum system The first kind is associ-ated with the systemrsquos wave property In optics it corresponds to the electricfield strength at a given phase angle ϑ The according (dimensionless) opera-tors are called the quadrature amplitudes Xϑ and have a continuous spectrumof eigenvalues Quadrature amplitudes are measured in very good approx-imation with a balanced homodyne detector using the interference with a

9

bright local oscillator beam see Fig 3 (a) In practice any measurement ofXϑ integrates over some sideband (Fourier) spectrum within the angular fre-quencies Ω plusmn ∆Ω2 The sideband information always needs to be quotedA straight forward but rather untypical way is by adding subscripts whichleads to Xϑ

Ω∆Ω The classical analogue of the quadrature amplitude operator

XϑΩ∆Ω is the modulation depth of the optical field at modulation phase angle

ϑ and at angular modulation frequency Ω measured over the band ∆Ω lt 2ΩThe uncertainties of the statersquos quadrature amplitudes at different phases ϑare limited by a Heisenberg uncertainty relation see section 3 The secondkind of measurement is associated with the systemrsquos particle property andis given by the photon number operator n associated with a measuring timeinterval ∆t Its precise measurement requires a photon counter ideally withsingle photon resolution The measurement result obviously has a discretespectrum Continuous as well as discrete observables are usually subject toquantum uncertainties and thus quantum noiseUsually the measurementrsquos integration time and frequency band actuallydefine the physical system that is characterized In quantum optics experi-ments the interrogated physical system is called a lsquomodersquo

21 Definition of a lsquosingle modersquo

Let us define a light field or generally any quantum system to be asingle mode if it corresponds to the lsquosmallest entity of a waversquo In this caseits spectral and temporal distributions as well as waist size and divergenceare at their Fourier limits and all other properties such as optical axis waistposition and polarization are well defined For instance a linearly polarizedlongitudinal resonance of an optical standing-wave cavity defines such a singlemode if the cavity finesse is high and transversal modes are non-degenerateThe complete photo-electrical detection of a cavity mode however is notstraight forward Most quantum optical experiments are instead performedon propagating light In this case single modes are defined by spatial filtersand by temporal-spectral measurement windows both being at the Fourierlimit Examples for single modes are a laser pulse and a spectraltemporalcutout from a continuous observation of a quasi-monochromatic continuous-wave light beam in the spatial TEM00 mode both at the Fourier limits

In classical physics the only remaining free parameter of a given singlemode is its excitation energy In quantum physics the situation is differentFor a given energy a single mode can be in many different quantum states

10

which differ in their quantum statistics Examples are coherent states num-ber (Fock) states and squeezed states

22 Observations on squeezed states using a single PIN photo-diode

An ideal PIN photo-diode absorbs the full energy of a light mode andproduces one photo electron for every absorbed photon energy It uses theinternal photo-electric effect inside a semiconductor such as silicon or In-GaAs In contrast to avalanche photo-diodes PIN photo-diodes operatewith unity gain lsquoPINrsquo stands for lsquopositiversquo lsquointrinsicrsquo and lsquonegativersquo and isdescribing the doping of the semiconductor layers A PIN photo-diode is op-timally suited for the continuous monitoring of a rather bright light field ofup to several tens of milliwatts An example is the photo-diode in the outputport of a gravitational-wave detector as shown in Fig 2 (a) The prominentwavelength of 1064 nm which is emitted by NdYAG lasers has an opticalfrequency of ν = ω(2π) = 282 middot 1014 Hz The period of the field oscillationis a few femtoseconds and cannot be directly resolved with photo-electric de-tectors However variations of the electric field around the averaged opticalfield oscillation on longer time-scales can be resolved Applying an electronicbandpass filter at the sideband angular frequency Ω plusmn ∆Ω2 to the photovoltage provides information about the lsquodepth of the lightrsquos amplitude mod-ulationrsquo which is also called the lsquoamplitude of the amplitude quadraturersquo Itcan also slowly vary in time and reads

Xϑ=0

Ω∆Ω(t) equiv XΩ∆Ω(t) equiv X (1)

The subscript is usually skipped as it is done with the time dependenceas indicated on the right Applying the electronic bandpass filter in factdefines the mode of the light being detected The structure of the defini-tion in Eq (1) forms the basis of interferometric signals and quantum noisealso in the semi-classical case of coherent states Lets take an example Inthe recent observation of gravitational waves [Fig 1 bottom row in Abbott(2016)] the time-frequency representation of the gravitational-wave signalcorresponded to the amplitude quadrature amplitude XΩ∆Ω(t) of the inter-ferometer output light Note that a larger value of ∆Ω allows for changes ofthe quadrature amplitude on shorter time scales

If the light fieldrsquos lsquomodulation modersquo does not contain any quanta simplybecause there are no photons that have a frequency difference of plusmnΩ with

11

respect to the carrier it is in its ground state In this case lsquovacuum noisersquois observed which originates from the ground state uncertainty Since thevacuum noise only becomes measurable as a beat with a bright light field itcan also be seen as the carrierrsquos band-path filtered shot noise A modulationmode in a displaced vacuum state (a coherent state) corresponds to nonzerocoherent modulationThe measured level of the vacuum noise generally depends on the power ofthe bright carrier light and on the electronic amplification In any case itprovides the reference for certifying lsquosqueezingrsquo Observations using a singlePIN photo-diode require an independent measurement to quantify vacuumnoise A necessary condition is that attenuating the total fieldrsquos light powerresults in the same attenuation of the measured XΩ∆Ω values If they showa stronger attenuation a coherent modulation or thermal noise might bepresent If they show a weaker attenuation the photo-diode and its electron-ics might be saturated

Fig 2 (b) illustrates how a broadband squeezed field improves the mea-surement of an amplitude modulation in time domain based on a PIN photo-diode Shown is a simulated time sequence of XΩ∆Ω-data sampled from thephotoelectric voltage In this simulation all sideband frequencies from zero(DC) to the cutoff frequency of the detector electronics (Ωcut) are included(Ω = ∆Ω2 = Ωcut2) No additional band pass filter is applied making it amaximally broadband detection Although the data in Fig 2 (bi) contains aclassical amplitude modulation of the detected light this signal is not visibledue to random noise here representing shot noise Fig 2 (bii) shows thesame situation but with shot noise that is squeezed over the full detectionband The quantum uncertainty of the modulation depth is squeezed andthe classical signal becomes visible

It needs to be noted that a single PIN photo-diode can only measure theamplitude of the amplitude quadrature XΩ∆Ω(t) but not the non-commutingobservable the lsquoamplitude of the phase quadraturersquo

Xϑ=90

Ω∆Ω (t) equiv YΩ∆Ω(t) equiv Y (2)

For values that are small compared to the field strength of the bright fieldthe quantity Y approximately describes the bright fieldrsquos lsquophase modulationdepthrsquo

12

23 Observations on squeezed states using a balanced homodyne detector

-72

-69

-66

-63(i)

(ii)

(iii)

(a)

LO

(Squeezed)signal input

Phaseshifter

PD1

PD2

5050

01 0200 03

-75

-60

-57Balanced homodyne detector

(b)

Time [s]

Noi

se p

ower

[dB

m]

Figure 3 Balanced homodyne detection (BHD) ndash (a) Setup The quadrature atchoosable angle ϑ of the signal field is measured by overlapping the latter with a localoscillator (LO) field of the same mode parameters on a balanced beam splitter and record-ing the difference voltage from two PIN photo-diodes as shown In order to meet theBHD approximation the LO needs to be much more intense than the signal field A closeto perfect mode overlap between LO and signal input field is crucial For a non-perfectoverlap the detector measures the input state with unwanted contributions of the vacuumstate (b) Noise power measurements (i) on an electronically amplified and band-passfiltered quadrature amplitude of the vacuum field (Xvac

Ω∆Ω) (signal input blocked) (ii) on

a squeezed quadrature (XsqzΩ∆Ω) of a squeezed vacuum state (ϑ = 0) and (iii) on re-

spective quadratures of the same squeezed state where the phase angle ϑ was continuouslyshifted by changing the optical path length of the LO The measurement data shows about5 dB of squeezing and was first published in [Chelkowski et al (2007)] Ω2π = 5 MHz∆Ω2π = 100 kHz

In contrast to a single PIN photo diode a balanced homodyne detector(BHD) is suitable to measure the quantum statistic of all types of modu-lations ie for all angles ϑ Such a detector consists of two identical PINphoto-diodes a balanced beam splitter and an external homodyne local os-cillator field that is much brighter than the signal beam and that has anadjustable phase The signal beam corresponds to the squeezed field whichin many experiments is in a squeezed vacuum field having an optical powerthat usually corresponds to just a few photons per mode The two beams areoverlapped on the balanced beam splitter with close to perfect mode match-ing and the two interference outputs are focussed onto the photo diodes seeFig 3 (left) The electric output signal of the BHD is the difference of the

13

photo diode voltages The LO takes over the role of the carrier light fieldbut with the possibility to choose the phase shift ϑ This way eigenvaluesof X Y or Xϑ can be measured where the latter is given by the followinglinear combination of the first two

Xϑ(t) = cos(ϑ) X(t) + sin(ϑ) Y(t) (3)

If the modulation depths of signal and local oscillator beams are weak com-pared to their coherent amplitudes |α| and |αLO| the output voltage of aBHD corresponds to eigenvalues of the following operator

V (t) prop 2cos(ϑ) |αLO||α|+ |αLO| Xϑ(t) + |α| XϑLO(t) (4)

The lsquohomodyne approximationrsquo further involves |αLO| |α| such that theterm on the right can be neglected even if the local oscillator shows someclassical quadrature excitation The output voltage of a BHD is usually spec-trally analysed or at least spectrally filtered which removes the DC part infull analogy to a single photo diode (see previous subsection) Sampling thefiltered voltage provides eigenvalues proportional to the generalized quadra-ture amplitude in Eq (3)

V BHDΩ∆Ω(t) prop |αLO| Xϑ

Ω∆Ω(t) (5)

Fig 3 (a) shows the setup of a balanced homodyne detector for the char-acterization of squeezed states Setting ϑ = 0 eigenvalues of the ampli-tude modulation depths can be sampled from the photo voltage accordingto Eq (5) Setting ϑ = 90 eigenvalues of the phase modulation depths aremeasured The datarsquos expectation values 〈Xϑ〉 provide the coherent displace-ment of the squeezed state The datarsquos variances

∆2Xϑ equiv 〈(Xϑ)2〉 minus 〈Xϑ〉2 (6)

provide the statersquos (quantum) noise A pure squeezed state as well as asqueezed state that experienced photon loss have Gaussian quantum statisticsand are thus fully described by the expectation values and variances (first andsecond moments) of two orthogonal quadratures but only if one quadraturereflects the lowest quadrature variance

14

In most experiments with squeezed light the photo electric voltage ac-cording to Eq (5) is not sampled with a data aquisition system but the signalis directly fed into a spectrum analyser measuring the noise power of the volt-age If the expectation value 〈Xϑ〉 is zero the noise power is proportionalto the variance ∆2Xϑ in Eq (6) The reference for quantifying the squeezefactor is measured by blocking the (squeezed) signal field in Fig 3 (a) Themeasured vacuum noise level corresponds to the LOrsquos (electronically ampli-fied) shot noise level

Traces (ii) and (iii) in Fig 3 (b) show measured noise powers of the mod-ulation mode (Ω2π = 5 MHz ∆Ω2π = 100 kHz) being in a squeezed vac-uum state (i) is proportional to the variance of the ground state uncertainty∆2Xvac

Ω∆Ω (ii) is proportional to the quantum noise variance of the squeezed

quadrature amplitude ∆2XsqzΩ∆Ω (iii) is proportional to the quantum noise

variance of the quadrature amplitude with scanned phase ∆2XΩ∆Ω(ϑ(t))

To fully characterize a quantum state ie to do quantum state tomogra-phy [Vogel and Risken (1989)] a BHD is a prerequisite But also interfero-metric measurements with balanced homodyne detectors instead of singlePIN photo-diodes have several advantages A correctly implemented BHDreadily provides the vacuum noise level when the signal beam is blockedWith a BHD the optimum operating point of the interferometer is preciselyat a dark fringe If a perfect dark fringe can practically be achieved ampli-tude noise of the laser does not couple into the signal port If the interfero-meter has balanced arm length also frequency noise of the laser then does notcouple into the signal port Some quantum non-demolition schemes with theprospect of evading quantum radiation pressure noise require the detection ofa non-canonical quadrature angle [Jaekel and Reynaud (1990) Kimble et al(2001)] Here the adjustable phase of a BHD provides a straight forwardapproach The experimental exploration of BHDs for gravitational-wave de-tectors only has started recently [Steinlechner et al (2015)]

A light field can be analysed with respect to many different modulationfrequencies Ω The result constitutes a spectrum [Breitenbach et al (1998)]where in principle every modulation mode can be in a different quantumstate Fig 4 shows spectra of squeezed states from 5 MHz to 100 MHz with∆Ω2π = 1 MHz The lower curve shows the spectrum of the most strongly

15

-14-12-10-8-6- 4-20 2 4 6 8

10 12 14 16 18

6 8 10 20 40 60 80 1005

Squeezed noise

Anti-squeezed noise

Vacuum noise

Frequency [MHz]

Nor

mal

ized

noi

se v

aria

nce

[dB

]

Figure 4 Spectrum of quadrature amplitude variances ndash Shown are the quantumnoise properties of a large number of modulation modes having a resolution bandwidth of∆Ω(2π) = 1 MHz For all traces the balanced homodyne detector output was analysedwith a spectrum analyser Squeezing of XΩ∆Ω (bottom trace) and anti-squeezing of

YΩ∆Ω (top trace) versus f = Ω(2π) are shown relative to the vacuum noise variance Thespectrum below 5 MHz is not shown since it contained less squeezing due to laser relaxationoscillation in the carrier field as well as disturbances from back-scattered light [Vahlbruchet al (2007)] Disturbances at frequencies above 70 MHz originated from relatively largedetector dark noise which was subtracted from all traces shown here The thin linerepresents a theoretical model that takes into account for the linewidth of the squeezingcavity The data was first presented in in Ref [Mehmet et al (2010)]

squeezed variances in this case the variances of XΩ∆Ω The upper spectrum

shows the variance in the orthogonal quadrature amplitude (YΩ∆Ω) Allvariances are normalized to those of the corresponding vacuum state Thesqueeze factor reduces towards higher frequencies due to the linewidth ofthe squeezing cavity The anti-squeezing is always higher than the absolutevalue of the squeezing due to Heisenbergrsquos uncertainty relation and due to thepresence of optical loss The curves do not represent pure squeezed states butmixed squeezed states with a significant contribution from vacuum statesdue to optical loss Pure squeezed states can only be produced by making

16

the influence of all decoherence processes negligibleThe choice of the resolution bandwidth (RBW ∆Ω) during data taking

and processing defines the spectral-temporal modulation modes includingtheir number within the detected spectrum For any setting of the RBWthe quantum mechanical properties of the quadrature amplitudes XΩ∆Ω and

YΩ∆Ω [Caves (1985)] fully correspond to those introduced for quadratures instandard text books and which are reviewed in Sec 3

24 Observations on two-mode squeezed states using balanced homodyne de-tectors

Two-mode squeezed states are composed of two subsystems lsquoArsquo and lsquoBrsquoand are bi-partite entangled states with a Gaussian quantum statistic Toavoid conflicts with different usage of the term lsquomodersquo they can synony-mously be named lsquobipartite Gaussian entangled statesrsquo or lsquobipartite squeezedstatesrsquo which will be mainly used in this Review In the same way multi-partite Gaussian entangled states correspond to multi-partite squeezed states

The measurement observables that prove or disprove the bi-partite squeez-ing property are XA

Ω∆Ω minus XBΩ∆Ω and Y A

Ω∆Ω + Y BΩ∆Ω where the minus and

plus signs may be swapped Bi-partite squeezed states are precisely thosestates that were discussed by Einstein Podolsky and Rosen (EPR) in theirseminal paper [Einstein et al (1935)] Fig 5 shows a measurement resulton bi-partite squeezed light [Eberle et al (2013)] The variances of bothjoined observables are squeezed as shown in the two lower traces They wererecorded consecutively by adding or subtracting the outputs of two balancedhomodyne detectors But by interfering the subsystems on a beam splitterone could even measure both joined observables simultaneously This pos-sibility is correctly described in quantum theory since their commutator iszero

The so-called EPR paradox arises as follows If we either measure XAΩ∆Ω

and XBΩ∆Ω or Y A

Ω∆Ω and Y BΩ∆Ω it is obvious from the data in Fig 5 that

we can always predict the measurement result at subsystem lsquoBrsquo when know-ing the result at subsystem lsquoArsquo This seems to suggest that both quantitiesat lsquoBrsquo are precisely defined simultaneously before the measurement on lsquoArsquowhich contradicts the rigorous (and correct) interpretation of their non-zerocommutator that they are not precisely defined simultaneously

To solve this paradox EPR conjectured that the wavefunction as definedby quantum theory does not provide the full information This led to a

17

discussion of whether hidden variables existed that needed to be included ina complete theory of quantum mechanics (see also Bell [Bell (1966)]) Theexperimentally observed violation of Bellrsquos inequality [Bell (1964) Aspectet al (1981) Giustina et al (2013) Hensen et al (2015)] however ruled outthe existence of (local) hidden variables

Based on that the EPR paradox needs to be solved in a different wayContrary to what EPR assumed it is in fact possible to predict the valueof an arbitrary observable of a physical system A with certainty via a mea-surement on system B although this observable was not defined before themeasurement Without any interaction a measurement on subsystem lsquoArsquo notonly creates lsquorealityrsquo of eg XA

Ω∆Ω simultaneously lsquorealityrsquo is also created

regarding the observable XBΩ∆Ω describing subsystem lsquoBrsquo Here the term

lsquorealityrsquo has the meaning as defined by EPR [Einstein et al (1935)] Simi-larly the detection of one photon of a two photon entangled number statenot only produces the reality of this photon but also that of a second oneA discussion of Einstein-Podolsky-Rosen entanglement can also be found in[Schnabel (2015)] Note that the EPR paradox can also be described aslsquoquantum steeringrsquo [Schrodinger (1935) Cavalcanti et al (2009) Handchenet al (2012)] It should also be mentioned that two-mode squeezing beingdetected with BHDs and not with photon counters cannot be used to violatea Bell inequality The latter topic is outside the scope of this Review

Bi-partite squeezed states were first characterized with balanced homo-dyne detectors by the group of J Kimble in 1992 [Ou et al (1992)] Gener-ally the EPR paradox becomes more pronounced the stronger the bi-partitesqueezing is A measure of the strength of EPR entanglement was introducedby M Reid [Reid and Walls (1985)] According to this measure the resultin Fig 5 can be quantified to ε2 = 00309 where the critical value is one Itcorresponds to the strongest Gaussian EPR entangled state generated so far

For a long time it looked like that two-mode squeezed states are notuseful for laser interferometers The reason for that belief was that a laserinterferometer as any other measurement device too is built to measureone observable It seems to be ideal already if the quantum noise in thissingle observable is squeezed The increased quantum noise in the orthog-onal observable is not harmful in this case and squeezing in two differentobservables useless Only recently realistic scenarios were discussed in whichtwo-mode squeezing in fact does improve the performance of a laser inter-

18

-12

-10

-8

-6

-4

-2

0

0 1 2 3 4 5 6 7 8 9 10

100

dB

109

dB

Δ (XAvac + X B

vac ) = Δ (Y

Δ (XA + X B)

Avacminus Y B

vac )2 2

2

Δ (YAminus Y B)2

Nor

mal

ized

noi

se v

aria

nce

[dB

]

Times [s]

Figure 5 Two-mode squeezing measurement ndash For this measurement the outputs oftwo balanced homodyne detectors are added or subtracted and the variances (noise pow-ers) of the results recorded The upper trace was measured with modes lsquoArsquo and lsquoBrsquo beingin their ground states This measurement served as a reference level Strong two-modesqueezing was observed as shown by the lower two traces The sideband frequency wasΩ(2π) = 8 MHz and the resolution bandwidth was ∆Ω(2π) = 200 kHz The measure-ment results were first published in Ref [Eberle et al (2013)]

ferometer [Steinlechner et al (2013)] The proof-of-principle experiment isreviewed in Sec 7

25 Observations using photon counters

Alternatively to field quadratures an optical mode in a squeezed statecan also be characterized at least partly by detecting its photon numberdistribution For a pure squeezed vacuum state such a measurement wouldreveal the existence of solely even photon numbers including a large prob-ability for zero photons The average photon numbers of squeezed vacuumstates with feasible squeeze factors are very small of the order of one persecond and bandwidth in hertz see Fig 13 (a) ndash (c) A distribution with closeto zero probability of odd photon numbers however has not been measuredso far The reason is the lack of ideal photon counters First of all theefficiency of these detectors ie their probability of converting one photoninto one click and no photon into no click must be almost perfect lsquoLostrsquophotons as well as dark counts wash out the oddeven oscillations Further-more most detectors available can only distinguish between zero and one

19

photon This problem can be solved by distributing the squeezed mode ontoa large number of single photon detectors using an array of beam splitterssuch that all paths have a low probability of carrying more than one pho-ton Photon number measurements on squeezed vacuum states neverthelessplay an extremely important role in quantum optics When the squeezingstrength is very low the probability of detecting more than 2 photons can beneglected and the detection of a photon heralds the existence of a second one

0

300

600

900

1200

-40 -20 0 20 400

5

10

15

20

25

Two-

fold

coi

ncid

ence

s [1

(4s

)](a) (b)

Non-degeneratetwo-mode squeezing

5050

APDA1

APDA2

APDHerald

Photon coincidencedetection

Delay [ns]

Thr

ee-f

old

coin

cide

nces

[1

(4s)

]1500

Figure 6 Coincidence clicks from non-degenerate photon pairs ndash The first suchexperiment was reported in Ref [Hong and Mandel (1986)] (a) shows a setup with threeavalanche photo-diodes (APDs) for proving the successful heralding of a single photonnumber state (b) Histograms of the two-fold coincidence detections at APDHerald andAPDA1 (red) and at APDHerald and APDA2 (yellow) with theoretical models (solid lines)If the two-mode squeezing just carried one photon in each spatial subsystem the three-foldcoincidence detection should be zero Indeed the according histogram (grey points righty axis) shows only a few events These are produced by false (dark) counts of the APDsThe delay for the three-fold coincidences is defined as the time between counts at lsquoA1rsquo andlsquoA2rsquo given that the trigger APDHerald detected a photon (within a 100 ns time window)The data was taken on photons that were up-converted from 1550 nm to 532 nm and itwas first published in Ref [Baune et al (2014)]

If a mode of light is always excited by either zero or two photons lsquocondi-tionalrsquo or lsquoheraldedrsquo one-photon Fock states can be realized (Measurementson an ensemble of the n-photon Fock state would always produce the mea-surement result n ie Fock states have a zero photon number uncertaintyThey are also called lsquonumber statesrsquo) The above concept of producing aone-photon Fock state obviously requires the deterministic and balanced dis-tribution of the down-converted signal and idler fields into two different paths

20

In order to achieve this the signal and idler fields need to be non-degenerateUsually a mode in a squeezed state is composed of degenerate signal andidler fields and this degeneracy thus needs to be removed Possible waysare producing the down-converted fields at well separated wavelengths [Vil-lar et al (2005) Su et al (2006) Li et al (2010) Samblowski et al (2011)]separating the upper and lower sidebands belonging to an ordinary squeezedmode by frequency filters [Schori et al (2002) Hage et al (2010)] and us-ing spatial filters [Hong et al (1987)] A frequently used approach is usingtype II parametric down-conversion where the photons within a pair are al-ways orthogonally polarized [Ou et al (1992) Kiess et al (1993) Kwiat et al(1995)]The list of experiments with conditional or heralded photon number states islong They showed for instance nonclassical g(2)-functions [Hong et al (1987)]and violations of Bell inequalities [Weihs et al (1998)] Fig (6) shows a re-sult from a more recent experiment in which a bipartite-squeezed state withsubsystems at 1550 nm and 810 nm was produced the subsystem at 1550 nmsubsequently up-converted to 532 nm and the lsquoquantum non-Gaussianityrsquo ofheralded up-converted single photons demonstrated [Baune et al (2014)]Squeezed states are also the resource for the conditional generation of super-positions of coherent states [Ourjoumtsev et al (2006) Neergaard-Nielsenet al (2006)] and so-called N00N-states [Afek et al (2010)]

The generation of nonclassical states mentioned in the paragraph aboveis not stationary but relies on a probabilistic trigger event The produc-tion of squeezed states themselves usually happens in a stationary fashionThis distinction has an important consequence for applications of nonclassi-cal states in measurement devices Only (stationary) squeezed states allowfor a continuous improvement of a measurement Avoiding any loss of mea-suring time is generally of high relevance for the detection of short-livedsignals with unknown arrival time as well as for the detection of long-livedquasi-monochromatic signals since the signal-to-noise-ratio (SN) improveswith measuring time

26 Conclusions

The detection of squeezed light produces measurement results that canbe considered as remarkable Let us focus on experiments where a mode in abright coherent state is overlapped with a mode in a squeezed vacuum state

21

as shown in Figs (1) and (3) In both setups the squeezed vacuum field caneasily be blocked which allows us to compare the measurement results ona bright coherent state with and without the interference with the squeezedvacuum state Without squeezing the photo-electric detectors measure alarge number of photon events with a large quantization noise (shot noise)The large noise reflects the fact that all photon events were independent fromeach other as shown in Fig 2 (bi) With squeezing the photo-electric de-tectors again measure a large number of photon events with an expectationvalue that is even slightly higher but nevertheless the quantization noise ofall detected photons is significantly reduced Fig 2 (bii)Based on the discussion of EPR entanglement in Subsec 24 the photo-electric detection of the output light of a squeezing-enhanced laser inter-ferometer (with αlowastα1) produces the reality of photons This way we cankeep the lsquowave picturersquo in which no photons exist when light travels alongthe interferometer arms and when it interferes at the beam splitter Whenthe energy of the beam is elevating electrons to the conductance band ofthe photo-diodersquos semi-conductor n photon events simultaneously appearwithin the measuring interval with probability P (n) What conclusion hasto be drawn if the probabilities resemble a sub-poissonian statistic ndash Theoccurrence of photon events is still truly random but in this case not forindividual photons The occurrence of photons is correlated in such a waythat the probability of detecting an additional photon in the same time in-terval reduces the larger the number of already detected photons is Whatfollows from the discussion of EPR entanglement for a photon counting ex-periment with pure squeezed vacuum and ideal photon counters Here theprobabilistic detection of one photon entails the detection of a second onewith certainty With some smaller probability a third photon is detectedwhich entails the detection of a fourth photon with certainty and so on

If a photon of a mode that was not interrogated by the environment beforeis absorbed its reality is created in this very moment If the photon belongsto a squeezed state this process instantaneously influences the probability ofother photons becoming reality

Of course a more general statement can be made based on the insightthat interaction with the environment creates the reality of any kind ofquanta including electrons atoms and molecules

22

3 Theoretical description of squeezed states

31 The quadrature amplitude operators

Consider a single mode of light at optical frequency ω Its Hamiltonoperator reads

Hω = ~ω(n+

1

2

)= ~ω

(adaggerωaω +

1

2

)= ~ω

(X2ω + Y 2

ω

) (7)

where n is the photon number operator and aω and adaggerω are the annihilationand creation operators which obey the commutation rule

[aω a

daggerω

]= 1 The

operator aω has a complex-valued dimensionless eigenvalue spectrum andcorresponds to the complex amplitude αω in classical optics Xω and Yω arethe hermitian amplitude and phase quadrature operators The eigenvaluesof the quadrature operators are also dimensionless and proportional to theelectric fields at the oscillationrsquos antinode and at the oscillationrsquos node Inthe above equation they are defined such that their variances are ∆2Xω =∆2Yω = 14 if the oscillator is in its ground state ie if 〈n〉 = 0

Although Eq (7) simply describes the energy of an harmonic oscillator itis the essence of quantum theory since it mathematically describes the wave-particle dualism Whereas the eigenvalues of n have a discrete spectrum theeigenvalues of Xω and Yω have a continuous spectrum In classical opticsthe phase quadrature is zero In quantum optics its expectation value is alsozero but its uncertainty contributes to the overall energy

Eq (7) describes a cavity mode as well as a section that is cut from apropagating quasi-monochromatic light beam The latter example is of highrelevance in actual experiments By setting the sectionrsquos time window ie themeasuring time interval the time-frequency (lsquomodulationrsquo) mode is defined

The quadrature operators introduced in Eq (7) and displayed in Fig 7 donot correspond to lsquoXrsquo and lsquoY rsquo that are of relevance in laser interferometry andin optical communication and which were already discussed in Subsec 22and 23 The optical frequency of visible and near-infrared light is far too highto be transferred to an oscillation of photoelectric voltage Quite general alaser interferometer targets signals at audio or radio band frequencies Ωi ωSuch a measurement is achieved as stated before by decomposing the photo-electric voltage from the photo diode at the interferometer output into asingle-sided spectrum (positive frequencies only) of intervals of Ωplusmn∆Ω2

23

(a)

(b)

(c)

(d)

2π 4π

2π 4π

2π 4π

2π 4π

Figure 7 Phase spaces and electric field oscillations of monochromatic lightndash Top Left Monochromatic light in a coherent state is represented by a phasor (whitearrow) including its quantum uncertainty (white dashed circle and fuzzy area) located inthe phase-space spanned by the quadratures Xω and Yω When the phase space rotateswith optical frequency ω2π the projection of the quantum phasor onto a fixed (vertical)axis corresponds to the electric field E(t) as shown on the right side (a) Weakly displacedcoherent state (b) Corresponding amplitude squeezed state The electric field uncertaintyaround the zero average field region is anti-squeezed (c) Vacuum state at the same opticalfrequency (d) Corresponding squeezed vacuum state The meaning of the uncertaintycould be carved out by supplementing them with monochromatic waves all having theoptical frequency ω2π Changing amplitudes then display amplitude quadrature noiseChanging shifts along the time axis model the electric field uncertainty at the expectedzero crossing They are not implemented in the graphics here however since any of thosewaves does not exist due to Heisenbergrsquos uncertainty relation

24

The signals as well as the quantum uncertainties carried by a beam oflight are thus described by a spectrum of pairs of non-commuting quadratureoperators Mathematically every such operator is defined by an integral overthe Fourier components within the bandwidth The spectral weighting of theFourier components is called the lsquowindow functionrsquo By going to sidebandintervals a spectrum of a new type of optical mode is defined which describesthe modulation of the electric field in the respective frequency interval Ω plusmn∆Ω2 In this Review we call it a lsquomodulation modersquo

The quadrature operators that are defined around a modulation frequencyΩ with a bandwidth of ∆Ω are the quadrature amplitude operators thatare relevant in laser interferometry Whenever they are not related to aspecific band we use the short form XΩ∆Ω(t) equiv X and YΩ∆Ω(t) equiv Y cfEqs (1) and (2) These operators can slowly vary with time where thetime dependence is limited by ∆Ω (The time dependence is not due toquantum uncertainty which usually is time independent but for instancedue to the time dependence of the signal eg a passing gravitational wave)Let us consider now a pair of quadrature operators for a particular sidebandΩ plusmn ∆Ω2 The Hamilton operator of the corresponding modulation modeis found by switching to the frame rotating at optical frequency ω Thetransition is done by applying the unitary transformation U = exp(iωadaggerat)generating a new Hamiltonian H = U daggerHωU minus i~UpartU daggerpartt The Hamiltonianof the modulation mode reads

H = ~Ω

(nΩ +

1

2

)= ~Ω

(adaggera+

1

2

)= ~Ω

(X2 + Y 2

) (8)

where nΩ is the (occupation) number operator for the modulation modeand a and adagger its annihilation and creation operators The commutation rule[a adagger

]= 1 is unchanged X and Y are the amplitude and phase quadra-

ture amplitude operators respectively They correspond to the depth of theamplitude modulation and for weak excitations to the depth of the phasemodulation respectively They are the conventional hermitian field operatorsin experimental quantum optics Note that modulation modes at angularfrequency Ω can be described by a superposition of three optical frequenciesa carrier at ω an upper sideband at ω + Ω and a lower sideband at ω minus ΩThe quantum mechanical description of modulation states in connection tooptical carrier and upper and lower sidebands is known as the lsquoTwo-PhotonFormalismrsquo [Caves and Schumaker (1985) Schumaker and Caves (1985)]

The quadrature amplitude operators in Eq (8) are again defined such

25

that the variances of the uncertainty of a modulation field in its ground stateor in a coherent state are

∆2Xvac = ∆2Yvac = 14 (9)

Generally quadrature operators X and Y as defined in Eqs (7) and (8) arethe real and imaginary parts of the annihilation operator

a = X + iY hArr adagger = X minus iY (10)

hArr X =1

2

(a+ adagger

) Y =

1

2i

(aminus adagger

) (11)

They satisfy the commutation relation[X Y

]=i

2 (12)

and their variances are limited by a Heisenberg uncertainty relation of thefollowing form

∆2X∆2Y ge 1

16 (13)

A quantum state is called a ldquosqueezed staterdquo [Bachor and Ralph (2004)]if ∆2Xϑ lt 14 for an arbitrary field quadrature Xϑ = X cosϑ + Y sinϑ seeEq (3) The angle of the lowest variance below 14 is called the squeeze angleθ The largest factor by which the variance is below 14 is called the squeezefactor often given on a decibel (dB) scale using the following transformation

minus 10 middot log10

(∆2Xθ

∆2Xvac

) (14)

The squeeze factor can also be described by the squeeze parameter r

eminus2r =∆2Xθ

∆2Xvac

(15)

When a squeezed state experiences optical loss it remains squeezed butthe squeeze factor is reduced Also the statersquos purity is reduced ie theproduct of the quadrature uncertainties increases above the minimum valueOptical loss corresponds to mixing the state with the vacuum state Let∆2Xϑ be the variance of a quadrature amplitude ∆2Xvac the variance of the

26

(quadrature angle independent) ground state uncertainty and (1 minus η2) therelative energy loss Then the resulting quadrature variance reads

∆2Xprime

ϑ = η2∆2Xϑ + (1minus η2)∆2Xvac (16)

To maximize the benefit from squeezed states in applications stronglysqueezed states need to be generated and optical loss minimized Opticalloss occurs due to absorption and scattering in the optical components inthe path of the squeezed beam including the squeezing resonator itself anddue to non-perfect matching to the interferometer mode non-perfect inter-ference contrast of the interferometer and non-perfect quantum efficiency ofthe photo diodes The sum of all losses including those outside the inter-ferometer need to be less then 10 to allow a nonclassical quantum noisesuppression of a factor of 10 in power ie 10 dB

32 Phase space representations of squeezed states

The Wigner function ndash The properties of squeezed states are nicely dis-played by the Wigner function W (X Y ) [Wigner (1932)] An example interms of a squeezed vacuum state is shown in Fig 8 It is a quasi-probabilitydistribution which contains the statersquos full information including its quan-tum statistic There are two ways how a Wigner function provides a sufficientcriterion for nonclassicality First by containing negative values second byfeatures that have a smaller (squeezed) width compared with the Wignerfunction of the ground state Integrating the Wigner function over Y pro-vides the probability density of measurement results ie of the eigenvaluesof the observable X and vice versa

infinintminusinfin

W (X Y )dY = p(X)

infinintminusinfin

W (X Y )dX = p(Y ) (17)

where p(X) and p(Y ) are the observed probability distributions also exem-plarily shown in Fig 8

The ground state coherent states as well as (quadrature) squeezed stateshave quadrature eigenvalue probability densities that are Gaussian TheirWigner functions are also Gaussian and thus entirely positive Wigner func-tions of other nonclassical states for instance Fock states exhibit negativevalues For this reason the Wigner function is called a quasi -probabilityfunction

27

X Y XY

Figure 8 Wigner function and its projections ndash Displayed is the full information ofa squeezed vacuum state The quasi probability density of the Wigner function (centre)is plotted along a third dimension and is color-coded Also shown are the actual mea-surement results from which the Wigner function is reconstructed They are representedby the squeezed and anti-squeezed Gaussian projections onto the X and Y axes Theirprobability densities p(X) and p(Y ) are given on the vertical axes The Gaussian measure-ment statistic of the first clearly shows squeezing compared to the ground state statistic(dashed) The squeeze factor is 116 dB and the squeeze parameter r = 1335 [Mehmetet al (2010)]

Fig 9 shows the Wigner functions for (a) a coherent state (b) the ground(vacuum) state (c) a displaced squeezed state and (d) a squeezed vacuumstate All Wigner functions describe a modulation of the carrier light atsideband frequency Ω integrated over the frequency interval ∆Ω The car-rier light is not part of these Wigner functions The displacement in (a)represents a classical amplitude modulation (b) corresponds to the absenceof any photons with a frequency offset of plusmnΩ from the local oscillator field(c) and (d) represent states whose amplitude modulation depth is more pre-cisely defined than that of the ground state Fig 10 shows Wigner functionspectrum for a broadband squeezed vacuum field Every Wigner functiondescribes the modulation field at some modulation frequency Ωi integrated

28

X

Y

X

(a) (b)

X X

(c) (d)

Y

Y Y

Figure 9 Simplified representation of Wigner functions ndash The darker the shadedareas the larger is the phase-space quasi-probability Shown are four different (time-independent) states of a modulation mode at frequency Ω for a specific resolution bandwidth ∆Ω Panel (a) represents a coherent state the displacement (α) corresponds to aclassical amplitude modulation Panel (b) represents the ground (vacuum) state (c) adisplaced squeezed state and (d) a squeezed vacuum state both with squeeze angle θ = 0The latter is in analogy to Fig 8 Again the light field that carries the modulation is notpart of the pictures

over the resolution bandwidth (RBW) of ∆Ω

The Glauber-Sudarshan P -function ndash The P -function [Glauber (1963)Sudarshan (1963)] is calculated by de-convoluting the Wigner function fromthe ground state uncertainty [Gerry and Knight (2005)] For displaced vac-uum states (coherent states) the P -function corresponds to a displaced δ-function The mathematical expression of the P -function of a squeezed statecontains infinitely high orders of derivatives of the δ-function [Vogel andWelsch (2006)] Such a function contains negativities but cannot be dis-played It is possible however to define a phase-space quasi probabilityfunction for squeezed states that can be displayed and that does show neg-

29

0

Ω1

Ω2

Ω

Single-sided spectrum

Y

X

Figure 10 Hint of a Wigner function spectrum ndash A single-sided spectrum (positivefrequencies only) with respect to the carrier field can be used to visualize a broadbandsqueezed field Shown are two examples displaying a squeezed vacuum state at Ω1 anda displaced amplitude squeezed state at Ω2 The individual Wigner functions cover theresolution bandwidth ∆Ω gt 0 (not shown) In general the squeezing strength as well asthe squeeze angle and the displacement are a function of sideband frequency

ativities as a sufficient and necessary condition for certifying the squeezingeffect This lsquononclassicality functionrsquo is calculated by de-convoluting theWigner function from an uncertainty distribution that is steeper than theGaussian distribution A pronounced negativity of a squeezed vacuum stateof up to 69 standard deviations was found [Kiesel et al (2011)]

The double-sided phasor picture ndash This phasor picture links quantumstates of modulations with the quantum states of the contributing opticalfields [Bachor and Ralph (2004)] and is mathematically described by thetwo-photon-formalism [Caves and Schumaker (1985) Schumaker and Caves(1985)] Generally a weak amplitude or phase modulation at frequency Ω ofa carrier field at optical frequency ω can be understood as the carrierrsquos beatwith two optical frequencies at ωplusmnΩ The double-sided phasor picture is ableto display a spectrum of different and independent modulation frequenciesin the rotating frame of the carrier field The carrier light field is time-independent but the upper and lower sidebands are not They rotate withplusmnΩi(2π) respectively around the frequency axis

30

ω0

Yω

Xω

Upper sideband

Lower sideband

ω0+Ω1

ω0+Ω2

ω0-Ω2

ω0-Ω1

ω

Upper sideband

Lower sideband

Yω

Xω

ω0+Ω1

ω0+Ω2

ω0-Ω2

ω0-Ω1

ω

ω0

Figure 11 Double-sided phasor pictures ndash Phase spaces at optical frequency ω0 plusmnΩirotate around the frequency axis with sideband frequency plusmnΩi Its sign determines thedirection of rotation with respect to the local oscillator in the rotating frame at ω0 Apair of phase spaces need to be superposed to provide a description of a modulation fieldat |Ωi| Top Amplitude quadrature squeezed field with a coherent displacement at |Ω2|The displacement corresponds to a classical amplitude modulation The uncertaintiesof all optical frequencies are circular and larger than that of the ground state (dashed)Quantum correlations are indicated by additional symbols More details are given in themain text Bottom Corresponding spectrum of (displaced) vacuum states which do nothave any quantum correlations

31

Fig 11 shows such a double-sided phase space picture where the carrierrsquosmodulation at Ω1 is in a squeezed vacuum state and where the modulationat Ω2 is in a displaced squeezed state The picture shows how a classicalamplitude modulation as well as the quantum statistic of a modulation fieldis decomposed into contributions from upper and lower sidebands For asqueezed modulation field the upper and lower sidebands show no squeezedbut circular thermally excited quantum uncertainties The uncertainties of apair of sidebands however show correlations as well as anti-correlations InFig 11 these (anti-) correlations are marked with times and + for the modulationfrequency Ω1 and with N and bull for the modulation frequency Ω2

33 Covariance matrix representation of (single-party) squeezed states

Since squeezed states have a Gaussian quantum statistic four numbersare sufficient for their full description These numbers are the second mo-ment of the quadrature amplitude showing the strongest squeezing and thesecond moment of its orthogonal quadrature amplitude as well as their firstmoments describing the displacement These four numbers are sufficientto calculate the Wigner function shown in Fig 8 In general the quadra-ture of strongest squeezing is not perfectly aligned with one of the axesof the measurementrsquos coordinate system The so-called covariance matrix(VXXVXY VY XVY Y ) [Simon et al (1994)] accounts for phase space rotationsand enables the calculation of how these states evolve within an interfero-metric arrangement Their components are normalized to the vacuum noisevariance ∆2Xvac = ∆2Yvac and read

VXY =

langXY + Y X

rangminus 2

langXranglang

Yrang

2∆2Xvac

(18)

The following examples represent the ground state a pure 10 dB am-plitude quadrature squeezed state and a pure 10 dB squeezed state with asqueeze angle of 45

Vvac =

(1 00 1

) V0

01 =

(01 00 10

) V45

01 =

(505 495495 505

) (19)

with V4501 =RT

45V001R45 where Rα = (cosα minussinα sinα cosα) is the rota-

tion matrix

32

34 Phase space representation of two-mode (bi-partite) squeezed states

XA

YA

XB

YB

Figure 12 Bi-partite squeezed vacuum state ndash Shown is a Wigner-function-basedphase space representation in close analogy to the single party version in Fig 9 (d) Thepicture describes a single modulation at frequency Ω with bandwidth ∆Ω Measurementsat party A and B reveal local Wigner functions that correspond to thermal states sincethe uncertainties (indicated by the color and by the large circles) are symmetric and largerthan that of the ground state (indicated by the small dashed circles) The uncertaintieshowever show correlations and anti-correlations here indicated by + and times respectivelyThe strength of these (anti-)correlations are indicated by ellipses Bi-partite squeezingie entanglement is present if the short axes of the ellipses are shorter than the diameterof the ground state uncertainty The picture in fact represents Einstein-Podolsky-Rosenentanglement [Einstein et al (1935)] From a measurement of XA or YA the correspondingmeasurement result at party B can always be inferred with an uncertainty that is smallerthan the ground state uncertainty

A bi-partite state enables a measurement on subsystem A and simulta-neous a measurement on subsystem B For a large number of simultaneousensemble measurements of the same quadrature amplitude Xϑ the followingtwo joint quadrature variance can be calculated

∆2(XAϑ plusmn XB

ϑ ) (20)

A state that is symmetrically shared between two parties (A and B) is calleda two-mode squeezed state if the variances of joint quadrature measurementsfulfill the following inequality [Duan et al (2000)] ie

∆2(XA minus XB)

∆2(XAvac minus XB

vac)+

∆2(Y A + Y B)

∆2(Y Avac + Y B

vac)lt 2 (21)

33

with ∆2(XAvac plusmn XB

vac) = ∆2(Y Avac plusmn Y B

vac) = 2∆2Xvac A lsquotwo-mode squeezedstatersquo reveals entanglement in the second moments of the measurement statis-tics It is thus a lsquobi-partite Gaussian entangled statersquo

Fig 12 displays a (pure) bi-partite squeezed vacuum state distributedbetween A and B The state shows full symmetry regarding its subsystemsat the two sites The large circles and the colored area represent Wignerfunctions of the subsystems Measurements of the quadrature amplitudesXA YA XB and YB show identical variances and the correlations and anti-correlations have identical strength since ∆2(XA minus XB) = ∆2(YA + YB) lt12 for our normalization of quadrature amplitudes having a ground statevariance of 14

Generally a symmetric bi-partite squeezed state fulfills another quantita-tive (Gaussian) entanglement criterion if less than 50 of the vacuum state issymmetrically mixed into the initially pure state Bi-partite squeezed statesare always entangled but in this case they are even Einstein-Podolsky-Rosen(EPR) entangled [Reid (1989)] allowing the demonstration of the quantumsteering effect [Einstein et al (1935) Schrodinger (1935) Reid (1989) Cav-alcanti et al (2009)] The first such experiment was performed by Ou etal [Ou et al (1992)] using type II parametric down-conversion (PDC) Laterexperiments produced bi-partite squeezed vacuum states by overlapping twosqueezed vacuum states each produced with type I PDC on a balanced beamsplitter and used the entangled output for the demonstration of quantumteleportation [Furusawa et al (1998) Bowen et al (2003ca)] The criterionin Eq (21) and the EPR criterion from [Reid (1989)] was experimentallycompared in Ref [Bowen et al (2003b)] The steering effect in asymmet-ric bi-partite squeezed states were recently experimentally characterized inRef [Handchen et al (2012)]

Fig 12 shows features similar to those in the top part of Fig 11 Thisis not a coincidence and shows that a bi-partite squeezed state can also begenerated by spatially splitting the upper and lower sideband of a (single-party) squeezed state This was first experimentally demonstrated by thegroup of E Polzik [Schori et al (2002)] and later used for EPR multiplex-ing of a single longitudinal mode of a squeezing resonator [Hage et al (2010)]

35 Covariance matrix representation of bi-partite squeezed states

Also the full information of bi-partite states including the entanglementcan be cast by the covariance matrix [Simon et al (1994)] which can be used

34

to calculate the propagation of these states in laser interferometers Againall variances are normalized to the vacuum noise variance in full analogy toEq (18) The generic bi-partite covariance matrix has dimension 4times4 andreads

Vbp =

VXAXA

VXAYA VXAXBVXAYB

VYAXAVYAYA VYAXB

VYAYBVXBXA

VXBYA VXBXBVXBYB

VYBXAVYBYA VYBXB

VYBYB

with

VXAYB =

langXAYB + YBXA

rangminus 2

langXA

ranglangYB

rang2∆2Xvac

ϑ

(22)

Due to the symmetry in Eq (22) the 4times4 covariance matrix is fully spec-ified by just ten independent coefficients If the phase spaces at A and Bare aligned along the strongest correlations and anti-correlations the ma-trix components referring to different quadrature amplitudes eg VXAYA arezero Such entangled states can be produced by overlapping two squeezedfields with a squeeze angle difference of 90 on a balanced beam splitter

A symmetric bi-partite squeezed vacuum state which is also called anlsquoS-classrsquo [DiGuglielmo et al (2007)] bi-partite squeezed vacuum state shows(anti-)correlations in two joint quadratures as defined in Eq (21) For a puresuch state of 10 dB squeezing the covariance matrix reads

Vbp10|10 =

505 0 495 0

0 505 0 minus495495 0 505 0

0 minus495 0 505

The following covariance matrix describes a so-called lsquoV-classrsquo 10 dB bi-partite squeezed vacuum state Here only one joint quadrature shows 10 dBsqueezing whereas the orthogonal joint quadrature shows vacuum noise Thestate is obtained by overlapping one 10 dB squeezed state with a vacuumstate on a balanced beam splitter

35

Vbp0|10 =

055 0 045 0

0 55 0 minus45045 0 055 0

0 minus45 0 55

The first measurement of all elements of such a covariance matrix was achievedin [DiGuglielmo et al (2007)]

36 Photon numbers of squeezed states

In contrast to the ground state squeezed vacuum states do have photonexcitations As said earlier quantum theory links the wave and the particlepictures Indeed the squeeze factor of a modulation mode is directly con-nected to a certain photon number excitation Squeezed states of light areproduced via spontaneous photon pair generation eg by parametric down-conversion The following operator S is called the lsquosqueeze operatorrsquo [Gerryand Knight (2005)] It creates and annihilates photon pairs

|r θ〉 = S(r θ) |0〉 (23)

where |r θ〉 is a squeezed vacuum state with squeeze parameter r and squeezeangle θ and |0〉 is the vacuum state The definition of the squeeze operatoris

S(r θ) = exp

[1

2

(reminusiθa2 minus reiθadagger 2

)] (24)

The following shows that this definition indeed results in a state with squeezedquadrature amplitude variances Lets set θ = 0

〈0| Sdagger(r 0) X S(r 0) |0〉 =1

2〈0| Sdagger(r 0)

(a+ adagger

)S(r 0) |0〉 (25)

〈0| Sdagger(r 0) Y S(r 0) |0〉 =i

2〈0| Sdagger(r 0)

(aminus adagger

)S(r 0) |0〉 (26)

Using the Baker-Hausdorff formula we get

Sdagger(r θ) a S(r θ) = a cosh r minus adaggereiθsinh r (27)

Sdagger(r θ) adagger S(r θ) = adaggercosh r minus aeminusiθsinh r (28)

36

Since 〈0| X |0〉 = 〈0| Y |0〉 = 0 also Eqs (25) and (26) are zero To finallycalculate the variances we need

〈0| Sdagger(r 0) X2 S(r 0) |0〉 =1

4〈0| Sdagger(r 0)

(a2 + adaggera+ aadagger + adagger2

)S(r 0) |0〉

〈0| Sdagger(r 0) Y 2 S(r 0) |0〉 = minus1

4〈0| Sdagger(r 0)

(a2 minus adaggeraminus aadagger + adagger2

)S(r 0) |0〉

Given that SSdagger is the identity and using again Eqs (27) and (28) we get theexpected variances

∆2X =1

4

[cosh2 r minus 2cosh r sinh r + sinh2 r

]=

1

4eminus2r

∆2Y =1

4

[cosh2 r + 2cosh r sinh r + sinh2 r

]=

1

4e2r

Since the squeeze operator can only create and annihilate photon pairsa squeezed vacuum state without photon loss must correspond to an evennumber of photons But not only photon loss also a coherent displacementleads to flattening out the odd-even oscillations The probability of detectingN photons in a pure displaced squeezed state are derived for instance in[Gerry and Knight (2005)] and read

P (N) = |〈N |α r θ〉|2 =(05 tanh r)N

N cosh rexp

[minus|α|2 minus 1

2(αlowast2eiθ + α2eminusiθ)tanh r

]times∣∣∣HN

[(αcosh r + αlowasteiθsinh r)

radiceiθsinh(2r)

]∣∣∣2 (29)

where HN is the N th Hermite polynomialFig 13 shows the photon number distributions for 5 different pure squeezed

states according to Eq (29) Panels (a) to (c) show squeezed vacuum stateswith 43 dB 86 dB and 172 dB of squeezing Panel (d) shows the moregeneral case of a squeezed state with a coherent displacement α 6= 0 Dueto θ = 0 the state is amplitude quadrature squeezed Panel (e) refers to thecorresponding phase quadrature squeezed state For comparison panel (f)shows the photon number distribution of the coherent state with the samedisplacement

The panels in Fig 13 represent the diagonal elements of the statersquos den-sity matrix in number basis Only the latter also contains the coherences

37

002

004

006

000 000

002

004

006

008

403020100 Photon number n

403020100Photon number n

Pro

babi

lity

403020100000

005

010

015

100 2 4 6 8100 2 4 6 800

04

06

02

100 2 4 6 800

03

02

01Pro

babi

lity

Pro

babi

lity

08

00

04

06

02

08

(a) (b)

(c) (d)

(e) (f)

Figure 13 Photon number distributions ndash All panels represent pure states (a) 43 dBsqueezed vacuum state (r = 05 α = 0) (b) 86 dB squeezed vacuum state (r = 1 α = 0)(c) 172 dB squeezed vacuum state (r = 2 α = 0) (d) displaced 86 dB squeezed state(r = 1 θ = 0 α = 4) (e) displaced 86 dB squeezed state (r = 1 θ = π2 α = 4) (f)Coherent state (r = 0 α = 4) The average photon numbers are about 027 14 131174 174 and exactly 16 see Eq (30)

between photon numbers [Gerry and Knight (2005)] Figures as shown heregenerally do not give full descriptions of the states

38

A squeezed vacuum state (r 6= 0) always has a non-zero photon numberand can not be the ground state The average photon number of a puresqueezed vacuum state can be calculated using Eq (8) With the maximallysqueezed quadrature variance ∆2Xsqz the average photon number is given by

n = 〈n〉|α=0θr〉 = ∆2Xsqz +(∆2Xsqz)

minus1

16minus 1

2=eminus2r

4+e2r

4minus 1

2 (30)

with the vacuum noise variance normalized to one quarter A coherent dis-placement further adds |α|2 photons on average

4 Squeezed-light generation

41 Overview

Squeezed light was first produced in 1985 by Slusher et al using four-wave-mixing in sodium atoms in an optical cavity [Slusher et al (1985)]Shortly after squeezed light also was generated by four-wave-mixing in an op-tical fibre [Shelby et al (1986)] and by degenerate parametric down-conversion(PDC) in a 2nd-order nonlinear crystal placed in an optical cavity [Wu et al(1986)] The pumped cavity was operated below its oscillation thresholdie the parametric gain did not fully compensate the round trip losses whichis also called lsquocavity-enhanced optical-parametric amplification (OPA)rsquo

The early day experiments achieved squeeze factors of a few percent up toabout 3 dB Today squeeze factors of more than 10 dB are directly observedin several experiments [Vahlbruch et al (2008) Eberle et al (2010) Stefszkyet al (2012) Vahlbruch et al (2016)] All of them are based on cavity-enhanced OPA (below threshold) The parametrically amplified mode isdegenerate ie signal and idler modes are identical In particular the down-conversion process is of lsquotype Irsquo which means that the amplified mode has awell-defined polarization Squeezed states can also be generated above oscil-lation threshold In Refs [Villar et al (2006) Jing et al (2006)] bi-partitesqueezing was generated with above-threshold PDC Both experiments usedtype II PDC which provides orthogonally polarized signal and idler fieldsType II PDC below threshold was also used to generate squeezed and bi-partite squeezed fields [Grangier et al (1987) Ou et al (1992)] All these

39

experiments were performed in the continuous-wave regime which is also thefocus of this Review Squeezed states of modulations of trains of laser pulseshowever have been also generated since the 1980s using either PDC or theoptical Kerr effect [Slusher et al (1987) Bergman and Haus (1991) Our-joumtsev et al (2006) Dong et al (2008)] For an overview of the develop-ments in squeezed-light generation in the continuous-wave as well as pulsedregime see Ref [Bachor and Ralph (2004)] Squeezed-light generation inopto-mechanical setups [Aspelmeyer et al (2014)] which use the intensitydependent phase shift from radiation pressure was discussed in Refs [Paceet al (1993) Rehbein et al (2005) Corbitt et al (2006)] and recently ex-perimentally achieved by several groups [Brooks et al (2012) Safavi-Naeiniet al (2013) Purdy et al (2013)]

42 Degenerate type I optical-parametric amplification (OPA)

This section provides a graphical description of how degenerate type IOPAPDC turns a vacuum state into a squeezed vacuum state and a coher-ent state into a displaced squeezed state The process requires a bright pumpfield and a 2nd-order nonlinear crystal For simplicity we set all nonlinearitiesabove 2nd-order to zero

Let us consider a short segment of the second-order nonlinear crystalpumped with light of optical frequency 2ν All other modes that enter thecrystal shall not contain any photons ie are in their vacuum states Ofthese the only mode of interest is that at optical frequency ν which spatiallyoverlaps with the pump mode Fig 14 shows the total electric field of theoptical input E in and the 2nd-order nonlinear dielectric polarisation of thecrystal P The latter is proportional to the total electric field of the outputEout The pump field at 2ν periodically drives the vacuum field at ν betweenregions of low and high polarisation This process transforms the vacuumstate into a squeezed vacuum state in the output [Bauchrowitz et al (2013)]The output further contains the hardly depleted pump field and frequencydoubled parts of the pump field at 4ν It is again emphasized that Fig 14displays OPA in a small segment of the crystal In reality the nonlinear effectaccumulates over the crystal length or even over several passages since thecrystal is usually put into an optical resonator A noticeable effect is achievedif all infinitesimal contributions constructively interfere This is achieved incase of phase matching ie if the wave fronts of the modes at 2ν and νpropagate with the same speed and thus do not run out of phase Note that

40

t

t

Squeezing

Figure 14 Optical parametric amplification of a vacuum state ndash The upper leftcorner shows the crystal polarization P(E) = ε0

(χ(1)E + χ(2)E2

) ie the separation of

charge carriers by the electric component of an optical field E The graph illustrates howan input quantum field (from below) is projected into an output quantum field (towardsthe right) In the example shown here the input field is composed of a classical pumpfield E in

2ν at frequency 2ν and zero-point fluctuations E inν of a field at frequency ν cf

Fig 7(c) The superposition E in of these two fields is transferred into a time-dependentdielectric polarization that is the source of (and thus directly proportional to) the elec-tric component of the output field Eout The quantum uncertainty of the output fieldshows a phase-dependent (parametric) amplification at frequency 2ν Spectral decompo-sition of the output field Eout reveals coherent amplitudes at frequencies 2ν and 4ν and asqueezed vacuum state Eout

sqzν cf to Fig 7(d) The concept of this figure was published inRef [Bauchrowitz et al (2013)]

41

t

t Amplitudequadraturesqueezing

Figure 15 Optical parametric amplification of a coherent state ndash The pictureshows how a displaced vacuum state is transformed into a displaced squeezed state Thepumprsquos electric field is maximal when the input field at fundamental frequency ν shows azero crossing The phase relation described results in an output state that is amplitudequadrature squeezed If the pump field at the input in phase was shifted by half of itswavelength the squeezed output field were a phase quadrature squeezed The squeezinggeneration displayed here corresponds to the transition from Fig 7(a) to Fig 7(b) but withan additional reduction of the displacement The concept of this figure was published inRef [Bauchrowitz et al (2013)]

in actual squeezing experiments the 4ν component is usually suppressed byphase miss-matching

Fig 15 shows the same process but now for an input field at frequencyν in a coherent state In this case the relative phase between the two inputstates ∆ϕ = ϕ2ν minus 2ϕν is relevant In Fig 15 the relative phase is set suchthat the expectation value of the field at frequency ν is zero when the pump

42

(a) (b)

(c) (d)

X

Y

X

YΔϕ = ndash50deg

X

Y

Δϕ = 0deg

X

Y

Δϕ = 90deg

Figure 16 Phase-space illustration of degenerate OPA ndash The (displaced) dashedcircle in each diagram represents the uncertainty of the initial state at optical frequencyν The (displaced) shaded area represents the state after degenerate optical parametricamplification All quantum uncertainties shown correspond to pure states The boldgreen arrow describes the bright second-harmonic pump field whose uncertainty can beneglected The phase between the 2nd harmonic pump and the initial state (∆ϕ = ϕ2ν minus2ϕν) determines the result of the parametric amplification

field reaches its maximum (∆ϕ = 90) The output at the fundamentalfrequency is then an amplitude squeezed state with a deamplified coherentamplitude

Fig 16 summarizes the squeezing operation on the vacuum state as wellas on displaced vacuum states for different phase relations ∆ϕ between thetwo input fields

43 Cavity-enhanced OPA

Placing the nonlinear crystal inside a cavity can greatly enhance thedown-conversion efficiency but not only that A cavity introduces a thresholdfor the pump power above which the parametric gain is infinite just limited

43

by the finite pump power In this case the vacuum uncertainty of the inputfield at frequency ν is amplified to a bright laser field at frequency ν Thedevice is then called an optical-parametric oscillator (OPO) For the gener-ation of squeezed states however the pump power is usually kept (slightly)below threshold Due to nonzero optical loss there exists a pump powersmaller than the threshold above which the tiny improvement of squeezingis not noticeable anymore Getting the pump power closer to the thresholdcould even reduce the observed squeeze factor if a fluctuating squeeze an-gle projects anti-squeezing into the observed quadrature amplitude [Franzenet al (2006) Suzuki et al (2006) Dwyer et al (2013)] The cavity has an-other important purpose It confines the transverse spatial mode usuallyto TEM00 This mode confinement is crucial for any efficient application ofthe squeezed state in laser interferometry since it allows the suppression ofanti-squeezing from other transversal modes The squeezing process requiresa nonlinear material that should show negligible absorption at both opticalfrequencies involved in particular at the wavelength of the squeezed modeIn Refs [Vahlbruch et al (2008) Mehmet et al (2009)] 10 dB and 116 dBof squeezing were achieved using MgOLiNbO3 The highest squeeze factorstoday are produced in (quasi phase matched) periodically poled KTP [Eberleet al (2010) Mehmet et al (2011) Stefszky et al (2012) Vahlbruch et al(2016)]

The optical cavity that is built around the nonlinear crystal is vital forsqueezed-light generation and it deserves a detailed consideration Gener-ally the mode propagating away from a cavity is the result of interferenceat the cavity coupling mirror One contribution is given by the intra-cavityfield attenuated by the amplitude transmission coefficient t of the outcouplingmirror The second contribution is given by the outside field that is reflectedby the same mirror with amplitude reflectivity r =

radic1minus t2 and spatially

overlapped with the first Also the mode from a squeezing resonator is suchan interference product

The impedance matched resonator

Let us consider first an empty optically stable and loss-less Fabry-Perotresonator built from two identical mirrors each with amplitude reflectivityr = r1 = r2 lt 1 A propagating field be perfectly mode-matched to one ofthe cavity resonances In this setup the resonator shows zero reflection andthe resonator is said to be impedance matched (for all such input fields)

44

|α〉

r1α

r12

= +

|α〉 r22 = r1

2

minusr1α equiv t1 αcav = 1minus r12 minusα sdot r1

1minus r12

0

XY

Figure 17 Empty impedance-matched resonator ndash Mode-matched and resonant lightthat enters the cavity from the left here displayed by a quantum phasor for a coherentstate is fully transmitted including its quantum uncertainty The back-reflected statedestructively interferes with itself for all frequencies well within the cavity linewidth The180 phase shift of the transmitted cavity field amplitude minusr1α is a necessary conditionin order to fulfill energy conservation on cavity resonance Impedance matching is forinstance achieved for a resonant cavity with matched mirror reflectivities (r2

2 = r21) and

zero optical loss The complex amplitude of the field traveling towards left inside the cavityis readily deduced from the figure and reads αcav = minusαr1(1minus r2

1)minus12 (highlighted by thedashed circle) In the displayed setup zero field uncertainties are reflected however alsothe vacuum state that enters the cavity from the right is fully transmitted (not shown)

r1α

XY

r12 r2

2 =1

= + |α〉

minus 1+ r1( )α equiv t1 αcav = 1minus r12 minus 1+ r1( )α

1minus r12

|α〉

Figure 18 Empty maximally overcoupled resonator ndash Maximal overcoupling isachieved for a resonant cavity with a perfect end-mirror reflectivity (r2

2 = 1) and zerooptical loss For a given input-mirror reflectivity r2

1 the intracavity light power is maximalMode-matched and resonating light entering the resonator from the left is fully reflectedThe complex amplitude of the field traveling towards left inside the cavity is readilydeduced from energy conservation to αcav = minusα(1 + r1)(1 minus r2

1)minus12 (dashed circle) Inthis setup no uncertainty from the right couples to the left side of the cavity

45

Obviously the interference described in the previous paragraph is fullydestructive The same resonator also shows zero reflection of the input fieldrsquosquantum uncertainty since the interference happens between parts of thesame quantum state The mode propagating away from such a resonatorhowever is not in a nonclassical but in a vacuum state because the vacuumstate that enters the cavity through the opposite site is also fully transmittedThe interference at the coupling mirror of an impedance matched resonatoris displayed in Fig 17

r1α

r12

+

=

OPA PDC

Xcavg r1(1+r1) Xcav

Δ2Xg 0 Δ2Yg

XY

r22 =1

|α〉

Figure 19 Squeezing resonator ndash Shown is the interference at the zero-loss squeezingresonator operated at threshold The lower line represents the perfectly squeezed modepropagating away from the cavity towards the left The parametric gain medium inside thecavity deamplifies the X quadrature of the cavity mode (Xcav) by the factor r1(1 + r1)which is the ratio of the intra-cavity field amplitudes of the two previous figures The Xquadrature of the field that is back-reflected towards the left destructively interferes withitself similar to the situation of the impedance matched cavity in Fig 17 The parametricpower gain per resonator round-trip (G) needs to mimic the effect of an end mirror withreflectivity r2

2 = r21 For this reason the deamplification of Xcav corresponds to a round-

trip deamplification factor of r1 equivradic

1G The round-trip amplification factor for Ycav

then is 1r1 equivradicG which exactly compensates for the outcoupling and thus determines

the parametric oscillation threshold (threshold for bright lasing) The variances of thequantum uncertainties ∆2Ycav and ∆2Y are thus infinite In this setup no field uncertaintyfrom the right couples to the left of the cavity and a perfectly X-quadrature-squeezed fieldoutside the squeezing resonator is produced

46

The perfectly over-coupled single-ended resonator

We now increase the reflectivity of the far mirror lsquo2rsquo to being perfect (r2 = 1)This way the counter-propagating vacuum state can not enter the cavityAgain a propagating field be perfectly mode-matched through mirror lsquo1rsquo toone of the cavity resonances For frequencies well inside the cavity linewidththe situation is displayed in Fig 18 The setup protects the left side of thecavity against vacuum fluctuations entering through mirror lsquo2rsquo but of coursedoes not squeeze quantum noise The intra-cavity built-up factor is too highfor achieving destructive interference below the vacuum uncertainty on theleft side of the resonator

The impedanced-matched single-ended squeezing resonator

Building on the two previous concepts the straight forward approach now isto start from the perfectly over-coupled single-ended resonator and insert anattenuator into the cavity that does not couple the cavity mode to any bathbut still results in a roundtrip efficiency of precisely r1(lt 1) in amplitudeOptical loss is not appropriate since it increases the coupling of the cavitymode to a thermal bath neither would any phase-insensitive atenuator beappropriate It is easy to show that a phase-insensitive attenuator adds ad-ditional uncertainty since otherwise the commutation relation [a adagger] = 1 isviolated The amplification process that matches our requirement is OPA Toachieve infinite squeezing in X on cavity resonance a second-order nonlinearcrystal needs to be put into the cavity and pumped such that the intra-cavityamplitude quadrature is attenuated by the factor (1 + r1)r1 (on cavity reso-nance) with respect to the empty cavity This factor is readily deduced fromFigs 17 and 18 Due to the symmetry in parametric amplification the intra-cavity phase quadrature is then amplified by (1 + r1)r1 and the round-tripgain has a value of 1r1 in amplitude In this situation not only infinitesqueezing but also the (laser) threshold of the resonator is achieved sincethe round-trip gain of the intra-cavity phase quadrature equals its roundtriploss here fully given by the incoupling mirror

The physical descriptions in Figs 17 to 19 are fully consistent with ob-servations in squeezing experiments The consideration above in particularshows that the intra-cavity field shows a finite squeezing strength while theexternal field shows infinite squeezing The strongest intra-cavity squeezefactor possible is (1 + r1)2r2

1 In the high reflectivity limit this factor corre-

47

sponds to 6 dB Averaged over the full cavity mode the squeeze factor of thecavity mode is in this limit even limited to 3 dB [Walls and Milburn (2008)]Higher intra-cavity squeeze factors are possible for lower mirror reflectivities

44 The generation of squeezed light for laser interferometry

With the insights gained in the previous subsection we now turn to ac-tual experiments The application of squeezed states in laser interferometrycertainly requires large squeeze factors (idealy accompanied with the highestpossible purity) to maximize the impact in terms of sensitivity improvementIn cavity-enhanced OPA the highest parametric gain is achieved on cavityresonance ie at zero sideband frequency But this is not the main reasonwhy this Subsection focusses on the generation of squeezed states at lowsideband frequencies The application of squeezed states in a laser inter-ferometer requires that their sideband frequencies cover the devicersquos signalband Ground-based gravitational wave (GW) detectors have a detectionband from about 10 Hz to 10 kHz frequencies which can be considered aslsquolowrsquo compared to typical frequencies in quantum optics experiments

Squeezing at MHz sideband frequencies is easier to observe than at acous-tic frequencies because the latter are often polluted with excess noise fromlight beams that serve as control beams [Bowen et al (2002) McKenzie et al(2004)] and parasitic interferences from back-scattered light [Vahlbruch et al(2007)] Furthermore the observation of squeezing at low sideband frequen-cies requires a more stable setup since larger measuring times are necessaryThe observation of strong squeezing at MHz frequencies however alreadysets an upper limit to the optical loss of the setup At least the same squeezefactor can be observed at lower frequencies

There are two different main topologies for squeezing resonators TheFabry-Perot-type standing-wave resonator consists of a minimum number ofmirror surfaces and has the advantage of being compact and thus robustagainst mechanical vibrations Usually one or even two mirror coatings aredirectly placed on the spherical and polished surfaces of the nonlinear crys-tal itself [Wu et al (1986) Grangier et al (1987) Breitenbach et al (1998)Vahlbruch et al (2008) Eberle et al (2010) Vahlbruch et al (2016)] TheBowtie traveling-wave resonator has the advantage of providing a separa-tely accessible counter propagating mode for cavity length control [Ou et al(1992) Takeno et al (2007)] It shows no direct back-reflection of incoupledlight which helps reducing parasitic interferences [Stefszky et al (2012)]

48

(a)

(b)

(c)

(d)Squeezingresonator

To inter-ferometer

OPA

SHG

BHD

Mode cleaner

LO

Laser

DBS

DBS

Figure 20 Generation of squeezed light ndash (a) Example of a 2nd-order nonlinearcrystal for the squeezed-light generation at 1064 nm Shown is a bi-convex 65 mm long7MgOLiNbO3 crystal whose polished surfaces also carry the mirror coatings of the res-onator The crystal thus realizes a monolithic squeezing resonator as it was used forthe first demonstration of 10 dB squeezing [Vahlbruch et al (2008)] (b) Optical con-figuration of a half-monolithic (hemilithic) standing-wave squeezing resonator Here thecavity length can be adjusted by displacing the coupling mirror The crystal surface insidethe cavity is anti-reflection coated The photograph shows a 10 mm long PPKTP crystalsqueezing resonator as used for the GEO 600 squeezed-light source [Abadie (2011)] (c)Mechanically stable housing of a standing-wave squeezing resonator The crystalrsquos temper-ature is stabilized at its phase matching condition using Peltier elements (d) Schematicfor the squeezed-light generation After spatial filtering of continuous-wave laser lighttwo hemilithic standing-wave resonators are employed The first generates second har-monic pump light (SHG) The second (OPA) generates a squeezed vacuum field at theinitial wavelength The squeezed states are observed by a balanced homodyne detector(BHD) or alternatively sent and mode-matched to the optical mode of an interferometerbeforehand LO local oscillator DBS dichroic beam splitter

49

Fig 20 (a) and (b) show photographs of typical nonlinear crystals used forsqueezed-light generation at near infra-red wavelengths The crystals shownhere form a monolithic standing-wave squeezing resonator (a) or are partof a half-monolithic standing-wave squeezing cavity (c) shows a tempera-ture stabilized and mechanically stable housing of the squeezing resonator(d) shows a schematic of a full setup for the generation of squeezed vacuumstates of light for an application in a laser interferometer The only brightinput required for the squeezing resonator (OPA) is the second-harmonicpump field The resonator mode at fundamental frequency is thus initiallynot excited by photons ie it is in its ground state characterized by vacuumfluctuations due to the zero point energy see Fig 7 (c) [Gerry and Knight(2005)] The pump field spontaneously decays in the degenerate pair of sig-nal and idler fields The combined down-converted field leaving the resonatorexhibits quantum correlations which give rise to a squeezed photon countingnoise when overlapped with a bright coherent local oscillator beam Thedetection is done either in a balanced homodyne detector (BHD) or with asingle photo diode The squeeze factor increases the closer the pump powerof the squeezing resonator gets to the oscillation threshold and the lower theoptical loss on down-converted photon pairs is

441 High squeeze factors ndash minimizing decoherence

Squeezed states of light have significant impact on the sensitivity of laserinterferometers if large squeeze factors can be produced Squeezing of 3 dBimproves the signal-normalized quantum-noise spectral density by a factorof 2 This factor corresponds to doubling the (coherent state) light powercirculating inside the interferometer Squeezing of 10 dB corresponds to aten-fold power increase The experimentally demonstrated squeeze factorswere considerably improved in recent years [Takeno et al (2007) Vahlbruchet al (2008) Polzik (2008) Eberle et al (2010) Stefszky et al (2012)]culminating in a value of as large as 150 dB [Vahlbruch et al (2016)] Thisvalue corresponds to the same reduction of signal-normalized quantum noisethat is achieved by increasing the light power by a factor of 32 (At this pointit is already noted that squeezing the quantum noise can simultaneouslyreduce quantum measurement noise (shot noise) as well as quantum backaction noise (radiation pressure noise) This is not possible with scaling thelight power of coherent states see Subsec 55)

50

Ideally a parametric squeezed-light source can produce an infinite squeez-ing level see Fig 19 fundamentally just limited by the energy provided bythe pump field In practice the limit is set by decoherence mechanismsThe by far most important one is optical loss Optical loss occurs duringsqueezed-light generation its propagation through the interferometric setupincluding imperfect mode matchings and finally the photo-electric detec-tion Also detector dark noise [Schneider et al (1998)] phase noise [Takenoet al (2007)] and excess noise [Bowen et al (2002)] impair the observablesqueezing strength

Optical loss is usually understood as coupling the squeezed mode to a zerotemperature bath ie overlapping it with a vacuum mode For any amountof loss the resulting state is still squeezed But to be able to directly observesay 10 dB of squeezing the total loss on the state needs to be less than 10in this example cf Eq (16) To minimize optical loss the nonlinear crys-tal as well as lenses and beam splitters in the interferometric path need toshow very low absorption and scattering at the wavelength of the squeezedlight PPKTP shows absorption of about 10minus4cm and below at near-infraredwavelengths Low OH content fused silica is a suitable material for all otheroptics Absorptions of less than 10minus6cm were measured [Hild (2007)] Coat-ings on crystal surfaces and on all other optical components should also showlowest optical loss Total loss of the 10minus6 level are available today Superpol-ished surfaces which show roughnesses with less than 1 A root mean square(integrated over spatial scales from approximately 1 micron to 100 microns)and thus very low scattering are necessary to achieve these low numbersMinimizing the total number of optical components is essential From thisperspective a monolithic squeezing resonator as shown in Fig 20 (a) is theoptimum choice The squeezed mode needs to be matched to the mode ofthe laser interferometer or to the mode of the balanced homodyne detectorVisibilities of up to 998 have been achieved [Eberle et al (2010)] whichcorresponds to a loss of about 04 Of great importance also is the quantumefficiency of the photo-diodes used for detecting the squeezed field (togetherwith the interferometric signal) Recently a quantum efficiency of photo-diodes in a squeezing experiment of (995plusmn 05) was measured [Vahlbruchet al (2016)] To minimize photon loss the photo-diodes had no protectionwindow an anti-reflection coating on the semi-conductor material and theremaining reflection was re-focussed with an external mirror

Also the dark-noise spectral density of the detection electronics reducesthe observable squeezing and needs to be as low as possible Similar to optical

51

noise it also provides a contribution to the observed variance The dark noiseof the detection electronics needs to be much lower than the detected photoncounting noise In [Vahlbruch et al (2016)] it was 28 dB below shot noisebut still reduced the observable squeeze factor from 153 dB to 150 dB

Excess noise emerges if the squeezed mode couples to a nonzero tem-perature bath or to a mode whose excitation is strongly fluctuating (Thecoupling process can always be understood as a beam splitter coupling andis physically described by overlapping electric fields Coupling to a zero tem-perature bath leads to Eq (16)) The captured excess noise variance thenneeds to be added to the initial squeezing variance which deteriorates theobserved squeezing stronger than just mixing in the vacuum mode Excessnoise is less likely to occur at MHz frequencies but can be significant ataudio-band sideband frequencies and below and is thus a serious issue ingravitational-wave detectors [Chua et al (2014)] The reason for that isthat acoustically or thermally excited motions of surfaces produce frequencyshifts of back-scattered light mainly at these low frequencies [Vahlbruch et al(2007)]

Phase noise corresponds to stochastic phase fluctuations between thesqueezed field and the local oscillator within the measuring time It cor-responds to mixing the squeezed mode with itself with a fluctuating squeezeangle [Suzuki et al (2006) Franzen et al (2006)] Phase noise in squeez-ing experiments typically is less of an issue than optical loss [Dwyer et al(2013) Oelker et al (2016) Vahlbruch et al (2016)] The setuprsquos phasenoise can be reduced by making the squeezing resonator more compact andthus mechanically more stable against acoustic and thermal fluctuations ofthe environment and by improving the quality of the servo loops for cavitylength and propagation length controls Operating a squeezed-light resonatorin vacuum might also be beneficial The ability to run a high performancesqueezed-light generator in vacuum was demonstrated in [Wade et al (2015)]

442 Squeezing in the gravitational-wave (GW) detection band

High squeeze factors have been first demonstrated at sideband frequen-cies of a few MHz and above where excess noise is generally negligible whenworking with visible or near-infra-red light Today we know that extendingthe squeezing spectrum towards the audio-band and even below is technicallynot always easy but straight forward once a high squeeze factor is achieved

52

at MHz frequencies In most squeezing experiments the main laser light

Figure 21 Photograph of the GEO 600 squeezed-light source ndash The breadboarddimensions are 135 cm times 113 cm The squeezing resonator is high-lighted by the white ar-row and is set up as a standing-wave hemilithic cavity containing a plano-convex PPKTPcrystal of about 10 mm length (see also Fig 20 b) It is pumped with continuous-wave532 nm light that is produced by frequency doubling of light (at angular frequency ω)from a commercial NdYAG laser Two more laser fields at about 1064 nm having fre-quency offsets of more than 10 MHz with respect to ω(2π) serve as optical control fieldsBoth fields are mode-matched and injected into the squeezing resonator together with thesecond-harmonic pump field

at the squeezing wavelength is accompanied by significant noise up to thelaser relaxation oscillation For this reason laser control fields at the opticalcarrier-frequency in the optical path of the squeezed mode need to be avoided[Bowen et al (2002) Schnabel et al (2004) McKenzie et al (2004)] and thesqueezing resonator length and the optical path stabilized by other means[McKenzie et al (2005) Vahlbruch et al (2006)] Furthermore and mostimportantly excess noise due to back-scattering is an issue Back-scattering(also called lsquoparasitic interferencesrsquo) is produced if DC light scatters out ofthe optical path hits a vibrating surface and re-scatters back into the opticalpath [Vahlbruch et al (2007)] Significant back-scattering can be produced

53

in interferometers for the detection of gravitational waves since light powersof several hundreds of kilowatts are used Even back-scattering from the mil-liwatt local oscillator of balanced homodyne detectors is an issue at acousticsideband frequencies and below The recipe for avoiding parasitic interfer-ences turns out to be threefold (i) avoiding scattering by using ultra-cleansuperpolished optics with close to perfect anti-reflex coatings (ii) avoidingback-scattering by carefully blocking all residual (faint) light fields and (iii)reduce the vibrationally and thermally excited motion of all mechanical andoptical parts that could potentially act as a re-scattering surface with re-spect to the optical path [Vahlbruch et al (2007) McKenzie et al (2007)]The insights described above led to the first demonstration of audio-bandsqueezing at frequencies down to 200 Hz [McKenzie et al (2004)] and laterto the first demonstration of squeezing over the full gravitational-wave de-tection band even from as low as 1 Hz [Vahlbruch et al (2007)] Whilea standing-wave squeezing resonator [Ou et al (1992)] can be built in avery compact way that is rather insensitive against mechanical vibrations[Chelkowski et al (2007)] a traveling-wave bow-tie squeezing resonator [Wuet al (1986)] is more tolerant against back-scattered light [Chua et al (2011)]The strongest squeezing in the audio-band of up to 116 dB was reported inRef [Stefszky et al (2012)]

443 The first squeezed-light source for GW detection

The first squeezed-light source for the continuous operation in GW de-tectors had been designed and completed between 2008 and 2010 [Vahlbruch(2008) Vahlbruch et al (2010)] Since then this source has been producingsqueezed vacuum states in a fully phase controlled way using co-propagatingfrequency-shifted bright control beams [Vahlbruch et al (2006)] as an inte-gral part of the GW detector GEO 600 The source is a turn-key device witha fully automated re-lock system [Vahlbruch et al (2010) Khalaidovski et al(2012)] Re-locking is required if the temperature of the environment changessignificantly which drives the actuators outside their dynamic ranges

Up to 9 dB of squeezing over the entire GW detection band was observedusing a balanced homodyne detector (BHD) located close to the squeezingresonator The squeeze factor has been limited by optical loss due to ab-sorption in the PPKTP crystal transmission of the back-surface and thenon-perfect AR-coating of the crystalsrsquos intra-cavity surface The adjustableair gap has been necessary to allow for an easy way to apply length control

54

-12

-8

-4

0

4

8

12

16

20

10 100 1k 10k

Rel

ativ

e no

ise

pow

er [

dB]

Frequency [Hz]

shot noise (a)

squeezed noise (b)

anti-squeezed noise (c)

~ 9dB

Figure 22 Broadband squeezing spectrum ndash Noise power spectra measured on theoutput of the GEO 600 squeezed-light source shown in Fig 21 with a balanced homodynedetector The traces correspond to the spectra of quadrature amplitude variances normal-ized to vacuum noise The resolution bandwidth used increases towards higher frequenciesto reduce the measurement time (a) Shot noise normalized to unity which serves as thereference level (0 dB) (b) Squeezed noise covering the complete detection band of ground-based GW detectors (c) Anti-squeezed noise Peaks at 50 Hz and 100 Hz are the electricmains frequency and its first harmonic The data was published in Ref [Vahlbruch et al(2010)]

Additional optical loss in the path to the balanced homodyne detector mainlyarose due to a Faraday isolator that eliminated parasitic interferences Fi-nally the mode missmatch to the BHD as well as its non-perfect quantumefficiency provided additional loss Inferring the squeeze factor without theBHD detection loss more than 10 dB of squeezing are provided by the sourceSince 2010 it has been used in basically all observational runs of the GEO 600GW detector see Section 6

444 Generation of two-mode (bi-partite) squeezing

lsquoTwo-mode squeezed lightrsquo or lsquobi-partite squeezed lightrsquo is light that allowsfor joint measurements at two locations A and B These joint quadraturemeasurements reveal correlations and anti-correlations with a remaining un-certainty smaller than the ground-state uncertainty which certifies the pres-

55

ence of entanglement cf Subsec 34 Bi-partite squeezed light has been gen-erated by type I and by type II parametric down-conversion In case of type Ithe squeezed fields from two squeezing resonators as described in Subsec 44are overlapped on a balanced beam splitter with a 90 phase shift The twooutput fields together represent the entangled mode [Furusawa et al (1998)Bowen et al (2003c) Eberle et al (2013)] In case of type II signal and idlerfields are non-degenerate regarding polarisation and a single cavity contain-ing an appropriate crystal and a polarising beam splitter are sufficient for theproduction of bi-partite squeezing Also in this case the measurements ofthe quadrature amplitudes of signal and idler fields show large uncertaintiestogether with bi-partite correlations and anti-correlations that are strongerthan the ground state uncertainty of individual subsystems [Ou et al (1992)Villar et al (2006) Jing et al (2006)]

To date the strongest entanglement of bi-partite squeezed light has beenproduced based on type I parametric down-conversion [Eberle et al (2013)]The requirements of producing strong entanglement are identical to those ofproducing strong squeezing outlined above The strength of bi-partite en-tanglement can be given in decibels in full analogy to the squeeze factorPractically the strength of bi-partite squeezing is always somewhat smallerthan that of single party squeezing since it requires an additional mode-matching that results in additional optical loss

45 Conclusions

The first observation of squeezed light was achieved in 1985 [Slusher et al(1985)] Shortly after cavity-enhanced optical parametric amplification forsqueezed-light generation was demonstrated [Wu et al (1986)] which todayenables the observation of up to 15 dB of squeezing [Vahlbruch et al (2016)]Quite generally the maximum squeezing level that is observed does not de-pend on the strength of the optical nonlinearity Squeezing cavities can easilybe operated at their oscillation threshold where they should provide infinitesqueezing if decoherence is zero The main limiting factor is optical lossincluding that of the photo-electric detection

Dedicated experimental research and development towards a squeezed-light source for applications in gravitational-wave detectors can be tracedback to 2002 [McKenzie et al (2002) Bowen et al (2002)] Since then a sur-prising amount of progress has been made culminating in the first squeezed-light source specifically built for the integration into a gravitational-wave

56

detector For the future squeeze factors above 15 dB will certainly be possi-ble by further reducing optical loss This statement is supported by the highdegree of matching between experimental data and a theoretical loss modelas presented in Fig 3 of [Vahlbruch et al (2016)]

5 Quantum noise in laser interferometers

51 Interferometric measurements

The purpose of a laser interferometer is the precise measurement of smallchanges of an optical path length with respect to a reference path Forthis the interferometer transfers the change of the phase difference betweentwo light fields into an amplitude quadrature change of the interferometerrsquosoutput light The latter can easily be detected by a single photo diode Ofgeneral interest are differential changes of the optical path length that aremuch smaller than the laser wavelength ie differential phase changes thatare much smaller than 2π In this case the differential phase change can bedescribed in very good approximation as a differential change of the phasequadrature instead

In order to transfer the phase quadrature signal with minimum loss ahigh interference contrast at the interferometerrsquos beam splitter is essentialAdditionally instrumental noise in terms of unwanted excitations of the out-putrsquos amplitude quadrature needs to be reduced to a minimum Noise arisesdue to power fluctuations of the input laser light back-scattered laser lightinside the interferometer thermally driven displacements of mirror surfacesand in many more ways The important measure of the sensitivity of aninterferometer obviously is its signal-to-noise-ratio The most useful measureis given in terms of the noise spectral density S(f=Ω2π) that is normalizedto the physical unit of the signal S(f) is in fact a lsquonoise-to-signal-ratiorsquoand can be seen as the signal-normalized variance of the photo diode out-put decomposed into spectral components versus sideband frequency f withthe resolution bandwidth of 1 Hz As an example S(100 Hz) = 10minus39 m2Hzmeans that the instrumental noise in the one hertz band around 100 Hz equalsa signal that would be produced if the mirror of one interferometer arm os-cillates with an amplitude of just

radic10 middot 10minus20 m in the very same band Such

small spectral densities are achieved by gravitational-wave detectors [Abbott(2016)]

57

52 Quantum measurement noise and shot noise

The most fundamental noise source in laser interferometers is due to thequantum noise of light which is in fact two-fold [Caves et al (1980)] Firstof all there is lsquoquantum measurement noisersquo which arises in the processof photo-electric detection For coherent states the quantum measurementnoise is the lsquophoton counting noisersquo of mutually independent photons andusually simply called lsquoshot-noisersquo Fig 2 (b i) shows a time series of suchnoise hiding the actual signal The frequency components of the shot noiseare well described by the quantum uncertainty of the output fieldrsquos amplitudequadratures XΩ∆Ω see Subsec 22 (Recall this quantity corresponds to the

differential phase quadrature YΩ∆Ω of the light beams in the interferometerarms) The photon counting noise has a white Fourier spectrum howeverthe lsquoshot noisersquo of an interferometer is usually normalized to the signal whosetransfer function is usually not white for instance due to the presence of armcavities or a signal-recycling cavity

All current and planned gravitational-wave detectors are Michelson-typelaser interferometers with operating points very close to a dark fringe Thelight power in the output port is just a couple of tens of mW which canbe handled by a single photo diode In this configuration the signal-to-shot-noise-ratio is actually maximized which can be shown in three steps [Bachorand Ralph (2004)] For the first step we use plane waves to describe theelectric field in the output port of a Michelson interferometer For perfectinterference contrast at the balanced beam splitter ie for perfect modematching and for defining φ = 0 as the dark port condition we get

Eout(t φ) =1

2E0 sin(ωt+ φ)minus 1

2E0 sin(ωt) (31)

where E0 is the amplitude of the total internal field whose two parts hasaccumulated a differential phase It directly follows for the squared fields

E2out(t φ) =

(sin

φ

2

)2

(E0 cos(ωt+ φ2))2 (32)

We now turn to a light beam with a localized transversal mode that can befocussed onto a photo-electric detector The photo diode has perfect quantumefficiency ie the rate of photo electrons is not only proportional to the rateof output field photons but also has a unity slope efficiency Since the optical

58

frequency is too high to be resolved we consider the averaged light power

P out(φ) =

(sin

φ

2

)2

P (33)

The next step is a formulation of the signal being the derivative of detectedphoton number versus phase Let n be the average value of the photonnumber per measuring time interval Eq (33) can then be rewritten as

nout(φ) =

(sin

φ

2

)2

n (34)

rArr dnout(φ) = n sinφ

2cos

φ

2dφ (35)

The final step is the calculation of the signal-to-shot-noise-ratio Shot noiserefers to coherent states which have a standard deviation of the photonnumber of σ(n) =

radicn

dnout(φ)

σ(nout)=n sinφ

2cosφ

2dφ

radicn sinφ

2

(36)

and find for a signal-to-noise ratio of unity for coherent states and for anon-zero but still small phase difference ∆φCoh 2π

1 =radicn cos

φ

2∆φCoh with φ 6= 0 (37)

In this equation the smallest measurable phase difference is given for φrarr 0

∆φCohmin =

1radicn (38)

This is the well-known shot-noise limit of high-precision phase sensing ∆φCohmin

is the smallest phase shift that can be measured with a signal-to-noise ra-tio of one when using n mutually independent photons per measuring time(those of a coherent state) when the loss of photons is assumed to be zeroThe typical purpose of a laser interferometer is the continuous sensing (moni-toring) of a continuously changing phase An illustrative example is the phasesignal produced by the black hole merger measured by Advanced LIGO onSept 14 in 2015 (Fig 1 in [Abbott (2016)]) The measuring interval should

59

be short to be able to resolve the time-evolution of the signal Generally themeasurement of an arbitrary signal that lasts for a finite time thus needs tobe understood as l subsequent measurement intervals using n photons eachIt can be shown that Eq (38) is also valid for interferometers operated athalf fringe ie when each output port contains the same light power In thiscase photo diodes need to be placed in both output ports and the actualsignal is provided by their difference voltage

Due to its importance the shot-noise limit deserves some remarksThe phase φ in Eq (31) is the phase difference of two mode-matched fieldsand might be accumulated by a single pass along the length L such as ina Mach-Zehnder interferometer or in a double pass such as in a (simple)Michelson interferometer or in four passes as realized in a Michelson inter-ferometer with folded arms [Grote (2005)] The shot-noise limit in Eq (38)and its scaling therefore holds independent of the number of passes Theclaim in Ref [Higgins et al (2007)] that the scaling according to Eq (38) canbe surpassed by multiple passes is not justifiedThe fact that Eq (38) is derived by approaching φ rarr 0 correctly describesthe actual operation point of gravitational-wave detectors which is close tobut not exactly at a dark port In practice a tiny offset from dark port ischosen at which the shot noise is well above the photo diodersquos electronic darknoiseEq (38) solely depends on the number of quanta but not on the lightrsquos wave-length λ Of course the shot-noise limit for the change of an optical pathlength ∆L does depend on the wavelength and ∆φCoh

min needs to be replacedby ∆φCoh

min = 2π∆LCohminλ

Finally an essential result of the shot-noise limit is that the ideal precisemeasurement should use lsquoas much quanta as possible per measuring inter-valrsquo which translates to lsquoas much light power in the interferometer armsas possiblersquo Eq (38) is indeed the one and only reason why gravitational-wave detectors use high power lasers power-recycling and arm resonatorsExtending the measuring time for a given light power can also improve thesensitivity but only if the signal repeats ie is periodic Let us assume thatone period of the signal is resolved by l intervals using n photons each Inthis case repeating the overall measurement k times improves Eq (38) by1radick The fundamental statement of Eq (38) however does not change

since the actual photon number n may then simply incorporate the factor k

60

For a given average photon number the shot-noise limit in Eq (38) canonly be surpassed by using photons that are quantum correlated ie by usingnonclassical states of light How is the shot-noise limit surpassed with thehelp of squeezed states A Michelson interferometer that is operated closeto a dark fringe acts like an almost perfect mirror for both input ports Allthe input light is back-reflected towards the laser source This also accountsfor the quantum uncertainty of the input light The quantum uncertaintythat impinges onto the photo diode thus (mainly) enters the interferometerthrough its (almost) dark port An interferometer that uses displaced co-herent states entering from one port can thus be improved by replacing theordinary vacuum entering the signal output port by a squeezed vacuum stateThis was the proposal by CM Caves in 1981 [Caves (1981)] which is labeledhere with lsquoCSVrsquo If the differential phase quadrature of the interferometer issqueezed Eq (38) then within the limit of large coherent state displacementα sinh2r improves to

∆φCSVmin asymp

eminusrradicn (39)

(The above expression is an approximation since the squeezing operationproduces a small number of photons that are not accounted for here) Ofcourse the mode of the squeezed vacuum needs to be precisely matchedto the mode of the interferometer The first experimental demonstrations ofsqueezed phase measurements used a Mach-Zehnder [Xiao et al (1987)] and apolarization interferometer [Grangier et al (1987)] Fig 2 shows how spatialdegeneracy between an externally generated squeezed mode and the signalmode in a Michelson interferometer is achieved using a polarizing beam split-ter and a Faraday rotator Again the limit in Eq (39) can only be achievedif optical loss is zero Optical loss not only reduces the signal but here alsoreduces the squeeze parameter see Eqs (15) and (16)

Let us consider an example The sensitivity of a laser interferometer thatuses coherent states with an excitation of 1023 photons per second can beimproved by a factor of

radic10 by either adding 09 middot 1024 photons per second

or by adding about just 2 photons per second and bandwidth in hertz thatbelong to the 10 dB squeezed vacuum confer Eq (30) Since the full sig-nal band of ground-based GW detectors covers sideband frequencies up to10 kHz just 2 middot 104 photons per second are necessary At a wavelength ofλ = 1064 nm these values correspond to a power increase by 168 kW and

61

37 fW respectively

The question arises whether a scaling of the sensitivity better than propradic1n is possible It was theoretically shown that in principle the scaling

can indeed considerably be improved yielding the so-called Heisenberg limitor Heisenberg scaling [Bondurant and Shapiro (1984) Yurke et al (1986)Braunstein (1992) Holland and Burnett (1993)]

∆φHLmin prop

1

n (40)

The Heisenberg scaling requires nonclassical states that have a certain num-ber of quanta similar to Fock states ie n = n The theoretically optimalstates describe a superposition of n (N) indistinguishable photons in oneinterferometer arm while having zero (0) photons in the second arm and viceversa and were named lsquoN00Nrsquo-states [Dowling (2008)] A specific propertyof these states is lsquosuper-resolutionrsquo The output ports of the interferometershow an n-times faster oscillation of the interference fringes when changingthe phase between the two interferometer arms Super-resolution correspondsto an n-times improved signal transfer function and was demonstrated forinstance in Refs [Rarity et al (1990) Kuzmich and Mandel (1998) Mitchellet al (2004) Afek et al (2010)] The presence of this nonclassical phe-nomenon however does not prove a sensitivity better than the semi-classicalbound according to Eq (38) Sensitivity is rather related to the signal-to-noise-ratio and needs to take into account all imperfections as well as theprobability of a successful detection of the sensing state [Thomas-Peter et al(2011)] All experiments so far used post-selection on particular measure-ment outcomes and neglected the typically large probability that nothingwas detected

Super-resolution was demonstrated with up to n = 5 [Afek et al (2010)]In addition to the fact that super-resolution does not prove a sensitivity bet-ter than the semi-classical bound photon numbers in state-of-the-art super-resolution experiments are extremely small compared to the photon numberof about 1023 (within a measuring interval of one second) of coherent statesused in Ref [Abbott (2016)] and of about 1022 using coherent states plussqueezed vacuum states used in Ref [Abadie (2011)]( which did prove a sen-sitivity better than the semi-classical bound)

Another interesting and related question is what the smallest phase is

62

that can be estimated in a single measurement again using a given numberof quanta Taking into account that no prior information about the phaseshift exists still a scaling proportional to 1n is possible In the limit of largen however an additional factor of π is required in the nominator of Eq (40)[Sanders and Milburn (1995) Berry and Wiseman (2000)] yielding

∆φHLmin asymp πn (41)

The state that can actually achieve this bound is different from the N00Nstate and was found in [Summy and Pegg (1990) Luis and Perina (1996)Berry and Wiseman (2000)] A N00N state is not the optimum state forphase estimation (via a single measurement) since it only provides one bit ofinformation A recent review on generell aspects on phase measurements isgiven by Ref [Demkowicz-Dobrzanski et al (2015)]

It is important to note that Eqs (38-40) do not consider photon lossExperiments that demonstrated super-resolution and aimed for achievingthe scaling in Eq (40) were conditioned on zero photon loss Let η gt 0 bethe average efficiency of detecting (all) photons Eq (38) then reads

∆φηCohmin =

radic1

η n (42)

Eq (39) turns into

∆φηCSVmin asymp

radicηeminus2r + 1minus η

η n(43)

and Eq (40) turns into [Demkowicz-Dobrzanski et al (2012 2013)]

∆φηmin =

radic1minus ηη n

for 0 lt η lt 1 (44)

For non-zero photon loss most interestingly the ultimate sensitivity of aphase measurement for a given photon number also shows a 1

radicn -scaling

The difference between the CSV strategy of using bright coherent states incombination with squeezed vacuum states which is bounded by Eq (43) andthe strategy of using the optimal nonclassical state which is bounded byEq (44) is marginal in practice [Demkowicz-Dobrzanski et al (2013)] Forgravitational-wave detectors and for any other laser interferometer using in-tense light there is no need for an alternative to the CSV strategy

63

We now turn back to the shot noise according to Eq (38) Generally noisecan be decomposed into its spectral contributions For a simple Michelsoninterferometer without arm resonators and without a signal-recycling cavitythe square-root of the single-sided shot-noise spectral density normalized tothe differential arm length change x in units of m

radicHz is given by [Saulson

(1994)] radicSMI

SNx =

radic~c2

2ωPprop 1radic

P (45)

where ω is the optical angular frequency of the quasi-monochromatic carrierlight and P the total light power in both arms including the built-ups fromcavities In combination with a squeezed vacuum whose relative phase gen-erates squeezing of the output lightrsquos amplitude quadrature the right handside reduces according to the factor eminusr Note that the single-sided spectraldensity is only defined for positive sideband frequencies and thus twice aslarge as the double-sided spectral density

The spectral density of the measurement of a GW induced strain is givenby the same expression but normalized to h = xL (If the gravitational waveis oriented in an optimal way with respect to the Michelson interferometerone arm is squeezed while the other is expanded by the same amount of ∆L =x2 and h then corresponds to the actual gravitational-wave amplitude)The square-root of the single-sided shot-noise spectral density normalized tostrain in units 1

radicHz is given by

radicSMI

SNh =

radic~c2

2L2ωP (46)

Equations (45) and (46) show that the smallest measurable signal (corre-sponding to unity signal-to-shot-noisendashratio) is inversely proportional to thesquare root of the laser power and has a white spectrum for sideband frequen-cies much smaller than the carrier frequency see horizontal line in Fig 23

All first- and second-generation GW detectors use power-recycling andadditional cavities to improve their sensitivities Fabry-Perot arm resonatorsdo not only increase the light power but additionally also increase the signalfor signal frequencies inside the resonator linewidth For lossless Fabry-Perot

64

arm resonators the spectral densities in Equations (45) and (46) need to bemultiplied by the following factor [Kimble et al (2001)]

HFP =

radicL2(γ2

FP + Ω2)

c2 (47)

where γFP = cTFP(4L) is the Fabry-Perot arm resonatorrsquos half bandwidthand TFP is the light power transmission of the input mirror The end mirrorsare assumed to have perfect reflectivity A similar expression can be derivedfor describing the improvement due to signal-recycling [Buonanno and Chen(2001)]

In summary shorter laser wavelengths higher light powers and squeezingof the amplitude quadrature of the interferometer output reduce shot noise ina broadband way ie for all signal frequencies Fabry-Perot arm resonatorsas well as signal-recycling provide improvements mainly for frequencies insidethe resonator linewidths

53 Quantum back-action and quantum radiation pressure noise

In laser interferometers quantum back-action noise results from the un-certainty of the lightrsquos radiation pressure force on the interferometer mirrorsand is also called lsquo(quantum) radiation pressure noisersquo (RPN) Its origin isthe quantum uncertainty of the differential amplitude quadrature XΩ∆Ω ofthe fields in the interferometer arms It results in an uncertain momentumtransfer to the mirrors and thus in an position uncertainty of the mirrors atfuture times with respect to their differential mode of motion [Caves et al(1980)] The physical mechanism of radiation pressure corresponds to anintensity dependent phase shift [Pace et al (1993)]The higher the light power in the arms of a laser interferometer the loweris its shot-noise spectral density see Eq (45) Unfortunately the spectraldensity of quantum back-action noise increases with light power The single-sided force noise spectral density reads

radicSRPNF =

radic8~ωPc2

(48)

Whereas the force noise of the quantum radiation pressure has a white spec-trum the RPN does not since the mirrorrsquos reaction to external periodicforces depends on frequency The link between the Fourier component of an

65

external force F (Ω) and the Fourier component of the displacement x(Ω) isgiven be the mechanical susceptibility HM It reads for an harmonic oscillatorwith mass M

HM(Ω) =1

M | minus Ω2 + Ω2M + iΩΩMQ|

(49)

where ΩM is the oscillatorrsquos resonance frequency and Q its quality factorThe square root of the single-sided spectral density of the RPN normalized

to the displacement of an harmonic oscillator with mass M is then given by

radicSRPNx = HM(Ω)

radic8~ωPc2

(50)

In GW detectors the test mass mirrors are suspended as pendula with highmechanical Q-factors and their centre of mass motion corresponds to that ofa harmonic oscillator The resonance frequencies of the pendula are lowerthan the detection band of interest The mechanical susceptibility is thereforeoften approximated for the so-called free-mass regime as H fm

M (Ω) = (mΩ2)minus1The square root of the single-sided spectral density of the RPN normalizedto differential displacement of two mirrors with each of mass M in a simpleMichelson interferometer is given by [Saulson (1994)]radic

SfmMIRPNx =

radic2~ωPc2m2Ω4

propradicP (Ω ΩM) (51)

where m = M2 is the mirrorsrsquo reduced mass In case of a simple Michel-son interferometer that is enhanced with arm cavities the spectral density inEq (51) needs to be multiplied with the expression given in Eq (47) In com-bination with a squeezed vacuum whose relative phase generates squeezing ofthe output lightrsquos phase quadrature the right hand side reduces according tothe factor eminusr Note if the radiation pressure noise is squeezed the shot noisemust be anti-squeezed or vice versa The radiation pressure noise calibratedto strain of space time is given by the right side of Eq (51) divided by theinterferometer arm length LIn summary heavier masses longer laser wavelengths lower light powersand squeezing of the amplitude quadrature in the interferometer arms re-duce radiation pressure noise

66

1 10 100 1000Frequency [Hz]

Radiation pressure noise

Shot noise

SQL

10-21

10-18

10-15

10-12

radicSx

[mradic

Hz]

ndashndash

ndashndash

Figure 23 Displacement-normalized quantum noise spectral densities ndash Consid-ered is a simple Michelson interferometer with neither arm cavities nor signal recyclingThe two end mirrors (m = 100 g) of the interferometer arms are suspended as pendulahaving a resonance frequency of ΩM2π = 1 Hz and a Q-factor of 107 The interferometeruses quasi-monochromatic light (in coherent states) with a total power of 4 kW Opticalloss and the offset from a dark output fringe is assumed to be negligible Wavelengthλ = 1550 nm The standard quantum limit (SQL) corresponds to the lowest noise achiev-able at a given sideband frequency when varying the light power without using quantumcorrelations

54 Interferometer total quantum noise and the standard quantum limit

Both shot noise and radiation pressure noise contribute to the total quan-tum noise of a given interferometer If they are not quantum correlatedwhich is the case for a conventional Michelson interferometer when detectingthe output lightrsquos amplitude quadrature their variances add up (The resultis not shown in Fig 23) It can easily be deduced from the previous sectionsthat changing the laser power will shift the two quantum noise contributionsHowever the total quantum noise never goes below the standard quantumlimit (SQL) [Braginsky and Manukin (1967)]

Let us consider Fig 23 for sideband frequencies much greater than thependulum resonance Here the test mass mirrors react as free masses whenexerted to external forces The SQL in this free-mass regime is calculated

67

by minimizing the sum of the squares of Eqs (45) and (51) [Saulson (1994)]

Sfmtotx =

~c2

2ω

[1

P+

4ω2

c4m2Ω4P

] (52)

Its derivative reads

dSfmtotx

dP=minus1

P 2+

4ω2

c4m2Ω4 (53)

Setting the above equation to zero provides the optimum laser power versussideband frequency in order to achieve the lowest total quantum noise

P fmopt =

c2mΩ2

2ω (54)

Inserting the optimal light power into Eq (57) provides the square root ofthe single-sided noise spectral density of the free-mass SQL in m

radicHzradic

SfmSQLx =

radic2~mΩ2

(55)

Again m is the reduced mass and dividing by the interferometer arm lengthL yields normalization to the GW-induced strain h Eq (55) shows that theSQL falls off with sideband frequency The corresponding equation for aMichelson interferometer that uses arm cavities readsradic

SfmFPSQLx =

radic~

mΩ2

(1

HFP

+HFP

) (56)

with HFP according to Eq (47)Using the expression for the SQL the square root of the total quantum

noise spectral density of a Michelson interferometer in the free-mass approx-imation can be written asradic

SfmFPtotx =

radicSfmFP

SQLx

2

[1

k+ k

] (57)

with the radiation pressure coupling parameter

k(Ω) =2ωP

mc2Ω2 (58)

68

For a fixed light power and fixed reduced mass of the mirrors the quantumnoise limited interferometer reaches the SQL when k = 1 which is realizedat the angular sideband frequency ΩSQL =

radic2ωP(mc2)

Note that neither squeezing the phase quadrature nor squeezing the am-plitude quadrature of the interferometer light leads to sub-SQL performance[Caves (1981)] also confer Ref [Schnabel (2005)] Fig 3 (left) As we willsee in the next sections the standard quantum limit can be surpassed if shotand radiation pressure noise are correlated Then the total quantum noise isnot given by the sum of the variances ie the sum of the squares in Eqs (45)and (51)

55 Squeezed light for surpassing the standard quantum limit

A measurement with sensitivity better than the standard quantum limit(SQL) is also called a lsquoquantum non-demolition (QND)rsquo measurement [Bra-ginsky and Khalili (1995 1996) Kimble et al (2001)] Several QND tech-niques for laser interferometers were proposed in recent decades [Jaekel andReynaud (1990) Kimble et al (2001) Purdue and Chen (2002) Chen (2003)McClelland et al (2011) Danilishin and Khalili (2012) Graf et al (2014)]What they have all in common is they exploit quantum correlations betweenobservable uncertainties

Arguably the most extensive way of introducing quantum correlationsand surpassing the SQL is the injection of squeezed states of light [Jaekeland Reynaud (1990)] If the squeezed quadrature angle of the injected statesis neither 0 nor 90 the quantum uncertainties of the amplitude and phasequadrature amplitudes that describe the differential field in the two interfe-rometer arms become correlatedLet us consider a very simplified setup that just consists of a quasi-mono-chromatic light field that is back-reflected from a quasi-free mirror Thelight power and the mass be such that reflected light in a coherent stateresults in a measurement of the mirror position with a noise spectral den-sity at the SQL at sideband angular frequency ΩSQL At this frequencyquantum measurement noise and back-action noise are of the same sizeie the uncertainty in X produces an equally large additional uncertaintyin Y Upon reflection the quadrature amplitude variances change from∆2X = ∆2Y = 14 to 2∆2X = ∆2Y = 12 This result corresponds tothe situation in Fig 23 at the crossing frequency of shot noise and radiation

69

pressure noise The coupling of the uncertainty variances can be describedby the matrix K = (1 minusk 0 1) where k = 1 at the SQL If the modulationstate at ΩSQL is the ground state its variances are transferred according to

KT

(1 00 1

)K =

(1 0minus1 1

)(1 00 1

)(1 minus10 1

)=

(1 minus1minus1 2

) (59)

In accordance with Fig 23 the variance of YΩSQLis twice as large as the vac-

uum noise varianceNow let the quantum noise of the light field be 10 dB squeezed at 45

(Eq (19)) The projection of the quantum uncertainty onto the X-observableproduces the radiation pressure noise by being transferred with the couplingfactor k = 1 at the SQL into the Y -observable in fact with negative signsince a larger value of X produces a larger optical path length and thus aretardation of the phase Due to the squeezing at 45 the initial uncertaintyin Y cancels with the additional uncertainty that originates from the onein X The following calculation shows that the strength of the cancellationcorresponds to the initial squeezing strength Upon reflection the quantumuncertainties transform in the following way(

1 0minus1 1

)(505 495495 505

)(1 minus10 1

)=

(505 minus01minus01 02

) (60)

The state of light after reflection has a squeezed phase quadrature amplitudeThe improvement in comparison to Eq (59) is exactly 10 dB The quantumnoise improvement corresponds to the input squeeze factor and is also a mea-sure by what factor the SQL is surpassed Squeezed vacuum injection thusallows surpassing the SQL upon measuring the conventional Y -quadrature(which is realized by a single photo diode in the interferometerrsquos output port)as first realized by [Unruh (1983) Yuen (1983) Jaekel and Reynaud (1990)]

In the example above the input squeeze angle is optimized for a sin-gle sideband frequency Injecting a broadband squeezed vacuum field withfrequency-independent squeeze angle of 45 would result in a rather bad inter-ferometer quantum-noise performance at frequencies far smaller or largerthan ΩSQL Fig 24 shows the quantum-noise performance if the input fieldhas squeeze angles that are optimized for every k(Ω) as given in Eq (58)Injected squeezing can thus lead to a broadband sub-SQL performance ifthe quantum measurement noise (shot noise) and the quantum back-actionnoise (radiation pressure noise) are correlated in an optimal way Due to the

70

10 50 100 500 1000 5000Frequency [Hz]

Dis

plac

emen

t noi

se sp

ectra

l den

sity

[mradic

Hz]

10-20

10-19

10-18

Total quantum noise 0dB

Total quantum noise -10dB

SQL

Y

X

Y

X

Y

X

Figure 24 Surpassing the SQL with squeezed-light injection ndash At shot-noise lim-ited sideband frequencies squeezing of the Y -quadrature amplitude improves the noisespectral density of the interferometer At radiation-pressure-noise limited sideband fre-quencies squeezing of the X-quadrature amplitude improves the noise spectral density ofthe interferometer If both kinds of quantum noise contribute equally (at the SQL markedwith a dot) a squeeze angle of 45 results in surpassing the SQL by the full squeeze factorsee Eq (60) In the graph here the squeeze angle is optimized for all frequencies result-ing in a broadband quantum noise reduction [Jaekel and Reynaud (1990)] Measurementsensitivities beyond the SQL (shaded area) are in the so-called quantum non-demolition(QND) regime [Kimble et al (2001)] Dashed horizontal lines represent the (squeezed)shot noise Dashed straight lines with negative slope represent the (squeezed) radiationpressure noise The calculations use 10 dB of squeezing a conventional Michelson inter-ferometer with neither arm resonators nor signal recycling a light power at the beamsplitter of 1 MW at a wavelength of λ = 1550 nm and mirror masses of 1 kg

correlation shot noise and radiation pressure noise can be squeezed simulta-neously

71

Light with a frequency-dependent squeeze angle

The discovery that shot noise and radiation pressure noise can be squeezedsimultaneously and thus a broadband reduction of quantum noise beyond theSQL be achieved required the insight that the spectral analysis of light definesa spectrum of many lsquosideband modulation modesrsquo that all can be in differ-ent quantum states An ordinary squeezing resonator which is on resonancefor light at twice the pump wavelength produces a spectrum of modulationmodes that all have the same squeeze angle A frequency-dependent squeezeangle can be introduced by reflecting such a field from a detuned single-ended filter cavity which was suggested by Kimble and coworkers [Kimbleet al (2001)] They showed that the optimal frequency dependence thatleads to the broadband improvement shown in Fig 24 can be realized byusing altogether two filter cavities as shown in Fig 25 Motivated by this re-sult research and development on filter cavities for optimizing the frequencydependence of broadband squeezed fields has been very active in recent years[Corbitt et al (2004) Chelkowski et al (2005) Dwyer et al (2013) Kweeet al (2014) Straniero et al (2015) Oelker et al (2016)]

Photo diode Squeezed vacuum

Faraday Rotator

Coherent light

YΩΔΩ

Figure 25 Frequency dependent squeezing injection ndash A broadband squeezed fieldwith a frequency-dependent squeeze angle that is optimal for gravitational-wave detectorsis produced by reflecting off an ordinary broadband squeezed field from two detuned opticalfilters [Kimble et al (2001)]

A light field with a frequency-dependent squeeze angle was first demon-strated in Ref [Chelkowski et al (2005)] see Figs 26 and 27 The experi-

72

Detuned filter cavityLockingphoto diode

Isolator

EOM R asymp 1

SHG

EOM

DBSOPA

LaserIsolator

Mode cleaner

LO

Homodyne detector

Figure 26 Generation of a frequency-dependent squeezing ndash A frequency-dependent orientation of the squeeze ellipse was first demonstrated in Ref [Chelkowskiet al (2005)] Initially a conventional spectrum of squeezed vacuum states of light was gen-erated in a squeezing resonator (lsquoOPArsquo) The squeezed vacuum was transmitted throughan optical isolator to a detuned filter cavity After reflection the squeezed vacuum stateswere absorbed in a balanced homodyne detector (BHD) The phase of the BHDrsquos localoscillator (LO) was changed for quantum state tomography of the squeezed states in diffe-rent regions of the spectrum The result showed a frequency-dependent orientation of thesqueeze ellipse see Fig 27 SHG second harmonic generation EOM electro-optical mod-ulator for applying phase modulation sidebands for cavity length control DBS dichroicbeam splitter R mirror reflectivity λ4 quarter wave plate for turning linear polarizedlight into circular polarised light and vice versa

ment consisted of a standing-wave squeezing resonator which produced ans-polarized broadband amplitude quadrature squeezed field accompanied bya dim continuous-wave DC control field with a wavelength of λ = 2πcω =1064 nm The squeeze bandwidth covered sideband frequencies up to aboutΩ(2π) = 30 MHz which corresponded to the linewidth of the squeezing res-onator The optical cavity for producing the frequency dependence of thesqueeze angle was a standing-wave cavity composed of a plane incouplingmirror of reflectivity r1 =

radic097 and a concave end mirror of reflectivity

r2 =radic

09995 The cavity length was L = 50 cm resulting in a linewidthof 147 MHz The squeezed field first passed a Faraday isolator to preventinterference effects between the filter cavity and the squeezing resonator A

73

Phase quadrature

min

max

-2 -1 0 1 2

141 MHz

-2

-1

0

1

2

Am

plit

ud

e q

uad

ratu

re

Figure 27 Frequency-dependent squeezing ndash Picture top right Reconstructed con-tour plot of the Wigner function of the sideband modulation at Ω(2π) = 141 MHz afterreflection from a 1515 MHz detuned filter cavity The state shows quantum correlationsbetween phase and amplitude quadratures ie squeezing at an angle of here about 40The white circle visualizes the standard deviation of the vacuum state uncertainty Thewhite ellipse represents the standard deviation of the squeezed uncertainty Small picturesMeasurement results on the same continuous-wave laser beam at various sideband frequen-cies around 15 MHz For each tomographic picture noise histograms of 100 equidistantquadrature angles were measured In each case the laser beam was phase locked to a ref-erence beam and the quadrature angle stably controlled and stepwise rotated The phasereference was given by a phase modulation at 198 MHz [Chelkowski et al (2005)] Thepicture was first published in Ref [Schnabel (2005)] (copyright ccopy2007 by Imperial CollegePress)

λ4-waveplate turned the s-polarized field into a circularly polarized beamwhich was then mode matched into the detuned cavity The retro-reflectedfield was analyzed by a balanced homodyne detector (BHD) for quantumstate tomography The filter cavity was electro-optically controlled to be de-tuned by 1515 MHz with respect to the DC control field The cavity length

74

control was achieved by the Pound-Drever-Hall (PDH) locking techniqueutilizing a circularly polarized laser beam that carried 15 MHz phase mod-ulation sidebands and was coupled into the filter cavity from the back Theoutput voltage of the BHD was characterized by a spectrum analyser as wellas used to perform quantum state tomography In the latter case the BHDoutput voltage was mixed down with an electronic local oscillator at differentradio-frequencies around 15 MHz and low-pass filtered to set the resolutionbandwidth to ∆Ω(2π) = 100 kHz The final electric signal corresponds to atime series of quadrature amplitude measurements XθΩi∆Ω Quantum statetomography is a method to reconstruct the phase space quasi-probability dis-tribution (Wigner function) of quadrature amplitudes from sets of measuredXθΩ∆Ω distributions when varying the angle θ [U Leonhardt (1997)] Forevery sideband frequency Ωi 100000 quadrature values were measured di-vided up on 100 equidistant quadrature angles Each quadrature angle wasstably controlled with a precision of plusmn1 Fig 27 shows the reconstructedWigner functions which were all measured on the same laser beam but atdifferent sideband frequencies For these measurements the detuned filtercavity was locked to the lower sideband at minus1515 MHz The result clearlyshows the frequency-dependent orientation of the squeeze ellipse In a morerecent experiment a frequency-dependent squeeze angle was also realized inthe kHz regime [Oelker et al (2016)]

56 Optomechanically induced (ponderomotive) squeezing

The radiation pressure of light when acting on a movable mirror re-sults in an intensity dependent phase shift [Pace et al (1993)] The cou-pling produces a so-called lsquoponderomotive effectrsquo [Braginsky and Manukin(1967)] which is of third order optical nonlinearity and which transforms abright coherent state inside an interferometer into a squeezed state of light[Vyatchanin and Matsko (1993)] This type of squeezed-light generationis usually called lsquoponderomotive squeezingrsquo or lsquooptomechanical squeezingrsquoConsequently even if no squeezed field is injected into the interferometercorrelations between the quadrature amplitudes are generated that allow forsurpassing the SQLPonderomotive squeezing as produced by the interferometer itself can onlybe exploited for evading back-action (radiation pressure noise) It can notbe used to squeeze the interferometer shot noise This is why ponderomotivesqueezing is fundamentally less extensive than injecting externally producedsqueezed states of light [Corbitt et al (2006)] suggested an external pon-

75

deromotive squeezing source for gravitational-wave detectors In this casedue to its external generation also the interferometerrsquos shot noise can besqueezed Recently ponderomotive squeezing was observed for the first time[Brooks et al (2012) Purdy et al (2013)] The achieved squeeze factors aremuch smaller than those produced by optical-parametric down-conversion[Vahlbruch et al (2016)]

Let us have a look again at Eq (59) Rotating the covariance matrix onthe right by arctan(minus

radic54minus 12) asymp minus58 indeed reveals squeezing

(cos 58 minussin 58

sin 58 cos 58

)(1 minus1minus1 2

)(cos 58 sin 58

minussin 58 cos 58

)asymp(

262 00 038

) (61)

The vacuum-noise normalized variance of 038 corresponds to about 42 dBof ponderomotive squeezing This is the general value that is produced atthe angular sideband frequency ΩSQL At higher frequencies the squeezefactor gets smaller at lower frequencies higher The squeezing strength of42 dB can be observed if the photo diode in the interferometer output portis replaced by a balanced homodyne detector using a local oscillator phase ofabout minus58 It can be shown however that the optimal signal-to-quantum-noise-ratio at the SQL is achieved for a local oscillator phase of exactly 45At this angle back-action is fully evaded

Full evasion of radiation pressure noise at all frequencies requires an op-timized frequency dependence of the relative local oscillator phase This canbe achieved by reflecting off the interferometer output field from two detunedfilter cavities [Kimble et al (2001)] The scheme was called lsquovariational out-putrsquo In the case of zero optical loss this scheme can fully evade radiationpressure noise just leaving the shot noise as the only quantum noise contri-bution

The variational-output scheme can be used to enhance the frequency-dependent squeezed input scheme The right site of Eq (60) shows thatthe output statersquos squeezing is not optimally detected in the Y -quadratureRather than with a single photo diode the detection should be done witha balanced homodyne detector with optimized phase of its local oscillatorIn this case the output lightrsquos quantum noise is solely given by squeezedshot noise The total quantum noise in Fig 24 would then be given by thelowest (dashed) horizontal line This combined scheme was called lsquosqueezed

76

variationalrsquo [Kimble et al (2001)] It can be realized by reflecting off theinterferometer output light from in total two optical filter cavities placed infront of the balanced homodyne detector

57 Conclusions

The highest quantum-noise-limited sensitivities of high-precision laserinterferometers are achieved by employing a large number of quanta tomaximize the signal strength in combination with strongly squeezed statesto minimize the quantum noise From this perspective it is clear that thequantum-noise-limited sensitivity of future gravitational-wave detectors willbe further improved ndash by increasing the light power and the squeeze factorTo be able to do so the optical loss in these devices needs to be reduced

In principle the optical loss in laser interferometers can be made smallbut never zero Recent theoretical research has shown that for any non-zero loss the sensitivity scales proportional to 1

radicn at best where n is the

average photon number per measurement This scaling is efficiently achievedby combining strongly displaced coherent states with squeezed vacuum statesof light

If a repeated measurement is not only limited by quantum measurementnoise but also by quantum back-action noise squeezed states of light can beused to simultaneously reduce both ie in the case of an interferometer shotnoise and radiation pressure noise

6 The first application of squeezed light in an operating gravita-tional-wave detector

Squeezed states of light have been successfully used to improve the sensi-tivity of the gravitational-wave detector GEO 600 from 2010 up to the pointwhen this Review was written [Abadie (2011) Grote et al (2013)] Afterdecades of proof-of-principle experiments [Xiao et al (1987) Grangier et al(1987) McKenzie et al (2002 2004) Vahlbruch et al (2005 2006 20072008) Goda et al (2008)] the implementation of a squeezed-light source inGEO 600 has resulted in the first sensitivity improvement beyond shot noiseof a measurement device that targets new observations in nature The im-plementation of squeezed states in GEO 600 was not done to provide anotherproof-of-principle demonstration but was realized because it offered a rela-tively cheap way of further improving the measurement sensitivity Of course

77

the sensitivity of GEO 600 can also be further increased by purely classicalmeans however the implementation of arm resonators to enable higher lightpowers without increasing the thermal load on the beam splitter or eventhe realization of longer interferometer arms are much more expensive Inthis respect the sensitivity improvement of GEO 600 with squeezed light canarguably be regarded as the first lsquotruersquo application that developed out of thefield of lsquononclassical (quantum) metrologyrsquo (Note that the term rsquoquantummetrologyrsquo is currently defined in different ways [Giovannetti et al (2006)Gobel and Siegner (2015)] and the term rsquononclassicalrsquo referring to a non-classical P-function gives a distinct description)

61 Gravitational waves

Einsteinrsquos General Theory of Relativity [Einstein (1916)] or simply lsquoGen-eral Relativityrsquo (GR) predicts that accelerating mass distributions producegravitational radiation analogous to electromagnetic radiation from accel-erating charges Experimental evidence of their existence is given by theobservation of the slow spiraling together of two neutron stars caused bythe loss of orbital energy to gravitational waves The inspiral rate exactlymatches the predictions of Einsteinrsquos theory [Weisberg and Taylor (2005)]Recently Advanced LIGO observed gravitational waves for the first time[Abbott (2016)] thereby giving the go-ahead for gravitational-wave astron-omy The gravitational-wave source was the final inspiraling and the mergerof two black holes 13 billion light years away from earth

Gravitational-waves evolve in the far field of the source propagate withthe speed of light and are measurable on earth with laser interferometersFig 28 displays a gravitational wave propagating along a certain directionGravitational waves are dynamical changes of space-time They are transver-sal and quadrupolar in nature and have two polarization states

A variety of known astrophysical and cosmological sources are predictedto emit gravitational radiation that should reach the Earth with a measur-able strength [Sathyaprakash and Schutz (2009)] The first gravitational waveevent detected was produced by two black holes of 36 and 29 solar massesDuring the final 02 seconds of their inspiraling they produced a peak gravi-tational strain in our solar system of 10minus21 covering frequencies up to 250 Hz[Abbott (2016)] Other predicted sources are mergers of neutron stars supernovae and background signals from the Big Bang According to GR GWsfrom complex astrophysical sources carry a plethora of information that willhave a major impact on gravitational physics astrophysics and cosmology

78

L +Δ L

Binary system

fBS

Figure 28 Space-time oscillation ndash Gravitational waves are dynamical deformations ofspace-time that form in the plane perpendicular to the direction of wave propagation Asa result distances between free-falling test masses in a transverse plane will change witha strain h = ∆LL For black hole or neutron star binary systems with orbital frequencyfBS distances will oscillate at frequency fGW = 2fBS The wavelength of this oscillation isgiven by λGW = cfGW where c is the speed of light The wave of orthogonal polarizationwith respect to the one shown is rotated by 45 around the propagation axis

62 Interferometric detection of gravitational waves

Current gravitational wave detectors are kilometre-scale laser interfero-meters [Dooley et al (2016) Aasi (2015) Acernese (2015) Aso et al (2013)]Continuous-wave laser light is split into two beams traveling in orthogonaldirections Both beams are reflected back towards the central beam splitterwhere they interfere Gravitational waves change the optical path lengthdifference and thus the light power directed towards the photo-diode thatis positioned in the signal output port of the beam splitter A gravitationalwave at frequency fGW = ΩGW(2π) reveals itself as a light-power modula-tion at the same frequency The spectral decomposition of the output signalis described by a spectrum of the quadrature amplitude YΩ∆Ω introduced inSec 3 It corresponds to the amplitude quadrature amplitude of the outputlight and relates to the differential phase quadrature of the interferometerarms

79

Photo diode

Squeezed vacuum

Faraday Rotator

Signal

Quantum noise

Coherent light XΩΔΩ

YΩΔΩ

(a) (b)

YΩΔΩFigure 29 Squeezed-light-enhanced interferometric measurement ndash (a) Michelsoninterferometer with arm cavities power recycling and signal recycling (see main text forexplanation) The interferometer is operated close to a dark fringe such that the quantumnoise entering from the dark port is back-reflected The squeezed field is mode-matched tothe signal output field (b) Phase space diagram of the gravitational-wave signal outputat sideband frequency Ω(2π) The quantum noise is squeezed below the ground stateuncertainty and thus the signal to quantum noise ratio improved

The first key ingredient of an interferometric gravitational-wave detectorare suspended heavy mirrors that can be regarded as quasi-free in the di-rection of laser light propagation thereby acting as test masses that probespacetime Being on ground current detectors are located in rather noisy en-vironments that allow the realization of undisturbed quasi-free mirrors onlyabove a sideband frequency of the order of 10 Hz Since sufficiently strongGW signals are expected up to a frequency of 10 kHz todayrsquos gravitationalwave detectors target at signals in the acoustic band from 10 Hz to 10 kHzThe quasi-free motion of the test mass mirrors in this frequency regime isachieved by suspending the mirrors as sophisticated multiple-stage pendulain vacuum chambers [Aasi (2015)] Far above the pendularsquos resonant frequen-cies which are typically around 1 Hz the centre of masses of the mirrors areisolated from vibrations of the ground and they react on frequency compo-nents of small external forces approximately as free masses The mirrors andtheir suspensions are built from materials having exquisitely high mechani-cal quality factors This helps to concentrate the thermal energy that causes

80

displacements of the mirror surface into well-defined vibrational frequencymodes At these particular very sharp frequencies no gravitational wavescan be detected

The second key ingredient of an interferometric gravitational-wave de-tector is laser light with a power of up to hundreds of kilowatts or evenmegawatts The light is quasi-monochromatic and needs to show very lowamplitude (quadrature) noise and phase (quadrature) noise at sideband fre-quencies within the detection band Low amplitude noise is necessary toprovide a shot noise limited output field It is also necessary to avoid classi-cal radiation pressure noise which becomes an issue if the light power or themirror masses in the two arms are not identical Low phase noise is requiredif the storage time of the light in the two arms is not identical This mightaccidentally occur due to different linewidths of the arm cavities or mightbe part of the interferometer design to allow for the length control schemeproposed by Schnupp [Heinzel et al (1998)] To maximize the light power in-side the interferometerrsquos cavities it should be produced in an almost perfecttransversal spatial distribution of a Gaussian TEM00 mode

Light sources of gravitational-wave detectors are ultra-stable NdYAGmaster-slave systems that provide up to 200 W of light at 1064 nm [Winkel-mann et al (2011) Kwee et al (2012)] The high power in the interferometerarms is achieved by cavity built-ups in the so-called power-recycling cavityand in the arm cavities Power recycling uses a partially reflective mirror thatis located between the light source and the interferometer beam splitter Itssurface is matched to the lightrsquos wave front and forms an optical cavity to-gether with the rest of the interferometer Since gravitational-wave detectorsare operated close to a dark fringe large power built-ups can be achievedThe highest power built-up is achieved for a mirror transmission equal to the(given) interferometer round trip loss In this case an impedance-matchedcavity is achieved The power-recycling cavity as well as the arm cavitiesare stabilized on resonance for the input light The difference between theirfunctionality is that the power-recycling cavity does not limit the detectionbandwidth of the interferometer GEO 600 as well as Advanced LIGO em-ploy a third type of cavity the so-called signal-recycling cavity Similarlyto power recycling a partially reflecting mirror that is placed between theoutput port of the beam splitter and the photodiode is used to resonantly en-hance the GW signal [Meers (1988)] The signal-recycling cavity resonantlyenhances the signal modulation fields within its linewidth without furtherenhancing the carrier light power In combination with low linewidth Fabry-

81

Perot arm resonators it can also be used to extract the signal by reducing theeffective finesse of the arm resonators for the signal sidebands This schemeis called resonant sideband extraction [Heinzel et al (1996)] The signal-recycling cavity has also been tested in a detuned setting in which just theupper or lower sideband is extracted or resonantly enhanced respectively[Heinzel et al (2002)] Current gravitational-wave detectors however usecarrier-tuned signal recycling

All these techniques are lsquoclassicalrsquo approaches for maximizing the signal-to-shot-noise ratio At frequencies above a few hundred Hertz howevershot-noise is still the limiting noise source in gravitational-wave detectorsFuture gravitational-wave detectors will therefore use even higher light pow-ers but further increasing the light power becomes more and more challeng-ing Optical materials with less light absorption need to be found to coun-teract an increasing thermal load inside the mirrors Mirror masses need tobe further increased to counteract the increasing radiation pressure noiseNonclassical approaches are superior and become more and more attractivethe farther classical approaches are pushed to the extremes Nonclassicalapproaches allow for simultaneously increasing the signal-to-shot-noise ratioand the signal-to-radiation-pressure-noise ratio without changing light poweror mirror masses see Fig 24 They also allow for a complete evasion of radia-tion pressure noise [Braginsky and Khalili (1995 1996) Kimble et al (2001)]see Subsec 56

63 Squeezed-light enhancement of the gravitational-wave detector GEO 600

In 2010 GEO 600 was equipped with the squeezed-light source shownin Fig 18 The location of the squeezed-light source close to the outputport is shown in Fig 30 It was known that GEO 600 was shot-noise limitedat sideband frequencies above about 700 Hz In this frequency regime thereplacement of the ordinary vacuum states that entered the interferometerfrom the output port by a spectrum of squeezed vacuum states was expectedto reduce the noise spectral density into the nonclassical regime It was notprecisely clear what squeezing factors could be expected since the opticalloss upon mode-matching an external field into the output port propagationalong the arms and the final photo-electric detection was not determined

Critical components were the quantum efficiency of the photo-diode aswell as the optical loss of Faraday rotator for overlapping the squeezed fieldwith the output mode Also the transversal output mode of GEO 600 was an

82

issue since it contained significant contributions from higher order modesto which a good mode matching of the squeezed mode was not possible Theimplementation of the squeezed-light source thus was accompanied with anew custom made InGaAs photo-diode with 3 mm diameter The goal was aquantum efficiency of greater 99 [Vahlbruch et al (2016)] Also the Fara-day rotator was custom-made and optimized for lowest optical loss which

Figure 30 GEO 600 ndash View into the central building of the British-German GW detectorlocated close to Hannover Germany The vacuum chambers contain the suspended beamsplitter power- and signal recycling mirrors additional input and output optics as well asmirrors to realize a double pass of the laser light through the 600 m long interferometerarms By courtesy of the AEI

83

involved a rather precise rotation of the polarisation of 45 plusmn 05 over anaperture of more than 15 mm Finally a ring cavity (output mode cleaner)was placed in front of the photo-diode which acted as a passive filter forhigher transversal modes Since GEO 600 was not limited by radiation pres-sure noise and since it used a carrier-tuned signal-recycling cavity a frequencyindependent orientation of the squeezing angle was optimum After several

100 200 300 500 700 1000 2000 3000 500010-22

10-21

10-20

Obs

erva

tory

noi

se c

alib

rate

d to

str

ain

[1radic

Hz]

10-19

10-18

10-17

Noise calibrated to test m

ass displacement [m

radicHz]

Sideband frequency f

Figure 31 Nonclassical reduction of the GEO 600 instrumental noise ndash Shownare the square roots of the noise spectral densities without squeezed vacuum states (top)and with squeezed vacuum states (bottom) [Abadie (2011)] Their calibrations [Affeldtet al (2014)] to gravitational-wave strain and differential arm length change are shownon the left and right y-axes respectively Note that both traces increase towards higherfrequencies due to the frequency-dependent signal enhancement of the signal-recyclingcavity The injection of squeezed vacuum states leads to a broadband noise reductionof up to 35 dB at shot-noise limited frequencies The spectral features are for instancecaused by excited violin modes of the mirror suspensions (600ndash700 Hz and harmonics)Data by courtesy of the AEI

months the combination of the squeezed-light source and the gravitational-wave detector succeeded The main laser of the squeezed-light source wasphase locked to the main laser of GEO 600 and a stable mode matching be-

84

tween the squeezed field and the interferometer output field was achieved[Abadie (2011)] The effect on GEO 600rsquos (strain normalized) noise spectraldensity is shown in Fig 31 At frequencies above about 700 Hz the squareroot of noise spectral density was reduced by up to one third This value cor-responds to a quantum noise variance of 045 (minus35 dB) with the shot noisevariance normalized to unity For isotropically distributed gravitational-wavesources this factor produces a detection rate increase by a factor of 153 asymp 34After its integration into GEO 600 the squeezed-light source was used in allscientific runs seeking for gravitational waves for instance in the observa-tional run S6eVSR4 that was undertaken from June 3rd to September 5th

in 2011 [Grote et al (2013)]Towards the end of 2011 right before the start of the detector upgrade

to Advanced LIGO a nonclassical sensitivity improvement was also demon-strated in one of the LIGO detectors [Aasi (2013)] In this experiment anonclassical sensitivity improvement corresponding to up to 215 dB abovefrequencies of about 150 Hz was achieved The successful test is a strongmotivation for a squeezed-light upgrade of Advanced LIGO Note that thedesign of Advanced LIGO was completed in 1999 and squeezed-light sourceswere not mature at those times

In the past years the squeezing enhanced GEO 600 detector was not onlyused for observations but also was the control of the injected squeezed modefurther improved Stabilizing the overlap between squeezed mode and brightmode of the interferometer to close to perfect is necessary to reduce theeffective optical loss and to maximize the measurable squeezing factor Re-cently superior methods for stabilizing the longitudinal phase of squeezedvacuum mode were found [Dooley et al (2015)] and the first automatic align-ment system for stabilizing and optimizing the transversal mode overlap wasdemonstrated [Schreiber et al (2016)]

64 Are squeezed states the optimal nonclassical resource ingravitational-wave detectors

For a given number of photons Eq (41) quotes the ultimately smallestphase change that can be measured with a signal-to-noise-ratio of one Thescaling with number of photons per measuring time of this Heisenberg limitseems appealing compared to the scaling achievable with coherent states orsqueezed states according to Eqs (38) and (39) respectively The Heisenberglimit however is only valid for precisely zero photon loss Since the non-classical states required to achieve Eqs (40) and (41) show an exponentially

85

increasing sensitiveness to loss when increasing the photon number the ac-tual scaling can not be deduced from Eq (40) Proposals to use Fock statesand the so-called N00N states for optimizing interferometer sensitivities [Hol-land and Burnett (1993) Dowling (1998) Mitchell et al (2004) Afek et al(2010)] are thus only applicable when the experiment is conditioned on zerophoton loss As discussed in recent publications the correct expression forthe fundamental sensitivity limit needs to consider not only the total photonnumber inside the interferometer but also the total photon loss [Dorner et al(2009) Ko lodynski and Demkowicz-Dobrzanski (2010) Knysh et al (2011)Escher et al (2011) Demkowicz-Dobrzanski et al (2012)]

Based on these earlier works Ref [Demkowicz-Dobrzanski et al (2013)]proved that the nonclassical sensitivity enhancement of GEO 600 reportedin Ref [Abadie (2011)] has been exceedingly close to fundamental quantuminterferometry bound under given energy constraints and photon loss lev-els More than that it was generally proven that the approach of com-bining displaced coherent states and squeezed vacuum states is optimal forgravitational-wave detectors

In Ref [Abadie (2011)] the gravitational-wave detector GEO 600 used aneffective number of photons per second of approximately n = 2 middot 1022 whichcorresponded to a total optical power inside the interferometer arms of P asymp37 kW at a wavelength of 1064 nm The total optical loss was 1minus η asymp 038The injected squeezing factor was eminus2r asymp 01 For these numbers the ratioof Eqs (43) and (44) is calculated to

∆φCSVmin

∆φgenmin

asymp

radicηeminus2r + 1minus η

1minus ηasymp 108 (62)

which is a good approximation within the limit of large coherent state dis-placements α sinh2r The quantum noise of GEO 600 including thesqueezed-light source was just 8 above the fundamental quantum inter-ferometry bound An increased squeezing strength of 16 dB (eminus2r asymp 0025)which is in reach would bring the approach based on coherent states andsqueezed vacuum states to within just 2 above the fundamental bound

Future GW detectors will have significantly reduced optical loss values(1 minus η) lsquoLossrsquo includes scattering and absorption at mirrors non-perfectfringe contrasts and the non-perfect quantum efficiency of the photo detec-tor Optical loss reduction is important for at least four reasons First itleads to an increased signal second it leads to a reduced quantum noise when

86

employing squeezed states third less absorption reduces the thermal load onthe test mass mirrors and fourth less scattering reduces the probability ofback-scattered light which produces disturbance signals [Billing et al (1979)Vahlbruch et al (2007) Punturo et al (2014)] The higher the finesse valuesof the arm and signal-recycling cavities are the more significant is opticalloss at mirror test masses the beam splitter and the signal-recycling mirrorThe finesse value of the power-recycling cavity and the loss of mirrors andlenses that guide the output field to the photo-diode are less critical Suitablephoto detectors of 995 quantum efficiency are available today [Vahlbruchet al (2016)] but achieving a total optical loss of 10 is still challengingThe reason for that is that first of all a measurement device aiming for bestabsolute sensitivity should use as much quanta (photons) as possible Highfinesse values for the enhancement cavities are thus essential but results inan unavoidable scaling-up of the effect of mirror losses A realistic exampleof future gravitational wave detectors thus considers η = 09 with a squeezingfactor of 20 dB (eminus2r = 001) In this case the quantum noise will be about4 above the ultimate fundamental bound for a given photon number

From Eq (62) it can be concluded that there is no need for any moresophisticated nonclassical states than squeezed states In particular non-classical states with a defined photon number such as N00N states are notrequired Within the approximation quoted this result is independent ofthe photon number This result is also independent of the physical systemused for interferometric phase estimation and can also be made for quantum-enhanced atomic clock calibration in the presence of dephasing Here the-oretical results indicate that the precision of Ramsey interferometry withspin-squeezed states is close to the optimal one in the asymptotic regime ofa large number of atoms [Huelga et al (1997) Ulam-Orgikh and Kitagawa(2001) Escher et al (2011)] as already stated in Ref [Demkowicz-Dobrzanskiet al (2013)] More sophisticated nonclassical states with fixed number ofn quanta might still be useful for the exceptional case when the absorptionof one quantum already results in zero measurement sensitivity anyway Anexample is an ensemble measurement where the absorption of a single photondemolishes the source of the phase change to be characterized A typicallyused approach of conditioning the measurement result on n clicks of n singlephoton counters conditions on precisely zero loss and is thus able to use theadvantage of Eq (40) over Eq (39)

87

65 Conclusions

Squeezed states of light will contribute to realizing gravitational-wave ob-servatories with much higher sensitivities than existing or planned ones Tobenefit from squeezed states in a most efficient way optical loss in terms ofabsorption and scattering must be minimized In particular the optical lossof mirror coatings and mirror substrates need to be minimized The rele-vant mirrors include the test masses the balanced beam splitter the signalrecyclingextraction mirror and all optical components between the latterand the photo diode Excellent spatial mode matching between the brightinterferometer field and the squeezed vacuum field is also of great impor-tance Achieving this requires further improvement of the surface figures ofall reflective optical components of the interferometer as well as improvedhomogeneity of all optical components that the light passes through

The quantum noise reduction achieved in a gravitational-wave detectoris of course always smaller than the highest squeeze factor provided by thesqueezed-light source As an example let us consider the observation of 15 dBof nonclassical noise suppression directly at the source If the squeezed fieldsenses an additional loss of 5 when propagating through the interferometerwhich is a very challenging number from todayrsquos point of view the remainingsqueezing level is about 11 dB see Eq (16)

7 The application of 2-mode-squeezed light in laser interferome-ters

71 Quantum Dense Metrology

At first glance the application of bi-partite (two-mode) squeezed statesto a device whose goal is measuring a single observable seems meaninglessSqueezing the uncertainty of that observable should be the optimum one cando This is indeed true when concerning just quantum noise but recentlyit was discovered that in the presence of classical disturbances bi-partitesqueezing can improve such measuring devices [Steinlechner et al (2013)]The concept was named quantum dense metrology (QDM) The potentialimprovement of a gravitational-wave detector with bi-partite squeezed statesis shown in Fig 32 (a) A description is given in the caption The pre-condition for a potential improvement can be best understood within a phasespace diagram Fig 32 (b) contains two different kinds of lsquosignalsrsquo The firstis the actual signal which always shows up as a phase space displacement

88

along the Y axis The second is a disturbance signal that can produce adisplacement in arbitrary direction in phase space A prominent example

Faraday Rotator

Coherent light

Photo detectors Balanced homodyne detectors

Squeezed vacuum

Squeezed vacuum

Quantum noise in

Y (A) and X (B)

XΩΔΩ(B)

YΩΔΩ(A)

(a) (b)

Disturbance projected

onto Y

and X

XΩΔΩ(B)

YΩΔΩ(A)

Example of (unknown) disturbance

Signal

Figure 32 Bi-partite-squeezed-light-enhanced measurement ndash (a) Setup for theapplication of bi-partite (two-mode) squeezed light in a laser interferometer on the basisof QDM Two squeezed vacuum fields are overlapped on a balanced beam splitter withprogrammable squeeze angles for instance with a relative angle of 90 which produces abi-partite state as shown in Fig 12 The beam splitter outputs are entangled for any rela-tive angle greater than zero One part is matched to the interferometer mode The secondpart is kept outside as a reference beam The interference of the interferometer outputand the reference beam is arranged with such a phase difference that it reproduces the twosqueezed inputs on the photo detectors The two squeezed beams are photo-electricallydetected measuring the respective squeezed quadrature (using balanced homodyne detec-tors) Both beams carry half of all interferometer induced modulations which includesignals as well as disturbances A single readout as shown in Fig 21 cannot distinguishbetween the two kinds The double readout shown here provides additional informationand allows for recognition of the disturbance [Steinlechner et al (2013)] as well as in prin-ciple a modeling of the disturbance and with a correct model an improvement of thenoise spectral density of the interferometer [Ast et al (2016)] (b) Phase space diagramdescribing phase quadrature readout A as well as amplitude quadrature readout B Bothshow squeezed quantum noise The amplitude quadrature readout does not contain anygravitational-wave signal ie any feature in this channel must be due to disturbancesThis information can be used to improve the interferometer

89

for such a disturbance is parasitic interference due to back-scattered laserlight [Vahlbruch et al (2007)] Back-scattering is a limiting noise at lowsignal frequencies of gravitational-wave detectors [Billing et al (1979) Vinetet al (1997) Hild (2007) Ottaway et al (2012) Punturo et al (2014)] Notethat all noise that couples in via unwanted motions of the test mass mirrorsso-called lsquodisplacement noisersquo always produces a phase space displacementalong the Y axis and cannot be tackled with QDM

Fig 33 shows measurement results obtained in Ref [Steinlechner et al(2013)] In a table-top experiment one part of a bi-partite squeezed state ofa continuous-wave mode at 1064 nm was mode-matched into the output portof a Michelson laser interferometer operated at its dark fringe in full analogyto Fig 32(a) A lsquosignalrsquo was produced by driving the piezo behind one of theend mirrors at a frequency of 555 MHz The lsquodisturbancersquo was introducedby re-injecting a small amount of light that leaked through the second endmirror with an additional piezo-mounted mirror The piezo was driven ata frequency of 517 MHz to produce a phase modulation An additionalDC voltage defined an arbitrary and unknown optical path length of thelight before being re-injected and as such the phase space orientation of thedisturbance signal This mechanism of a parasitic interference is realizednaturally in any interferometric device due to back-scattering of quanta frommoving surfaces in the environment

The interferometer output consisted of the signal as well as the distur-bance with a quantum uncertainty given by one subsystem of the bi-partiteentanglement It was overlapped with the second subsystem of the entan-gled state on a balanced beam splitter and the two outputs were analysedwith balanced homodyne detectors The phases of the bi-partite entangle-ment and the BHD local oscillators were controlled to resemble Fig 32(b)ie both BHDs measured a squeezed uncertainty regardless of the phase ofthe (generally unknown) disturbance

The beam splitter that combines interferometer output and the entangledreference beam unavoidably splits the signal as well as the disturbance intotwo paths For a balanced beam splitter this generally reduces the signaland disturbance power by 3 dB for both quadrature measurements Fig 33shows however that both BHDs performed about 6 dB below shot noisewhich demonstrates the usefulness of the scheme The squeeze factor can inprinciple be infinite which thus qualifies the lsquo3 dB penaltyrsquo In the abovefigure the additional information from the second BHD output was used torecognize the parasitic interference in the first BHD output providing a lsquovetorsquo

90

50 51 52 53 54 55 56 57

50 51 52 53 54 55 56 57-70

-75

-80

-85

-90

-70

-75

-80

-85

-90

Y (A) (Ω2π)Ω∆Ω

Frequency ( f )

Noi

se p

ower

(dB

m)

Noi

se p

ower

(dB

m)

X (B) (Ω2π)Ω∆Ω

~ ndash6 dB

~ ndash6 dB

Shot noise reference

Shot noise reference

Signal Projected disturbance

Projected disturbance

Vet

o

No signal

Figure 33 Bi-partite-squeezed-light-enhanced measurement ndash The result wasachieved in a table-top setup [Steinlechner et al (2013)] In the two panels the lower

(blue) traces show the squeezed quadrature noise-power spectra ∆2Y(A)Ω∆Ω(Ω2π) (top) and

∆2X(B)Ω∆Ω(Ω2π) (bottom) as simultaneously measured with balanced homodyne detectors

lsquoArsquo and lsquoBrsquo respectively The conventional Y -measurement (top) cannot distinguish be-tween signal and disturbances The additional X-measurement (bottom) does not detectany phase quadrature signal thus any feature in this measurement is a parasitic signal dueto a disturbance The respective projection onto the Y -measurement can thus be lsquovetoedrsquoIn a more sophisticated approach the X-data might be used to model and then to elim-inate the disturbance as well as its projections on both quadrature measurements Theresult is a reduced spectral density of the actual phase quadrature measurement [Ast et al(2016)] Traces shown here are slightly sloped due to the decreasing transfer functions ofthe balanced homodyne detectors The resolution bandwidth was ∆Ω(2π) = 10 kHz thevideo bandwidth was 100 Hz All traces were averaged three times

signal to trigger its removal from the data streamThe question arose whether the additional information can be used to

reduce the actual noise spectral density of the first measurement ie to re-cover signals that were buried by parasitic interferences Very recently it wasshown that this is indeed possible Ref [Ast et al (2016)] reports a table-top proof-of-principle experiment in which the additional information in the

91

QDM approach could be used for improving the sensitivity of an interfero-meter The measurement sensitivity was improved from above-shot-noise tosub-shot-noise (sub-Poissonian) performance This result was possible notbecause the way the parasitic interference arose was known but because theadditional information provided by QDM allowed for fitting a model of theexcess noise to the readout data

Quantum dense metrology (QDM) as shown in Fig 32 improves a mea-surement by simultaneously reading out two conjugate observables Bothreadout observables show a squeezed quantum noise and act as estimatorsof independent physical quantities This situation was recently described aslsquoquantum-mechanics freersquo [Tsang and Caves (2012)] QDM is based on anEinstein-Podolsky-Rosen (EPR) entangled [Einstein et al (1935)] bi-partitesystem as described in Subsec 34 EPR entanglement was previously con-sidered for the quantum-informational task of dense coding which doublesthe capacity of quantum communication channels [Bennett et al (1992)Braunstein and Kimble (2000)] The application of EPR entanglement inmetrology was first proposed by DrsquoAriano et al [DrsquoAriano et al (2001)]

72 Conclusions

A single beam that carries an optimized spectrum of squeezed vacuumstates and that is injected into the interferometerrsquos dark port provides themost efficient and practically optimal approach to reduce the quantum noisein laser interferometers by means of nonclassical states (see Section 5) Theconclusion of the section here is that two entangled beams provide a superiorapproach if the interferometerrsquos sensitivity is limited by classical noise thatis not exclusively restricted to the actual observable which is the phasequadrature amplitude Y Parasitic interferences due to laser light that isbackscattered from vibrating surfaces are an example Current gravitational-wave detectors use light fluxes of about 1024 photons per second [Abbott(2016)] Just a single photon per second and hertz which leaves the mainlight beam and is backscattered from a vibrating surface and in this way getsfrequency shifted into the detection band produces a significant disturbancesignal The lsquoquantum-dense metrologyrsquo approach might provide a powerfultechnique to tackle this problem

Very recently it turned out that QDM is not the only technique that mayexploit EPR entanglement to improve phase measurements Ref [Ma et al(2017)] proposes to use EPR entanglement to simultaneously suppress shotnoise and radiation pressure noise in a gravitational-wave detector without

92

the need for an additional filter cavity (confer subsection 55) In this caseEPR entanglement is exploited that is carried by one broadband squeezedbeam and that is present between quadrature amplitudes defined with re-spect to different optical frequencies ω and ωprime as investigated in Ref [Hageet al (2010)] Such lsquofrequency multiplexedrsquo EPR entanglement might resultin considerably lower costs of building a gravitational-wave detector with abroadband simultaneous squeezing of shot noise and radiation pressure noiseAlso this proposal does not lead to a fundamentally lower quantum noise butrather improves on classical aspects of an interferometer

8 Summary and Outlook

In many cases experiments that involve interference of quantum statescan be described in a semi-classical way This description uses the classicalwave picture for the interference part of the experiment and subsequently theclassical particle picture when the states transfer their energy to a detectoror more generally to a thermal bath This semi-classical description is notpossible when using the specific class of lsquononclassicalrsquo states Squeezed statesof light are a prominent example of these Squeezed states and other nonclas-sical states allow for observations that made Einstein Podolsky and Rosenformulate their critical and seminal paper on quantum theory [Einstein et al(1935)]

In the review here it is argued that after many successful proof-of-principle experiments with nonclassical states in the past decades the routineuse of squeezed-light in observational runs of the gravitational-wave detectorGEO 600 goes beyond proof-of-principle and is a true application of nonclassi-cal light Since 2010 the squeezed-light source has improved the measurementsensitivity of GEO 600 in basically every observational run [Abadie (2011)Grote et al (2013) Dooley et al (2016)] At quantum noise limited frequen-cies ie above a few hundreds of hertz the sensitivity has been improvedcorresponding to a squeezing strength in the noise spectral density of up to37 dB which corresponds to an increase of the average gravitational-wavedetection rate by a factor of 043minus32 = 36 This success is a strong motiva-tion to also equip the Advanced LIGO Virgo and Kagra gravitational-wavedetectors with squeezed light Similar improvement factors even down tolower signal frequencies are expected [LSC (2013)] The achievable improve-ment factors are mainly limited by the optical loss on the squeezed states

93

and much higher factors are achievable in principleUp to now squeezed states have not been used to reduce the radiation-

pressure noise in gravitational-wave detectors The reason is that so far othernoise sources are larger than radiation pressure noise and such an effect can-not be observed It is expected however that future gravitational-wavedetectors will eventually be partly limited by radiation pressure noise Fromthis point on squeezed light will be used to simultaneously reduce shot noiseand radiation pressure noiseSqueezed states are the optimum nonclassical states for gravitational-wavedetectors or more generally for all laser interferometers operating with largeaverage photon numbers per measuring interval [Demkowicz-Dobrzanski et al(2013)] In addition to using higher light powers and heavier test mass mir-rors higher squeeze factors will thus contribute to mitigate the lightrsquos quan-tum noise in laser interferometers

Two-mode (bi-partite) squeezed light has not been used in gravitational-wave detectors so far They are not capable of further reducing the quantumnoise in laser interferometers but they can be used to mitigate classicalnoise that originates from fluctuating phase space displacements A well-known such noise source is back-scattered light Proof-of-principle experi-ments were performed recently [Steinlechner et al (2013) Ast et al (2016)]This new technique could turn out to be valuable in next generations ofgravitational-wave detectors in particular in those targeting high sensitivi-ties at low sub-audio signal frequencies and using high light powers Suchan implementation in gravitational-wave detectors does not require any newtechnology Compared to a squeezed-light enhanced interferometer just asecond squeezed-light source is required

It is certainly remarkable that those quantum states that made EinsteinPodolsky and Rosen falsely think quantum theory incomplete are now ex-ploited as new technologies in measurement devices targeting new observa-tions in nature

Acknowledgements

RS thanks M Ast J Bauchrowitz C Baune S Chelkowski J DiGugliel-mo A Franzen B Hage J Harms A Khalaidovski L Kleybolte NLastzka M Mehmet S Steinlechner and H Vahlbruch for their contri-butions many fruitful discussions and their support with the figures and JFiurasek for many valuable comments on the manuscript Thanks are also

94

due to Y Chen F Khalili and H Miao for fruitful discussions within thequantum noise working group of the LIGO Scientific Collaboration (LSC)Special thanks are due to H Vahlbruch and H Grote together with theGEO 600 team for their pioneering work on the squeezed-light implementa-tion in GEO 600 RS is supported by the Deutsche Forschungsgemeinschaft(Grant No SCHN 757-6) and by the European Research Council (ERC)project lsquoMassQrsquo (Grant No 339897)

References

References

Aasi J et al Jul 2013 Enhanced sensitivity of the LIGO gravitationalwave detector by using squeezed states of light Nature Photonics 7 (8)613ndash619URL httpwwwnaturecomdoifinder101038nphoton2013177

Aasi J et al Apr 2015 Advanced LIGO Classical and Quantum Gravity32 (7) 074001URL httparxivorgabs14114547httpstacks

ioporg0264-938132i=7a=074001key=crossref

20895763c84bce3f8929251031b2475c

Abadie J et al Sep 2011 A gravitational wave observatory operatingbeyond the quantum shot-noise limit Nature Physics 7 (12) 962ndash965URL httparxivorgabs11092295httpwwwnaturecom

doifinder101038nphys2083

Abbott B P et al Feb 2016 Observation of gravitational waves from abinary black hole merger Phys Rev Lett 116 061102URL httplinkapsorgdoi101103PhysRevLett116061102

Acernese F et al 2015 Advanced virgo a second-generation interferome-tric gravitational wave detector Classical and Quantum Gravity 32 (2)024001URL httpstacksioporg0264-938132i=2a=024001

95

but also a partially negative Wigner function (A cat state is a quantum su-perposition of two macroscopically distinct states [Monroe (2002)] referringto Schrodingerrsquos-cat gedanken experiment [Schrodinger (1935)]) In practicesqueezed states can even be regarded as superior nonclassical states becausethey represent the only nonclassical state that has been produced in a steadystate fashionIn almost all experiments so far the generation of Fock states and cat statesinvolves a probabilistic event such as the detection of a photon in anotherbeam path to herald these states In fact squeezed states provide the non-classical resource for the probabilistic preparation of Fock states as well as catstates But only the squeezed states themselves show a nonclassical effect in astationary way Limited only by the time duration and the frequency span ofthe mode that is in a squeezed state the squeezing effect can be continuouslyobserved independently of the time when the measurement is performed andalso independently of the measurement integration time This fact is of greatimportance for applications of squeezed states in measurement devices sincea squeezed-light-enhanced measurement remains unconditional and the ef-fective measurement time is not reduced

In past decades squeezed states of light were used in many proof-of-principle experiments to research their potential for improving the sensitivityof laser interferometers [Grangier et al (1987) Xiao et al (1987) McKen-zie et al (2002) Vahlbruch et al (2005) Goda et al (2008) Taylor et al(2013)] or the performance of imaging beyond the shot-noise limit [Lugiatoet al (2002) Treps et al (2003)] both accompanied by a huge number oftheoretical works Potential applications in secure optical communication(quantum key distribution) were also proposed and proof-of-principle ex-periments demonstrated [Ralph (1999) Furrer et al (2012) Gehring et al(2015)] This review restricts itself to the improvement of laser interfero-meters since only here has the application of squeezed light gone beyondproof-of-principle The gravitational-wave detector (GWD) GEO 600 hasoperated with squeezed light now for more than seven years starting in 2010[Abadie (2011) Grote et al (2013)] GEO 600 is a 600 m long Michelsonlaser interferometer built for the detection of gravitational waves Thesewaves are audio-band and sub-audio-band changes of space-time curvatureoriginating from cosmic events such as the merger of neutron stars or blackholes as detected recently [Abbott (2016)] In GWDs such as GEO 600 [Doo-ley et al (2016)] Advanced LIGO [Aasi (2015)] Advanced Virgo [Acernese

8

(2015)] and KAGRA [Aso et al (2013)] conventional laser technology hasbeen pushed to extremes over the past decades Noise spectral densities nor-malized to space-time strain of less than 10minus23 Hzminus12 have been measured[Abbott (2016)] Progress will continue and based on the successful appli-cation in GEO 600 squeezed light is now widely accepted to provide a newadditional technology to contribute to the new field of gravitational-waveastronomy It was also successfully tested in one of the LIGO detectors in2013 [LSC (2013)] and is an integral part of the European design study forthe 10 km Einstein-Telescope [Punturo et al (2010)]

GEO 600 has already taken several years of lsquosqueezedrsquo observational datawhich has increased its sensitivity at signal frequencies above 500 Hz Withthe implementation of a squeezed light source in GEO 600 the applicationof nonclassical states in metrology has been pushed beyond merely proof-of-principle

lsquoTwo-mode squeezed statesrsquo show a squeezed uncertainty in at least onejoint continuous variable of two subsystems lsquoArsquo and lsquoBrsquo Examples of jointvariables are differences and sums of phase-space observables of A and BTwo-mode squeezed states not only belong to the class of nonclassical statesbut due to their bi-partite character also to the class of lsquoinseparablersquo orlsquoentangledrsquo states They are the ideal states to demonstrate the Einstein-Podolsky-Rosen paradox [Einstein et al (1935)] as first achieved in [Ouet al (1992)] Apart from fundamental research on quantum mechanicsrecent proof-of-principle experiments demonstrated their usefulness in inter-ferometric measurements that go beyond the application of simple squeezedstates [Steinlechner et al (2013) Ast et al (2016)] This experiment is thefinal topic of this review

2 Observations on light fields in squeezed states

Generally there are two different kinds of observables that can be subjectof a measurement performed on a quantum system The first kind is associ-ated with the systemrsquos wave property In optics it corresponds to the electricfield strength at a given phase angle ϑ The according (dimensionless) opera-tors are called the quadrature amplitudes Xϑ and have a continuous spectrumof eigenvalues Quadrature amplitudes are measured in very good approx-imation with a balanced homodyne detector using the interference with a

9

bright local oscillator beam see Fig 3 (a) In practice any measurement ofXϑ integrates over some sideband (Fourier) spectrum within the angular fre-quencies Ω plusmn ∆Ω2 The sideband information always needs to be quotedA straight forward but rather untypical way is by adding subscripts whichleads to Xϑ

Ω∆Ω The classical analogue of the quadrature amplitude operator

XϑΩ∆Ω is the modulation depth of the optical field at modulation phase angle

ϑ and at angular modulation frequency Ω measured over the band ∆Ω lt 2ΩThe uncertainties of the statersquos quadrature amplitudes at different phases ϑare limited by a Heisenberg uncertainty relation see section 3 The secondkind of measurement is associated with the systemrsquos particle property andis given by the photon number operator n associated with a measuring timeinterval ∆t Its precise measurement requires a photon counter ideally withsingle photon resolution The measurement result obviously has a discretespectrum Continuous as well as discrete observables are usually subject toquantum uncertainties and thus quantum noiseUsually the measurementrsquos integration time and frequency band actuallydefine the physical system that is characterized In quantum optics experi-ments the interrogated physical system is called a lsquomodersquo

21 Definition of a lsquosingle modersquo

Let us define a light field or generally any quantum system to be asingle mode if it corresponds to the lsquosmallest entity of a waversquo In this caseits spectral and temporal distributions as well as waist size and divergenceare at their Fourier limits and all other properties such as optical axis waistposition and polarization are well defined For instance a linearly polarizedlongitudinal resonance of an optical standing-wave cavity defines such a singlemode if the cavity finesse is high and transversal modes are non-degenerateThe complete photo-electrical detection of a cavity mode however is notstraight forward Most quantum optical experiments are instead performedon propagating light In this case single modes are defined by spatial filtersand by temporal-spectral measurement windows both being at the Fourierlimit Examples for single modes are a laser pulse and a spectraltemporalcutout from a continuous observation of a quasi-monochromatic continuous-wave light beam in the spatial TEM00 mode both at the Fourier limits

In classical physics the only remaining free parameter of a given singlemode is its excitation energy In quantum physics the situation is differentFor a given energy a single mode can be in many different quantum states

10

which differ in their quantum statistics Examples are coherent states num-ber (Fock) states and squeezed states

22 Observations on squeezed states using a single PIN photo-diode

An ideal PIN photo-diode absorbs the full energy of a light mode andproduces one photo electron for every absorbed photon energy It uses theinternal photo-electric effect inside a semiconductor such as silicon or In-GaAs In contrast to avalanche photo-diodes PIN photo-diodes operatewith unity gain lsquoPINrsquo stands for lsquopositiversquo lsquointrinsicrsquo and lsquonegativersquo and isdescribing the doping of the semiconductor layers A PIN photo-diode is op-timally suited for the continuous monitoring of a rather bright light field ofup to several tens of milliwatts An example is the photo-diode in the outputport of a gravitational-wave detector as shown in Fig 2 (a) The prominentwavelength of 1064 nm which is emitted by NdYAG lasers has an opticalfrequency of ν = ω(2π) = 282 middot 1014 Hz The period of the field oscillationis a few femtoseconds and cannot be directly resolved with photo-electric de-tectors However variations of the electric field around the averaged opticalfield oscillation on longer time-scales can be resolved Applying an electronicbandpass filter at the sideband angular frequency Ω plusmn ∆Ω2 to the photovoltage provides information about the lsquodepth of the lightrsquos amplitude mod-ulationrsquo which is also called the lsquoamplitude of the amplitude quadraturersquo Itcan also slowly vary in time and reads

Xϑ=0

Ω∆Ω(t) equiv XΩ∆Ω(t) equiv X (1)

The subscript is usually skipped as it is done with the time dependenceas indicated on the right Applying the electronic bandpass filter in factdefines the mode of the light being detected The structure of the defini-tion in Eq (1) forms the basis of interferometric signals and quantum noisealso in the semi-classical case of coherent states Lets take an example Inthe recent observation of gravitational waves [Fig 1 bottom row in Abbott(2016)] the time-frequency representation of the gravitational-wave signalcorresponded to the amplitude quadrature amplitude XΩ∆Ω(t) of the inter-ferometer output light Note that a larger value of ∆Ω allows for changes ofthe quadrature amplitude on shorter time scales

If the light fieldrsquos lsquomodulation modersquo does not contain any quanta simplybecause there are no photons that have a frequency difference of plusmnΩ with

11

respect to the carrier it is in its ground state In this case lsquovacuum noisersquois observed which originates from the ground state uncertainty Since thevacuum noise only becomes measurable as a beat with a bright light field itcan also be seen as the carrierrsquos band-path filtered shot noise A modulationmode in a displaced vacuum state (a coherent state) corresponds to nonzerocoherent modulationThe measured level of the vacuum noise generally depends on the power ofthe bright carrier light and on the electronic amplification In any case itprovides the reference for certifying lsquosqueezingrsquo Observations using a singlePIN photo-diode require an independent measurement to quantify vacuumnoise A necessary condition is that attenuating the total fieldrsquos light powerresults in the same attenuation of the measured XΩ∆Ω values If they showa stronger attenuation a coherent modulation or thermal noise might bepresent If they show a weaker attenuation the photo-diode and its electron-ics might be saturated

Fig 2 (b) illustrates how a broadband squeezed field improves the mea-surement of an amplitude modulation in time domain based on a PIN photo-diode Shown is a simulated time sequence of XΩ∆Ω-data sampled from thephotoelectric voltage In this simulation all sideband frequencies from zero(DC) to the cutoff frequency of the detector electronics (Ωcut) are included(Ω = ∆Ω2 = Ωcut2) No additional band pass filter is applied making it amaximally broadband detection Although the data in Fig 2 (bi) contains aclassical amplitude modulation of the detected light this signal is not visibledue to random noise here representing shot noise Fig 2 (bii) shows thesame situation but with shot noise that is squeezed over the full detectionband The quantum uncertainty of the modulation depth is squeezed andthe classical signal becomes visible

It needs to be noted that a single PIN photo-diode can only measure theamplitude of the amplitude quadrature XΩ∆Ω(t) but not the non-commutingobservable the lsquoamplitude of the phase quadraturersquo

Xϑ=90

Ω∆Ω (t) equiv YΩ∆Ω(t) equiv Y (2)

For values that are small compared to the field strength of the bright fieldthe quantity Y approximately describes the bright fieldrsquos lsquophase modulationdepthrsquo

12

23 Observations on squeezed states using a balanced homodyne detector

-72

-69

-66

-63(i)

(ii)

(iii)

(a)

LO

(Squeezed)signal input

Phaseshifter

PD1

PD2

5050

01 0200 03

-75

-60

-57Balanced homodyne detector

(b)

Time [s]

Noi

se p

ower

[dB

m]

Figure 3 Balanced homodyne detection (BHD) ndash (a) Setup The quadrature atchoosable angle ϑ of the signal field is measured by overlapping the latter with a localoscillator (LO) field of the same mode parameters on a balanced beam splitter and record-ing the difference voltage from two PIN photo-diodes as shown In order to meet theBHD approximation the LO needs to be much more intense than the signal field A closeto perfect mode overlap between LO and signal input field is crucial For a non-perfectoverlap the detector measures the input state with unwanted contributions of the vacuumstate (b) Noise power measurements (i) on an electronically amplified and band-passfiltered quadrature amplitude of the vacuum field (Xvac

Ω∆Ω) (signal input blocked) (ii) on

a squeezed quadrature (XsqzΩ∆Ω) of a squeezed vacuum state (ϑ = 0) and (iii) on re-

spective quadratures of the same squeezed state where the phase angle ϑ was continuouslyshifted by changing the optical path length of the LO The measurement data shows about5 dB of squeezing and was first published in [Chelkowski et al (2007)] Ω2π = 5 MHz∆Ω2π = 100 kHz

In contrast to a single PIN photo diode a balanced homodyne detector(BHD) is suitable to measure the quantum statistic of all types of modu-lations ie for all angles ϑ Such a detector consists of two identical PINphoto-diodes a balanced beam splitter and an external homodyne local os-cillator field that is much brighter than the signal beam and that has anadjustable phase The signal beam corresponds to the squeezed field whichin many experiments is in a squeezed vacuum field having an optical powerthat usually corresponds to just a few photons per mode The two beams areoverlapped on the balanced beam splitter with close to perfect mode match-ing and the two interference outputs are focussed onto the photo diodes seeFig 3 (left) The electric output signal of the BHD is the difference of the

13

photo diode voltages The LO takes over the role of the carrier light fieldbut with the possibility to choose the phase shift ϑ This way eigenvaluesof X Y or Xϑ can be measured where the latter is given by the followinglinear combination of the first two

Xϑ(t) = cos(ϑ) X(t) + sin(ϑ) Y(t) (3)

If the modulation depths of signal and local oscillator beams are weak com-pared to their coherent amplitudes |α| and |αLO| the output voltage of aBHD corresponds to eigenvalues of the following operator

V (t) prop 2cos(ϑ) |αLO||α|+ |αLO| Xϑ(t) + |α| XϑLO(t) (4)

The lsquohomodyne approximationrsquo further involves |αLO| |α| such that theterm on the right can be neglected even if the local oscillator shows someclassical quadrature excitation The output voltage of a BHD is usually spec-trally analysed or at least spectrally filtered which removes the DC part infull analogy to a single photo diode (see previous subsection) Sampling thefiltered voltage provides eigenvalues proportional to the generalized quadra-ture amplitude in Eq (3)

V BHDΩ∆Ω(t) prop |αLO| Xϑ

Ω∆Ω(t) (5)

Fig 3 (a) shows the setup of a balanced homodyne detector for the char-acterization of squeezed states Setting ϑ = 0 eigenvalues of the ampli-tude modulation depths can be sampled from the photo voltage accordingto Eq (5) Setting ϑ = 90 eigenvalues of the phase modulation depths aremeasured The datarsquos expectation values 〈Xϑ〉 provide the coherent displace-ment of the squeezed state The datarsquos variances

∆2Xϑ equiv 〈(Xϑ)2〉 minus 〈Xϑ〉2 (6)

provide the statersquos (quantum) noise A pure squeezed state as well as asqueezed state that experienced photon loss have Gaussian quantum statisticsand are thus fully described by the expectation values and variances (first andsecond moments) of two orthogonal quadratures but only if one quadraturereflects the lowest quadrature variance

14

In most experiments with squeezed light the photo electric voltage ac-cording to Eq (5) is not sampled with a data aquisition system but the signalis directly fed into a spectrum analyser measuring the noise power of the volt-age If the expectation value 〈Xϑ〉 is zero the noise power is proportionalto the variance ∆2Xϑ in Eq (6) The reference for quantifying the squeezefactor is measured by blocking the (squeezed) signal field in Fig 3 (a) Themeasured vacuum noise level corresponds to the LOrsquos (electronically ampli-fied) shot noise level

Traces (ii) and (iii) in Fig 3 (b) show measured noise powers of the mod-ulation mode (Ω2π = 5 MHz ∆Ω2π = 100 kHz) being in a squeezed vac-uum state (i) is proportional to the variance of the ground state uncertainty∆2Xvac

Ω∆Ω (ii) is proportional to the quantum noise variance of the squeezed

quadrature amplitude ∆2XsqzΩ∆Ω (iii) is proportional to the quantum noise

variance of the quadrature amplitude with scanned phase ∆2XΩ∆Ω(ϑ(t))

To fully characterize a quantum state ie to do quantum state tomogra-phy [Vogel and Risken (1989)] a BHD is a prerequisite But also interfero-metric measurements with balanced homodyne detectors instead of singlePIN photo-diodes have several advantages A correctly implemented BHDreadily provides the vacuum noise level when the signal beam is blockedWith a BHD the optimum operating point of the interferometer is preciselyat a dark fringe If a perfect dark fringe can practically be achieved ampli-tude noise of the laser does not couple into the signal port If the interfero-meter has balanced arm length also frequency noise of the laser then does notcouple into the signal port Some quantum non-demolition schemes with theprospect of evading quantum radiation pressure noise require the detection ofa non-canonical quadrature angle [Jaekel and Reynaud (1990) Kimble et al(2001)] Here the adjustable phase of a BHD provides a straight forwardapproach The experimental exploration of BHDs for gravitational-wave de-tectors only has started recently [Steinlechner et al (2015)]

A light field can be analysed with respect to many different modulationfrequencies Ω The result constitutes a spectrum [Breitenbach et al (1998)]where in principle every modulation mode can be in a different quantumstate Fig 4 shows spectra of squeezed states from 5 MHz to 100 MHz with∆Ω2π = 1 MHz The lower curve shows the spectrum of the most strongly

15

-14-12-10-8-6- 4-20 2 4 6 8

10 12 14 16 18

6 8 10 20 40 60 80 1005

Squeezed noise

Anti-squeezed noise

Vacuum noise

Frequency [MHz]

Nor

mal

ized

noi

se v

aria

nce

[dB

]

Figure 4 Spectrum of quadrature amplitude variances ndash Shown are the quantumnoise properties of a large number of modulation modes having a resolution bandwidth of∆Ω(2π) = 1 MHz For all traces the balanced homodyne detector output was analysedwith a spectrum analyser Squeezing of XΩ∆Ω (bottom trace) and anti-squeezing of

YΩ∆Ω (top trace) versus f = Ω(2π) are shown relative to the vacuum noise variance Thespectrum below 5 MHz is not shown since it contained less squeezing due to laser relaxationoscillation in the carrier field as well as disturbances from back-scattered light [Vahlbruchet al (2007)] Disturbances at frequencies above 70 MHz originated from relatively largedetector dark noise which was subtracted from all traces shown here The thin linerepresents a theoretical model that takes into account for the linewidth of the squeezingcavity The data was first presented in in Ref [Mehmet et al (2010)]

squeezed variances in this case the variances of XΩ∆Ω The upper spectrum

shows the variance in the orthogonal quadrature amplitude (YΩ∆Ω) Allvariances are normalized to those of the corresponding vacuum state Thesqueeze factor reduces towards higher frequencies due to the linewidth ofthe squeezing cavity The anti-squeezing is always higher than the absolutevalue of the squeezing due to Heisenbergrsquos uncertainty relation and due to thepresence of optical loss The curves do not represent pure squeezed states butmixed squeezed states with a significant contribution from vacuum statesdue to optical loss Pure squeezed states can only be produced by making

16

the influence of all decoherence processes negligibleThe choice of the resolution bandwidth (RBW ∆Ω) during data taking

and processing defines the spectral-temporal modulation modes includingtheir number within the detected spectrum For any setting of the RBWthe quantum mechanical properties of the quadrature amplitudes XΩ∆Ω and

YΩ∆Ω [Caves (1985)] fully correspond to those introduced for quadratures instandard text books and which are reviewed in Sec 3

24 Observations on two-mode squeezed states using balanced homodyne de-tectors

Two-mode squeezed states are composed of two subsystems lsquoArsquo and lsquoBrsquoand are bi-partite entangled states with a Gaussian quantum statistic Toavoid conflicts with different usage of the term lsquomodersquo they can synony-mously be named lsquobipartite Gaussian entangled statesrsquo or lsquobipartite squeezedstatesrsquo which will be mainly used in this Review In the same way multi-partite Gaussian entangled states correspond to multi-partite squeezed states

The measurement observables that prove or disprove the bi-partite squeez-ing property are XA

Ω∆Ω minus XBΩ∆Ω and Y A

Ω∆Ω + Y BΩ∆Ω where the minus and

plus signs may be swapped Bi-partite squeezed states are precisely thosestates that were discussed by Einstein Podolsky and Rosen (EPR) in theirseminal paper [Einstein et al (1935)] Fig 5 shows a measurement resulton bi-partite squeezed light [Eberle et al (2013)] The variances of bothjoined observables are squeezed as shown in the two lower traces They wererecorded consecutively by adding or subtracting the outputs of two balancedhomodyne detectors But by interfering the subsystems on a beam splitterone could even measure both joined observables simultaneously This pos-sibility is correctly described in quantum theory since their commutator iszero

The so-called EPR paradox arises as follows If we either measure XAΩ∆Ω

and XBΩ∆Ω or Y A

Ω∆Ω and Y BΩ∆Ω it is obvious from the data in Fig 5 that

we can always predict the measurement result at subsystem lsquoBrsquo when know-ing the result at subsystem lsquoArsquo This seems to suggest that both quantitiesat lsquoBrsquo are precisely defined simultaneously before the measurement on lsquoArsquowhich contradicts the rigorous (and correct) interpretation of their non-zerocommutator that they are not precisely defined simultaneously

To solve this paradox EPR conjectured that the wavefunction as definedby quantum theory does not provide the full information This led to a

17

discussion of whether hidden variables existed that needed to be included ina complete theory of quantum mechanics (see also Bell [Bell (1966)]) Theexperimentally observed violation of Bellrsquos inequality [Bell (1964) Aspectet al (1981) Giustina et al (2013) Hensen et al (2015)] however ruled outthe existence of (local) hidden variables

Based on that the EPR paradox needs to be solved in a different wayContrary to what EPR assumed it is in fact possible to predict the valueof an arbitrary observable of a physical system A with certainty via a mea-surement on system B although this observable was not defined before themeasurement Without any interaction a measurement on subsystem lsquoArsquo notonly creates lsquorealityrsquo of eg XA

Ω∆Ω simultaneously lsquorealityrsquo is also created

regarding the observable XBΩ∆Ω describing subsystem lsquoBrsquo Here the term

lsquorealityrsquo has the meaning as defined by EPR [Einstein et al (1935)] Simi-larly the detection of one photon of a two photon entangled number statenot only produces the reality of this photon but also that of a second oneA discussion of Einstein-Podolsky-Rosen entanglement can also be found in[Schnabel (2015)] Note that the EPR paradox can also be described aslsquoquantum steeringrsquo [Schrodinger (1935) Cavalcanti et al (2009) Handchenet al (2012)] It should also be mentioned that two-mode squeezing beingdetected with BHDs and not with photon counters cannot be used to violatea Bell inequality The latter topic is outside the scope of this Review

Bi-partite squeezed states were first characterized with balanced homo-dyne detectors by the group of J Kimble in 1992 [Ou et al (1992)] Gener-ally the EPR paradox becomes more pronounced the stronger the bi-partitesqueezing is A measure of the strength of EPR entanglement was introducedby M Reid [Reid and Walls (1985)] According to this measure the resultin Fig 5 can be quantified to ε2 = 00309 where the critical value is one Itcorresponds to the strongest Gaussian EPR entangled state generated so far

For a long time it looked like that two-mode squeezed states are notuseful for laser interferometers The reason for that belief was that a laserinterferometer as any other measurement device too is built to measureone observable It seems to be ideal already if the quantum noise in thissingle observable is squeezed The increased quantum noise in the orthog-onal observable is not harmful in this case and squeezing in two differentobservables useless Only recently realistic scenarios were discussed in whichtwo-mode squeezing in fact does improve the performance of a laser inter-

18

-12

-10

-8

-6

-4

-2

0

0 1 2 3 4 5 6 7 8 9 10

100

dB

109

dB

Δ (XAvac + X B

vac ) = Δ (Y

Δ (XA + X B)

Avacminus Y B

vac )2 2

2

Δ (YAminus Y B)2

Nor

mal

ized

noi

se v

aria

nce

[dB

]

Times [s]

Figure 5 Two-mode squeezing measurement ndash For this measurement the outputs oftwo balanced homodyne detectors are added or subtracted and the variances (noise pow-ers) of the results recorded The upper trace was measured with modes lsquoArsquo and lsquoBrsquo beingin their ground states This measurement served as a reference level Strong two-modesqueezing was observed as shown by the lower two traces The sideband frequency wasΩ(2π) = 8 MHz and the resolution bandwidth was ∆Ω(2π) = 200 kHz The measure-ment results were first published in Ref [Eberle et al (2013)]

ferometer [Steinlechner et al (2013)] The proof-of-principle experiment isreviewed in Sec 7

25 Observations using photon counters

Alternatively to field quadratures an optical mode in a squeezed statecan also be characterized at least partly by detecting its photon numberdistribution For a pure squeezed vacuum state such a measurement wouldreveal the existence of solely even photon numbers including a large prob-ability for zero photons The average photon numbers of squeezed vacuumstates with feasible squeeze factors are very small of the order of one persecond and bandwidth in hertz see Fig 13 (a) ndash (c) A distribution with closeto zero probability of odd photon numbers however has not been measuredso far The reason is the lack of ideal photon counters First of all theefficiency of these detectors ie their probability of converting one photoninto one click and no photon into no click must be almost perfect lsquoLostrsquophotons as well as dark counts wash out the oddeven oscillations Further-more most detectors available can only distinguish between zero and one

19

photon This problem can be solved by distributing the squeezed mode ontoa large number of single photon detectors using an array of beam splitterssuch that all paths have a low probability of carrying more than one pho-ton Photon number measurements on squeezed vacuum states neverthelessplay an extremely important role in quantum optics When the squeezingstrength is very low the probability of detecting more than 2 photons can beneglected and the detection of a photon heralds the existence of a second one

0

300

600

900

1200

-40 -20 0 20 400

5

10

15

20

25

Two-

fold

coi

ncid

ence

s [1

(4s

)](a) (b)

Non-degeneratetwo-mode squeezing

5050

APDA1

APDA2

APDHerald

Photon coincidencedetection

Delay [ns]

Thr

ee-f

old

coin

cide

nces

[1

(4s)

]1500

Figure 6 Coincidence clicks from non-degenerate photon pairs ndash The first suchexperiment was reported in Ref [Hong and Mandel (1986)] (a) shows a setup with threeavalanche photo-diodes (APDs) for proving the successful heralding of a single photonnumber state (b) Histograms of the two-fold coincidence detections at APDHerald andAPDA1 (red) and at APDHerald and APDA2 (yellow) with theoretical models (solid lines)If the two-mode squeezing just carried one photon in each spatial subsystem the three-foldcoincidence detection should be zero Indeed the according histogram (grey points righty axis) shows only a few events These are produced by false (dark) counts of the APDsThe delay for the three-fold coincidences is defined as the time between counts at lsquoA1rsquo andlsquoA2rsquo given that the trigger APDHerald detected a photon (within a 100 ns time window)The data was taken on photons that were up-converted from 1550 nm to 532 nm and itwas first published in Ref [Baune et al (2014)]

If a mode of light is always excited by either zero or two photons lsquocondi-tionalrsquo or lsquoheraldedrsquo one-photon Fock states can be realized (Measurementson an ensemble of the n-photon Fock state would always produce the mea-surement result n ie Fock states have a zero photon number uncertaintyThey are also called lsquonumber statesrsquo) The above concept of producing aone-photon Fock state obviously requires the deterministic and balanced dis-tribution of the down-converted signal and idler fields into two different paths

20

In order to achieve this the signal and idler fields need to be non-degenerateUsually a mode in a squeezed state is composed of degenerate signal andidler fields and this degeneracy thus needs to be removed Possible waysare producing the down-converted fields at well separated wavelengths [Vil-lar et al (2005) Su et al (2006) Li et al (2010) Samblowski et al (2011)]separating the upper and lower sidebands belonging to an ordinary squeezedmode by frequency filters [Schori et al (2002) Hage et al (2010)] and us-ing spatial filters [Hong et al (1987)] A frequently used approach is usingtype II parametric down-conversion where the photons within a pair are al-ways orthogonally polarized [Ou et al (1992) Kiess et al (1993) Kwiat et al(1995)]The list of experiments with conditional or heralded photon number states islong They showed for instance nonclassical g(2)-functions [Hong et al (1987)]and violations of Bell inequalities [Weihs et al (1998)] Fig (6) shows a re-sult from a more recent experiment in which a bipartite-squeezed state withsubsystems at 1550 nm and 810 nm was produced the subsystem at 1550 nmsubsequently up-converted to 532 nm and the lsquoquantum non-Gaussianityrsquo ofheralded up-converted single photons demonstrated [Baune et al (2014)]Squeezed states are also the resource for the conditional generation of super-positions of coherent states [Ourjoumtsev et al (2006) Neergaard-Nielsenet al (2006)] and so-called N00N-states [Afek et al (2010)]

The generation of nonclassical states mentioned in the paragraph aboveis not stationary but relies on a probabilistic trigger event The produc-tion of squeezed states themselves usually happens in a stationary fashionThis distinction has an important consequence for applications of nonclassi-cal states in measurement devices Only (stationary) squeezed states allowfor a continuous improvement of a measurement Avoiding any loss of mea-suring time is generally of high relevance for the detection of short-livedsignals with unknown arrival time as well as for the detection of long-livedquasi-monochromatic signals since the signal-to-noise-ratio (SN) improveswith measuring time

26 Conclusions

The detection of squeezed light produces measurement results that canbe considered as remarkable Let us focus on experiments where a mode in abright coherent state is overlapped with a mode in a squeezed vacuum state

21

as shown in Figs (1) and (3) In both setups the squeezed vacuum field caneasily be blocked which allows us to compare the measurement results ona bright coherent state with and without the interference with the squeezedvacuum state Without squeezing the photo-electric detectors measure alarge number of photon events with a large quantization noise (shot noise)The large noise reflects the fact that all photon events were independent fromeach other as shown in Fig 2 (bi) With squeezing the photo-electric de-tectors again measure a large number of photon events with an expectationvalue that is even slightly higher but nevertheless the quantization noise ofall detected photons is significantly reduced Fig 2 (bii)Based on the discussion of EPR entanglement in Subsec 24 the photo-electric detection of the output light of a squeezing-enhanced laser inter-ferometer (with αlowastα1) produces the reality of photons This way we cankeep the lsquowave picturersquo in which no photons exist when light travels alongthe interferometer arms and when it interferes at the beam splitter Whenthe energy of the beam is elevating electrons to the conductance band ofthe photo-diodersquos semi-conductor n photon events simultaneously appearwithin the measuring interval with probability P (n) What conclusion hasto be drawn if the probabilities resemble a sub-poissonian statistic ndash Theoccurrence of photon events is still truly random but in this case not forindividual photons The occurrence of photons is correlated in such a waythat the probability of detecting an additional photon in the same time in-terval reduces the larger the number of already detected photons is Whatfollows from the discussion of EPR entanglement for a photon counting ex-periment with pure squeezed vacuum and ideal photon counters Here theprobabilistic detection of one photon entails the detection of a second onewith certainty With some smaller probability a third photon is detectedwhich entails the detection of a fourth photon with certainty and so on

If a photon of a mode that was not interrogated by the environment beforeis absorbed its reality is created in this very moment If the photon belongsto a squeezed state this process instantaneously influences the probability ofother photons becoming reality

Of course a more general statement can be made based on the insightthat interaction with the environment creates the reality of any kind ofquanta including electrons atoms and molecules

22

3 Theoretical description of squeezed states

31 The quadrature amplitude operators

Consider a single mode of light at optical frequency ω Its Hamiltonoperator reads

Hω = ~ω(n+

1

2

)= ~ω

(adaggerωaω +

1

2

)= ~ω

(X2ω + Y 2

ω

) (7)

where n is the photon number operator and aω and adaggerω are the annihilationand creation operators which obey the commutation rule

[aω a

daggerω

]= 1 The

operator aω has a complex-valued dimensionless eigenvalue spectrum andcorresponds to the complex amplitude αω in classical optics Xω and Yω arethe hermitian amplitude and phase quadrature operators The eigenvaluesof the quadrature operators are also dimensionless and proportional to theelectric fields at the oscillationrsquos antinode and at the oscillationrsquos node Inthe above equation they are defined such that their variances are ∆2Xω =∆2Yω = 14 if the oscillator is in its ground state ie if 〈n〉 = 0

Although Eq (7) simply describes the energy of an harmonic oscillator itis the essence of quantum theory since it mathematically describes the wave-particle dualism Whereas the eigenvalues of n have a discrete spectrum theeigenvalues of Xω and Yω have a continuous spectrum In classical opticsthe phase quadrature is zero In quantum optics its expectation value is alsozero but its uncertainty contributes to the overall energy

Eq (7) describes a cavity mode as well as a section that is cut from apropagating quasi-monochromatic light beam The latter example is of highrelevance in actual experiments By setting the sectionrsquos time window ie themeasuring time interval the time-frequency (lsquomodulationrsquo) mode is defined

The quadrature operators introduced in Eq (7) and displayed in Fig 7 donot correspond to lsquoXrsquo and lsquoY rsquo that are of relevance in laser interferometry andin optical communication and which were already discussed in Subsec 22and 23 The optical frequency of visible and near-infrared light is far too highto be transferred to an oscillation of photoelectric voltage Quite general alaser interferometer targets signals at audio or radio band frequencies Ωi ωSuch a measurement is achieved as stated before by decomposing the photo-electric voltage from the photo diode at the interferometer output into asingle-sided spectrum (positive frequencies only) of intervals of Ωplusmn∆Ω2

23

(a)

(b)

(c)

(d)

2π 4π

2π 4π

2π 4π

2π 4π

Figure 7 Phase spaces and electric field oscillations of monochromatic lightndash Top Left Monochromatic light in a coherent state is represented by a phasor (whitearrow) including its quantum uncertainty (white dashed circle and fuzzy area) located inthe phase-space spanned by the quadratures Xω and Yω When the phase space rotateswith optical frequency ω2π the projection of the quantum phasor onto a fixed (vertical)axis corresponds to the electric field E(t) as shown on the right side (a) Weakly displacedcoherent state (b) Corresponding amplitude squeezed state The electric field uncertaintyaround the zero average field region is anti-squeezed (c) Vacuum state at the same opticalfrequency (d) Corresponding squeezed vacuum state The meaning of the uncertaintycould be carved out by supplementing them with monochromatic waves all having theoptical frequency ω2π Changing amplitudes then display amplitude quadrature noiseChanging shifts along the time axis model the electric field uncertainty at the expectedzero crossing They are not implemented in the graphics here however since any of thosewaves does not exist due to Heisenbergrsquos uncertainty relation

24

The signals as well as the quantum uncertainties carried by a beam oflight are thus described by a spectrum of pairs of non-commuting quadratureoperators Mathematically every such operator is defined by an integral overthe Fourier components within the bandwidth The spectral weighting of theFourier components is called the lsquowindow functionrsquo By going to sidebandintervals a spectrum of a new type of optical mode is defined which describesthe modulation of the electric field in the respective frequency interval Ω plusmn∆Ω2 In this Review we call it a lsquomodulation modersquo

The quadrature operators that are defined around a modulation frequencyΩ with a bandwidth of ∆Ω are the quadrature amplitude operators thatare relevant in laser interferometry Whenever they are not related to aspecific band we use the short form XΩ∆Ω(t) equiv X and YΩ∆Ω(t) equiv Y cfEqs (1) and (2) These operators can slowly vary with time where thetime dependence is limited by ∆Ω (The time dependence is not due toquantum uncertainty which usually is time independent but for instancedue to the time dependence of the signal eg a passing gravitational wave)Let us consider now a pair of quadrature operators for a particular sidebandΩ plusmn ∆Ω2 The Hamilton operator of the corresponding modulation modeis found by switching to the frame rotating at optical frequency ω Thetransition is done by applying the unitary transformation U = exp(iωadaggerat)generating a new Hamiltonian H = U daggerHωU minus i~UpartU daggerpartt The Hamiltonianof the modulation mode reads

H = ~Ω

(nΩ +

1

2

)= ~Ω

(adaggera+

1

2

)= ~Ω

(X2 + Y 2

) (8)

where nΩ is the (occupation) number operator for the modulation modeand a and adagger its annihilation and creation operators The commutation rule[a adagger

]= 1 is unchanged X and Y are the amplitude and phase quadra-

ture amplitude operators respectively They correspond to the depth of theamplitude modulation and for weak excitations to the depth of the phasemodulation respectively They are the conventional hermitian field operatorsin experimental quantum optics Note that modulation modes at angularfrequency Ω can be described by a superposition of three optical frequenciesa carrier at ω an upper sideband at ω + Ω and a lower sideband at ω minus ΩThe quantum mechanical description of modulation states in connection tooptical carrier and upper and lower sidebands is known as the lsquoTwo-PhotonFormalismrsquo [Caves and Schumaker (1985) Schumaker and Caves (1985)]

The quadrature amplitude operators in Eq (8) are again defined such

25

that the variances of the uncertainty of a modulation field in its ground stateor in a coherent state are

∆2Xvac = ∆2Yvac = 14 (9)

Generally quadrature operators X and Y as defined in Eqs (7) and (8) arethe real and imaginary parts of the annihilation operator

a = X + iY hArr adagger = X minus iY (10)

hArr X =1

2

(a+ adagger

) Y =

1

2i

(aminus adagger

) (11)

They satisfy the commutation relation[X Y

]=i

2 (12)

and their variances are limited by a Heisenberg uncertainty relation of thefollowing form

∆2X∆2Y ge 1

16 (13)

A quantum state is called a ldquosqueezed staterdquo [Bachor and Ralph (2004)]if ∆2Xϑ lt 14 for an arbitrary field quadrature Xϑ = X cosϑ + Y sinϑ seeEq (3) The angle of the lowest variance below 14 is called the squeeze angleθ The largest factor by which the variance is below 14 is called the squeezefactor often given on a decibel (dB) scale using the following transformation

minus 10 middot log10

(∆2Xθ

∆2Xvac

) (14)

The squeeze factor can also be described by the squeeze parameter r

eminus2r =∆2Xθ

∆2Xvac

(15)

When a squeezed state experiences optical loss it remains squeezed butthe squeeze factor is reduced Also the statersquos purity is reduced ie theproduct of the quadrature uncertainties increases above the minimum valueOptical loss corresponds to mixing the state with the vacuum state Let∆2Xϑ be the variance of a quadrature amplitude ∆2Xvac the variance of the

26

(quadrature angle independent) ground state uncertainty and (1 minus η2) therelative energy loss Then the resulting quadrature variance reads

∆2Xprime

ϑ = η2∆2Xϑ + (1minus η2)∆2Xvac (16)

To maximize the benefit from squeezed states in applications stronglysqueezed states need to be generated and optical loss minimized Opticalloss occurs due to absorption and scattering in the optical components inthe path of the squeezed beam including the squeezing resonator itself anddue to non-perfect matching to the interferometer mode non-perfect inter-ference contrast of the interferometer and non-perfect quantum efficiency ofthe photo diodes The sum of all losses including those outside the inter-ferometer need to be less then 10 to allow a nonclassical quantum noisesuppression of a factor of 10 in power ie 10 dB

32 Phase space representations of squeezed states

The Wigner function ndash The properties of squeezed states are nicely dis-played by the Wigner function W (X Y ) [Wigner (1932)] An example interms of a squeezed vacuum state is shown in Fig 8 It is a quasi-probabilitydistribution which contains the statersquos full information including its quan-tum statistic There are two ways how a Wigner function provides a sufficientcriterion for nonclassicality First by containing negative values second byfeatures that have a smaller (squeezed) width compared with the Wignerfunction of the ground state Integrating the Wigner function over Y pro-vides the probability density of measurement results ie of the eigenvaluesof the observable X and vice versa

infinintminusinfin

W (X Y )dY = p(X)

infinintminusinfin

W (X Y )dX = p(Y ) (17)

where p(X) and p(Y ) are the observed probability distributions also exem-plarily shown in Fig 8

The ground state coherent states as well as (quadrature) squeezed stateshave quadrature eigenvalue probability densities that are Gaussian TheirWigner functions are also Gaussian and thus entirely positive Wigner func-tions of other nonclassical states for instance Fock states exhibit negativevalues For this reason the Wigner function is called a quasi -probabilityfunction

27

X Y XY

Figure 8 Wigner function and its projections ndash Displayed is the full information ofa squeezed vacuum state The quasi probability density of the Wigner function (centre)is plotted along a third dimension and is color-coded Also shown are the actual mea-surement results from which the Wigner function is reconstructed They are representedby the squeezed and anti-squeezed Gaussian projections onto the X and Y axes Theirprobability densities p(X) and p(Y ) are given on the vertical axes The Gaussian measure-ment statistic of the first clearly shows squeezing compared to the ground state statistic(dashed) The squeeze factor is 116 dB and the squeeze parameter r = 1335 [Mehmetet al (2010)]

Fig 9 shows the Wigner functions for (a) a coherent state (b) the ground(vacuum) state (c) a displaced squeezed state and (d) a squeezed vacuumstate All Wigner functions describe a modulation of the carrier light atsideband frequency Ω integrated over the frequency interval ∆Ω The car-rier light is not part of these Wigner functions The displacement in (a)represents a classical amplitude modulation (b) corresponds to the absenceof any photons with a frequency offset of plusmnΩ from the local oscillator field(c) and (d) represent states whose amplitude modulation depth is more pre-cisely defined than that of the ground state Fig 10 shows Wigner functionspectrum for a broadband squeezed vacuum field Every Wigner functiondescribes the modulation field at some modulation frequency Ωi integrated

28

X

Y

X

(a) (b)

X X

(c) (d)

Y

Y Y

Figure 9 Simplified representation of Wigner functions ndash The darker the shadedareas the larger is the phase-space quasi-probability Shown are four different (time-independent) states of a modulation mode at frequency Ω for a specific resolution bandwidth ∆Ω Panel (a) represents a coherent state the displacement (α) corresponds to aclassical amplitude modulation Panel (b) represents the ground (vacuum) state (c) adisplaced squeezed state and (d) a squeezed vacuum state both with squeeze angle θ = 0The latter is in analogy to Fig 8 Again the light field that carries the modulation is notpart of the pictures

over the resolution bandwidth (RBW) of ∆Ω

The Glauber-Sudarshan P -function ndash The P -function [Glauber (1963)Sudarshan (1963)] is calculated by de-convoluting the Wigner function fromthe ground state uncertainty [Gerry and Knight (2005)] For displaced vac-uum states (coherent states) the P -function corresponds to a displaced δ-function The mathematical expression of the P -function of a squeezed statecontains infinitely high orders of derivatives of the δ-function [Vogel andWelsch (2006)] Such a function contains negativities but cannot be dis-played It is possible however to define a phase-space quasi probabilityfunction for squeezed states that can be displayed and that does show neg-

29

0

Ω1

Ω2

Ω

Single-sided spectrum

Y

X

Figure 10 Hint of a Wigner function spectrum ndash A single-sided spectrum (positivefrequencies only) with respect to the carrier field can be used to visualize a broadbandsqueezed field Shown are two examples displaying a squeezed vacuum state at Ω1 anda displaced amplitude squeezed state at Ω2 The individual Wigner functions cover theresolution bandwidth ∆Ω gt 0 (not shown) In general the squeezing strength as well asthe squeeze angle and the displacement are a function of sideband frequency

ativities as a sufficient and necessary condition for certifying the squeezingeffect This lsquononclassicality functionrsquo is calculated by de-convoluting theWigner function from an uncertainty distribution that is steeper than theGaussian distribution A pronounced negativity of a squeezed vacuum stateof up to 69 standard deviations was found [Kiesel et al (2011)]

The double-sided phasor picture ndash This phasor picture links quantumstates of modulations with the quantum states of the contributing opticalfields [Bachor and Ralph (2004)] and is mathematically described by thetwo-photon-formalism [Caves and Schumaker (1985) Schumaker and Caves(1985)] Generally a weak amplitude or phase modulation at frequency Ω ofa carrier field at optical frequency ω can be understood as the carrierrsquos beatwith two optical frequencies at ωplusmnΩ The double-sided phasor picture is ableto display a spectrum of different and independent modulation frequenciesin the rotating frame of the carrier field The carrier light field is time-independent but the upper and lower sidebands are not They rotate withplusmnΩi(2π) respectively around the frequency axis

30

ω0

Yω

Xω

Upper sideband

Lower sideband

ω0+Ω1

ω0+Ω2

ω0-Ω2

ω0-Ω1

ω

Upper sideband

Lower sideband

Yω

Xω

ω0+Ω1

ω0+Ω2

ω0-Ω2

ω0-Ω1

ω

ω0

Figure 11 Double-sided phasor pictures ndash Phase spaces at optical frequency ω0 plusmnΩirotate around the frequency axis with sideband frequency plusmnΩi Its sign determines thedirection of rotation with respect to the local oscillator in the rotating frame at ω0 Apair of phase spaces need to be superposed to provide a description of a modulation fieldat |Ωi| Top Amplitude quadrature squeezed field with a coherent displacement at |Ω2|The displacement corresponds to a classical amplitude modulation The uncertaintiesof all optical frequencies are circular and larger than that of the ground state (dashed)Quantum correlations are indicated by additional symbols More details are given in themain text Bottom Corresponding spectrum of (displaced) vacuum states which do nothave any quantum correlations

31

Fig 11 shows such a double-sided phase space picture where the carrierrsquosmodulation at Ω1 is in a squeezed vacuum state and where the modulationat Ω2 is in a displaced squeezed state The picture shows how a classicalamplitude modulation as well as the quantum statistic of a modulation fieldis decomposed into contributions from upper and lower sidebands For asqueezed modulation field the upper and lower sidebands show no squeezedbut circular thermally excited quantum uncertainties The uncertainties of apair of sidebands however show correlations as well as anti-correlations InFig 11 these (anti-) correlations are marked with times and + for the modulationfrequency Ω1 and with N and bull for the modulation frequency Ω2

33 Covariance matrix representation of (single-party) squeezed states

Since squeezed states have a Gaussian quantum statistic four numbersare sufficient for their full description These numbers are the second mo-ment of the quadrature amplitude showing the strongest squeezing and thesecond moment of its orthogonal quadrature amplitude as well as their firstmoments describing the displacement These four numbers are sufficientto calculate the Wigner function shown in Fig 8 In general the quadra-ture of strongest squeezing is not perfectly aligned with one of the axesof the measurementrsquos coordinate system The so-called covariance matrix(VXXVXY VY XVY Y ) [Simon et al (1994)] accounts for phase space rotationsand enables the calculation of how these states evolve within an interfero-metric arrangement Their components are normalized to the vacuum noisevariance ∆2Xvac = ∆2Yvac and read

VXY =

langXY + Y X

rangminus 2

langXranglang

Yrang

2∆2Xvac

(18)

The following examples represent the ground state a pure 10 dB am-plitude quadrature squeezed state and a pure 10 dB squeezed state with asqueeze angle of 45

Vvac =

(1 00 1

) V0

01 =

(01 00 10

) V45

01 =

(505 495495 505

) (19)

with V4501 =RT

45V001R45 where Rα = (cosα minussinα sinα cosα) is the rota-

tion matrix

32

34 Phase space representation of two-mode (bi-partite) squeezed states

XA

YA

XB

YB

Figure 12 Bi-partite squeezed vacuum state ndash Shown is a Wigner-function-basedphase space representation in close analogy to the single party version in Fig 9 (d) Thepicture describes a single modulation at frequency Ω with bandwidth ∆Ω Measurementsat party A and B reveal local Wigner functions that correspond to thermal states sincethe uncertainties (indicated by the color and by the large circles) are symmetric and largerthan that of the ground state (indicated by the small dashed circles) The uncertaintieshowever show correlations and anti-correlations here indicated by + and times respectivelyThe strength of these (anti-)correlations are indicated by ellipses Bi-partite squeezingie entanglement is present if the short axes of the ellipses are shorter than the diameterof the ground state uncertainty The picture in fact represents Einstein-Podolsky-Rosenentanglement [Einstein et al (1935)] From a measurement of XA or YA the correspondingmeasurement result at party B can always be inferred with an uncertainty that is smallerthan the ground state uncertainty

A bi-partite state enables a measurement on subsystem A and simulta-neous a measurement on subsystem B For a large number of simultaneousensemble measurements of the same quadrature amplitude Xϑ the followingtwo joint quadrature variance can be calculated

∆2(XAϑ plusmn XB

ϑ ) (20)

A state that is symmetrically shared between two parties (A and B) is calleda two-mode squeezed state if the variances of joint quadrature measurementsfulfill the following inequality [Duan et al (2000)] ie

∆2(XA minus XB)

∆2(XAvac minus XB

vac)+

∆2(Y A + Y B)

∆2(Y Avac + Y B

vac)lt 2 (21)

33

with ∆2(XAvac plusmn XB

vac) = ∆2(Y Avac plusmn Y B

vac) = 2∆2Xvac A lsquotwo-mode squeezedstatersquo reveals entanglement in the second moments of the measurement statis-tics It is thus a lsquobi-partite Gaussian entangled statersquo

Fig 12 displays a (pure) bi-partite squeezed vacuum state distributedbetween A and B The state shows full symmetry regarding its subsystemsat the two sites The large circles and the colored area represent Wignerfunctions of the subsystems Measurements of the quadrature amplitudesXA YA XB and YB show identical variances and the correlations and anti-correlations have identical strength since ∆2(XA minus XB) = ∆2(YA + YB) lt12 for our normalization of quadrature amplitudes having a ground statevariance of 14

Generally a symmetric bi-partite squeezed state fulfills another quantita-tive (Gaussian) entanglement criterion if less than 50 of the vacuum state issymmetrically mixed into the initially pure state Bi-partite squeezed statesare always entangled but in this case they are even Einstein-Podolsky-Rosen(EPR) entangled [Reid (1989)] allowing the demonstration of the quantumsteering effect [Einstein et al (1935) Schrodinger (1935) Reid (1989) Cav-alcanti et al (2009)] The first such experiment was performed by Ou etal [Ou et al (1992)] using type II parametric down-conversion (PDC) Laterexperiments produced bi-partite squeezed vacuum states by overlapping twosqueezed vacuum states each produced with type I PDC on a balanced beamsplitter and used the entangled output for the demonstration of quantumteleportation [Furusawa et al (1998) Bowen et al (2003ca)] The criterionin Eq (21) and the EPR criterion from [Reid (1989)] was experimentallycompared in Ref [Bowen et al (2003b)] The steering effect in asymmet-ric bi-partite squeezed states were recently experimentally characterized inRef [Handchen et al (2012)]

Fig 12 shows features similar to those in the top part of Fig 11 Thisis not a coincidence and shows that a bi-partite squeezed state can also begenerated by spatially splitting the upper and lower sideband of a (single-party) squeezed state This was first experimentally demonstrated by thegroup of E Polzik [Schori et al (2002)] and later used for EPR multiplex-ing of a single longitudinal mode of a squeezing resonator [Hage et al (2010)]

35 Covariance matrix representation of bi-partite squeezed states

Also the full information of bi-partite states including the entanglementcan be cast by the covariance matrix [Simon et al (1994)] which can be used

34

to calculate the propagation of these states in laser interferometers Againall variances are normalized to the vacuum noise variance in full analogy toEq (18) The generic bi-partite covariance matrix has dimension 4times4 andreads

Vbp =

VXAXA

VXAYA VXAXBVXAYB

VYAXAVYAYA VYAXB

VYAYBVXBXA

VXBYA VXBXBVXBYB

VYBXAVYBYA VYBXB

VYBYB

with

VXAYB =

langXAYB + YBXA

rangminus 2

langXA

ranglangYB

rang2∆2Xvac

ϑ

(22)

Due to the symmetry in Eq (22) the 4times4 covariance matrix is fully spec-ified by just ten independent coefficients If the phase spaces at A and Bare aligned along the strongest correlations and anti-correlations the ma-trix components referring to different quadrature amplitudes eg VXAYA arezero Such entangled states can be produced by overlapping two squeezedfields with a squeeze angle difference of 90 on a balanced beam splitter

A symmetric bi-partite squeezed vacuum state which is also called anlsquoS-classrsquo [DiGuglielmo et al (2007)] bi-partite squeezed vacuum state shows(anti-)correlations in two joint quadratures as defined in Eq (21) For a puresuch state of 10 dB squeezing the covariance matrix reads

Vbp10|10 =

505 0 495 0

0 505 0 minus495495 0 505 0

0 minus495 0 505

The following covariance matrix describes a so-called lsquoV-classrsquo 10 dB bi-partite squeezed vacuum state Here only one joint quadrature shows 10 dBsqueezing whereas the orthogonal joint quadrature shows vacuum noise Thestate is obtained by overlapping one 10 dB squeezed state with a vacuumstate on a balanced beam splitter

35

Vbp0|10 =

055 0 045 0

0 55 0 minus45045 0 055 0

0 minus45 0 55

The first measurement of all elements of such a covariance matrix was achievedin [DiGuglielmo et al (2007)]

36 Photon numbers of squeezed states

In contrast to the ground state squeezed vacuum states do have photonexcitations As said earlier quantum theory links the wave and the particlepictures Indeed the squeeze factor of a modulation mode is directly con-nected to a certain photon number excitation Squeezed states of light areproduced via spontaneous photon pair generation eg by parametric down-conversion The following operator S is called the lsquosqueeze operatorrsquo [Gerryand Knight (2005)] It creates and annihilates photon pairs

|r θ〉 = S(r θ) |0〉 (23)

where |r θ〉 is a squeezed vacuum state with squeeze parameter r and squeezeangle θ and |0〉 is the vacuum state The definition of the squeeze operatoris

S(r θ) = exp

[1

2

(reminusiθa2 minus reiθadagger 2

)] (24)

The following shows that this definition indeed results in a state with squeezedquadrature amplitude variances Lets set θ = 0

〈0| Sdagger(r 0) X S(r 0) |0〉 =1

2〈0| Sdagger(r 0)

(a+ adagger

)S(r 0) |0〉 (25)

〈0| Sdagger(r 0) Y S(r 0) |0〉 =i

2〈0| Sdagger(r 0)

(aminus adagger

)S(r 0) |0〉 (26)

Using the Baker-Hausdorff formula we get

Sdagger(r θ) a S(r θ) = a cosh r minus adaggereiθsinh r (27)

Sdagger(r θ) adagger S(r θ) = adaggercosh r minus aeminusiθsinh r (28)

36

Since 〈0| X |0〉 = 〈0| Y |0〉 = 0 also Eqs (25) and (26) are zero To finallycalculate the variances we need

〈0| Sdagger(r 0) X2 S(r 0) |0〉 =1

4〈0| Sdagger(r 0)

(a2 + adaggera+ aadagger + adagger2

)S(r 0) |0〉

〈0| Sdagger(r 0) Y 2 S(r 0) |0〉 = minus1

4〈0| Sdagger(r 0)

(a2 minus adaggeraminus aadagger + adagger2

)S(r 0) |0〉

Given that SSdagger is the identity and using again Eqs (27) and (28) we get theexpected variances

∆2X =1

4

[cosh2 r minus 2cosh r sinh r + sinh2 r

]=

1

4eminus2r

∆2Y =1

4

[cosh2 r + 2cosh r sinh r + sinh2 r

]=

1

4e2r

Since the squeeze operator can only create and annihilate photon pairsa squeezed vacuum state without photon loss must correspond to an evennumber of photons But not only photon loss also a coherent displacementleads to flattening out the odd-even oscillations The probability of detectingN photons in a pure displaced squeezed state are derived for instance in[Gerry and Knight (2005)] and read

P (N) = |〈N |α r θ〉|2 =(05 tanh r)N

N cosh rexp

[minus|α|2 minus 1

2(αlowast2eiθ + α2eminusiθ)tanh r

]times∣∣∣HN

[(αcosh r + αlowasteiθsinh r)

radiceiθsinh(2r)

]∣∣∣2 (29)

where HN is the N th Hermite polynomialFig 13 shows the photon number distributions for 5 different pure squeezed

states according to Eq (29) Panels (a) to (c) show squeezed vacuum stateswith 43 dB 86 dB and 172 dB of squeezing Panel (d) shows the moregeneral case of a squeezed state with a coherent displacement α 6= 0 Dueto θ = 0 the state is amplitude quadrature squeezed Panel (e) refers to thecorresponding phase quadrature squeezed state For comparison panel (f)shows the photon number distribution of the coherent state with the samedisplacement

The panels in Fig 13 represent the diagonal elements of the statersquos den-sity matrix in number basis Only the latter also contains the coherences

37

002

004

006

000 000

002

004

006

008

403020100 Photon number n

403020100Photon number n

Pro

babi

lity

403020100000

005

010

015

100 2 4 6 8100 2 4 6 800

04

06

02

100 2 4 6 800

03

02

01Pro

babi

lity

Pro

babi

lity

08

00

04

06

02

08

(a) (b)

(c) (d)

(e) (f)

Figure 13 Photon number distributions ndash All panels represent pure states (a) 43 dBsqueezed vacuum state (r = 05 α = 0) (b) 86 dB squeezed vacuum state (r = 1 α = 0)(c) 172 dB squeezed vacuum state (r = 2 α = 0) (d) displaced 86 dB squeezed state(r = 1 θ = 0 α = 4) (e) displaced 86 dB squeezed state (r = 1 θ = π2 α = 4) (f)Coherent state (r = 0 α = 4) The average photon numbers are about 027 14 131174 174 and exactly 16 see Eq (30)

between photon numbers [Gerry and Knight (2005)] Figures as shown heregenerally do not give full descriptions of the states

38

A squeezed vacuum state (r 6= 0) always has a non-zero photon numberand can not be the ground state The average photon number of a puresqueezed vacuum state can be calculated using Eq (8) With the maximallysqueezed quadrature variance ∆2Xsqz the average photon number is given by

n = 〈n〉|α=0θr〉 = ∆2Xsqz +(∆2Xsqz)

minus1

16minus 1

2=eminus2r

4+e2r

4minus 1

2 (30)

with the vacuum noise variance normalized to one quarter A coherent dis-placement further adds |α|2 photons on average

4 Squeezed-light generation

41 Overview

Squeezed light was first produced in 1985 by Slusher et al using four-wave-mixing in sodium atoms in an optical cavity [Slusher et al (1985)]Shortly after squeezed light also was generated by four-wave-mixing in an op-tical fibre [Shelby et al (1986)] and by degenerate parametric down-conversion(PDC) in a 2nd-order nonlinear crystal placed in an optical cavity [Wu et al(1986)] The pumped cavity was operated below its oscillation thresholdie the parametric gain did not fully compensate the round trip losses whichis also called lsquocavity-enhanced optical-parametric amplification (OPA)rsquo

The early day experiments achieved squeeze factors of a few percent up toabout 3 dB Today squeeze factors of more than 10 dB are directly observedin several experiments [Vahlbruch et al (2008) Eberle et al (2010) Stefszkyet al (2012) Vahlbruch et al (2016)] All of them are based on cavity-enhanced OPA (below threshold) The parametrically amplified mode isdegenerate ie signal and idler modes are identical In particular the down-conversion process is of lsquotype Irsquo which means that the amplified mode has awell-defined polarization Squeezed states can also be generated above oscil-lation threshold In Refs [Villar et al (2006) Jing et al (2006)] bi-partitesqueezing was generated with above-threshold PDC Both experiments usedtype II PDC which provides orthogonally polarized signal and idler fieldsType II PDC below threshold was also used to generate squeezed and bi-partite squeezed fields [Grangier et al (1987) Ou et al (1992)] All these

39

experiments were performed in the continuous-wave regime which is also thefocus of this Review Squeezed states of modulations of trains of laser pulseshowever have been also generated since the 1980s using either PDC or theoptical Kerr effect [Slusher et al (1987) Bergman and Haus (1991) Our-joumtsev et al (2006) Dong et al (2008)] For an overview of the develop-ments in squeezed-light generation in the continuous-wave as well as pulsedregime see Ref [Bachor and Ralph (2004)] Squeezed-light generation inopto-mechanical setups [Aspelmeyer et al (2014)] which use the intensitydependent phase shift from radiation pressure was discussed in Refs [Paceet al (1993) Rehbein et al (2005) Corbitt et al (2006)] and recently ex-perimentally achieved by several groups [Brooks et al (2012) Safavi-Naeiniet al (2013) Purdy et al (2013)]

42 Degenerate type I optical-parametric amplification (OPA)

This section provides a graphical description of how degenerate type IOPAPDC turns a vacuum state into a squeezed vacuum state and a coher-ent state into a displaced squeezed state The process requires a bright pumpfield and a 2nd-order nonlinear crystal For simplicity we set all nonlinearitiesabove 2nd-order to zero

Let us consider a short segment of the second-order nonlinear crystalpumped with light of optical frequency 2ν All other modes that enter thecrystal shall not contain any photons ie are in their vacuum states Ofthese the only mode of interest is that at optical frequency ν which spatiallyoverlaps with the pump mode Fig 14 shows the total electric field of theoptical input E in and the 2nd-order nonlinear dielectric polarisation of thecrystal P The latter is proportional to the total electric field of the outputEout The pump field at 2ν periodically drives the vacuum field at ν betweenregions of low and high polarisation This process transforms the vacuumstate into a squeezed vacuum state in the output [Bauchrowitz et al (2013)]The output further contains the hardly depleted pump field and frequencydoubled parts of the pump field at 4ν It is again emphasized that Fig 14displays OPA in a small segment of the crystal In reality the nonlinear effectaccumulates over the crystal length or even over several passages since thecrystal is usually put into an optical resonator A noticeable effect is achievedif all infinitesimal contributions constructively interfere This is achieved incase of phase matching ie if the wave fronts of the modes at 2ν and νpropagate with the same speed and thus do not run out of phase Note that

40

t

t

Squeezing

Figure 14 Optical parametric amplification of a vacuum state ndash The upper leftcorner shows the crystal polarization P(E) = ε0

(χ(1)E + χ(2)E2

) ie the separation of

charge carriers by the electric component of an optical field E The graph illustrates howan input quantum field (from below) is projected into an output quantum field (towardsthe right) In the example shown here the input field is composed of a classical pumpfield E in

2ν at frequency 2ν and zero-point fluctuations E inν of a field at frequency ν cf

Fig 7(c) The superposition E in of these two fields is transferred into a time-dependentdielectric polarization that is the source of (and thus directly proportional to) the elec-tric component of the output field Eout The quantum uncertainty of the output fieldshows a phase-dependent (parametric) amplification at frequency 2ν Spectral decompo-sition of the output field Eout reveals coherent amplitudes at frequencies 2ν and 4ν and asqueezed vacuum state Eout

sqzν cf to Fig 7(d) The concept of this figure was published inRef [Bauchrowitz et al (2013)]

41

t

t Amplitudequadraturesqueezing

Figure 15 Optical parametric amplification of a coherent state ndash The pictureshows how a displaced vacuum state is transformed into a displaced squeezed state Thepumprsquos electric field is maximal when the input field at fundamental frequency ν shows azero crossing The phase relation described results in an output state that is amplitudequadrature squeezed If the pump field at the input in phase was shifted by half of itswavelength the squeezed output field were a phase quadrature squeezed The squeezinggeneration displayed here corresponds to the transition from Fig 7(a) to Fig 7(b) but withan additional reduction of the displacement The concept of this figure was published inRef [Bauchrowitz et al (2013)]

in actual squeezing experiments the 4ν component is usually suppressed byphase miss-matching

Fig 15 shows the same process but now for an input field at frequencyν in a coherent state In this case the relative phase between the two inputstates ∆ϕ = ϕ2ν minus 2ϕν is relevant In Fig 15 the relative phase is set suchthat the expectation value of the field at frequency ν is zero when the pump

42

(a) (b)

(c) (d)

X

Y

X

YΔϕ = ndash50deg

X

Y

Δϕ = 0deg

X

Y

Δϕ = 90deg

Figure 16 Phase-space illustration of degenerate OPA ndash The (displaced) dashedcircle in each diagram represents the uncertainty of the initial state at optical frequencyν The (displaced) shaded area represents the state after degenerate optical parametricamplification All quantum uncertainties shown correspond to pure states The boldgreen arrow describes the bright second-harmonic pump field whose uncertainty can beneglected The phase between the 2nd harmonic pump and the initial state (∆ϕ = ϕ2ν minus2ϕν) determines the result of the parametric amplification

field reaches its maximum (∆ϕ = 90) The output at the fundamentalfrequency is then an amplitude squeezed state with a deamplified coherentamplitude

Fig 16 summarizes the squeezing operation on the vacuum state as wellas on displaced vacuum states for different phase relations ∆ϕ between thetwo input fields

43 Cavity-enhanced OPA

Placing the nonlinear crystal inside a cavity can greatly enhance thedown-conversion efficiency but not only that A cavity introduces a thresholdfor the pump power above which the parametric gain is infinite just limited

43

by the finite pump power In this case the vacuum uncertainty of the inputfield at frequency ν is amplified to a bright laser field at frequency ν Thedevice is then called an optical-parametric oscillator (OPO) For the gener-ation of squeezed states however the pump power is usually kept (slightly)below threshold Due to nonzero optical loss there exists a pump powersmaller than the threshold above which the tiny improvement of squeezingis not noticeable anymore Getting the pump power closer to the thresholdcould even reduce the observed squeeze factor if a fluctuating squeeze an-gle projects anti-squeezing into the observed quadrature amplitude [Franzenet al (2006) Suzuki et al (2006) Dwyer et al (2013)] The cavity has an-other important purpose It confines the transverse spatial mode usuallyto TEM00 This mode confinement is crucial for any efficient application ofthe squeezed state in laser interferometry since it allows the suppression ofanti-squeezing from other transversal modes The squeezing process requiresa nonlinear material that should show negligible absorption at both opticalfrequencies involved in particular at the wavelength of the squeezed modeIn Refs [Vahlbruch et al (2008) Mehmet et al (2009)] 10 dB and 116 dBof squeezing were achieved using MgOLiNbO3 The highest squeeze factorstoday are produced in (quasi phase matched) periodically poled KTP [Eberleet al (2010) Mehmet et al (2011) Stefszky et al (2012) Vahlbruch et al(2016)]

The optical cavity that is built around the nonlinear crystal is vital forsqueezed-light generation and it deserves a detailed consideration Gener-ally the mode propagating away from a cavity is the result of interferenceat the cavity coupling mirror One contribution is given by the intra-cavityfield attenuated by the amplitude transmission coefficient t of the outcouplingmirror The second contribution is given by the outside field that is reflectedby the same mirror with amplitude reflectivity r =

radic1minus t2 and spatially

overlapped with the first Also the mode from a squeezing resonator is suchan interference product

The impedance matched resonator

Let us consider first an empty optically stable and loss-less Fabry-Perotresonator built from two identical mirrors each with amplitude reflectivityr = r1 = r2 lt 1 A propagating field be perfectly mode-matched to one ofthe cavity resonances In this setup the resonator shows zero reflection andthe resonator is said to be impedance matched (for all such input fields)

44

|α〉

r1α

r12

= +

|α〉 r22 = r1

2

minusr1α equiv t1 αcav = 1minus r12 minusα sdot r1

1minus r12

0

XY

Figure 17 Empty impedance-matched resonator ndash Mode-matched and resonant lightthat enters the cavity from the left here displayed by a quantum phasor for a coherentstate is fully transmitted including its quantum uncertainty The back-reflected statedestructively interferes with itself for all frequencies well within the cavity linewidth The180 phase shift of the transmitted cavity field amplitude minusr1α is a necessary conditionin order to fulfill energy conservation on cavity resonance Impedance matching is forinstance achieved for a resonant cavity with matched mirror reflectivities (r2

2 = r21) and

zero optical loss The complex amplitude of the field traveling towards left inside the cavityis readily deduced from the figure and reads αcav = minusαr1(1minus r2

1)minus12 (highlighted by thedashed circle) In the displayed setup zero field uncertainties are reflected however alsothe vacuum state that enters the cavity from the right is fully transmitted (not shown)

r1α

XY

r12 r2

2 =1

= + |α〉

minus 1+ r1( )α equiv t1 αcav = 1minus r12 minus 1+ r1( )α

1minus r12

|α〉

Figure 18 Empty maximally overcoupled resonator ndash Maximal overcoupling isachieved for a resonant cavity with a perfect end-mirror reflectivity (r2

2 = 1) and zerooptical loss For a given input-mirror reflectivity r2

1 the intracavity light power is maximalMode-matched and resonating light entering the resonator from the left is fully reflectedThe complex amplitude of the field traveling towards left inside the cavity is readilydeduced from energy conservation to αcav = minusα(1 + r1)(1 minus r2

1)minus12 (dashed circle) Inthis setup no uncertainty from the right couples to the left side of the cavity

45

Obviously the interference described in the previous paragraph is fullydestructive The same resonator also shows zero reflection of the input fieldrsquosquantum uncertainty since the interference happens between parts of thesame quantum state The mode propagating away from such a resonatorhowever is not in a nonclassical but in a vacuum state because the vacuumstate that enters the cavity through the opposite site is also fully transmittedThe interference at the coupling mirror of an impedance matched resonatoris displayed in Fig 17

r1α

r12

+

=

OPA PDC

Xcavg r1(1+r1) Xcav

Δ2Xg 0 Δ2Yg

XY

r22 =1

|α〉

Figure 19 Squeezing resonator ndash Shown is the interference at the zero-loss squeezingresonator operated at threshold The lower line represents the perfectly squeezed modepropagating away from the cavity towards the left The parametric gain medium inside thecavity deamplifies the X quadrature of the cavity mode (Xcav) by the factor r1(1 + r1)which is the ratio of the intra-cavity field amplitudes of the two previous figures The Xquadrature of the field that is back-reflected towards the left destructively interferes withitself similar to the situation of the impedance matched cavity in Fig 17 The parametricpower gain per resonator round-trip (G) needs to mimic the effect of an end mirror withreflectivity r2

2 = r21 For this reason the deamplification of Xcav corresponds to a round-

trip deamplification factor of r1 equivradic

1G The round-trip amplification factor for Ycav

then is 1r1 equivradicG which exactly compensates for the outcoupling and thus determines

the parametric oscillation threshold (threshold for bright lasing) The variances of thequantum uncertainties ∆2Ycav and ∆2Y are thus infinite In this setup no field uncertaintyfrom the right couples to the left of the cavity and a perfectly X-quadrature-squeezed fieldoutside the squeezing resonator is produced

46

The perfectly over-coupled single-ended resonator

We now increase the reflectivity of the far mirror lsquo2rsquo to being perfect (r2 = 1)This way the counter-propagating vacuum state can not enter the cavityAgain a propagating field be perfectly mode-matched through mirror lsquo1rsquo toone of the cavity resonances For frequencies well inside the cavity linewidththe situation is displayed in Fig 18 The setup protects the left side of thecavity against vacuum fluctuations entering through mirror lsquo2rsquo but of coursedoes not squeeze quantum noise The intra-cavity built-up factor is too highfor achieving destructive interference below the vacuum uncertainty on theleft side of the resonator

The impedanced-matched single-ended squeezing resonator

Building on the two previous concepts the straight forward approach now isto start from the perfectly over-coupled single-ended resonator and insert anattenuator into the cavity that does not couple the cavity mode to any bathbut still results in a roundtrip efficiency of precisely r1(lt 1) in amplitudeOptical loss is not appropriate since it increases the coupling of the cavitymode to a thermal bath neither would any phase-insensitive atenuator beappropriate It is easy to show that a phase-insensitive attenuator adds ad-ditional uncertainty since otherwise the commutation relation [a adagger] = 1 isviolated The amplification process that matches our requirement is OPA Toachieve infinite squeezing in X on cavity resonance a second-order nonlinearcrystal needs to be put into the cavity and pumped such that the intra-cavityamplitude quadrature is attenuated by the factor (1 + r1)r1 (on cavity reso-nance) with respect to the empty cavity This factor is readily deduced fromFigs 17 and 18 Due to the symmetry in parametric amplification the intra-cavity phase quadrature is then amplified by (1 + r1)r1 and the round-tripgain has a value of 1r1 in amplitude In this situation not only infinitesqueezing but also the (laser) threshold of the resonator is achieved sincethe round-trip gain of the intra-cavity phase quadrature equals its roundtriploss here fully given by the incoupling mirror

The physical descriptions in Figs 17 to 19 are fully consistent with ob-servations in squeezing experiments The consideration above in particularshows that the intra-cavity field shows a finite squeezing strength while theexternal field shows infinite squeezing The strongest intra-cavity squeezefactor possible is (1 + r1)2r2

1 In the high reflectivity limit this factor corre-

47

sponds to 6 dB Averaged over the full cavity mode the squeeze factor of thecavity mode is in this limit even limited to 3 dB [Walls and Milburn (2008)]Higher intra-cavity squeeze factors are possible for lower mirror reflectivities

44 The generation of squeezed light for laser interferometry

With the insights gained in the previous subsection we now turn to ac-tual experiments The application of squeezed states in laser interferometrycertainly requires large squeeze factors (idealy accompanied with the highestpossible purity) to maximize the impact in terms of sensitivity improvementIn cavity-enhanced OPA the highest parametric gain is achieved on cavityresonance ie at zero sideband frequency But this is not the main reasonwhy this Subsection focusses on the generation of squeezed states at lowsideband frequencies The application of squeezed states in a laser inter-ferometer requires that their sideband frequencies cover the devicersquos signalband Ground-based gravitational wave (GW) detectors have a detectionband from about 10 Hz to 10 kHz frequencies which can be considered aslsquolowrsquo compared to typical frequencies in quantum optics experiments

Squeezing at MHz sideband frequencies is easier to observe than at acous-tic frequencies because the latter are often polluted with excess noise fromlight beams that serve as control beams [Bowen et al (2002) McKenzie et al(2004)] and parasitic interferences from back-scattered light [Vahlbruch et al(2007)] Furthermore the observation of squeezing at low sideband frequen-cies requires a more stable setup since larger measuring times are necessaryThe observation of strong squeezing at MHz frequencies however alreadysets an upper limit to the optical loss of the setup At least the same squeezefactor can be observed at lower frequencies

There are two different main topologies for squeezing resonators TheFabry-Perot-type standing-wave resonator consists of a minimum number ofmirror surfaces and has the advantage of being compact and thus robustagainst mechanical vibrations Usually one or even two mirror coatings aredirectly placed on the spherical and polished surfaces of the nonlinear crys-tal itself [Wu et al (1986) Grangier et al (1987) Breitenbach et al (1998)Vahlbruch et al (2008) Eberle et al (2010) Vahlbruch et al (2016)] TheBowtie traveling-wave resonator has the advantage of providing a separa-tely accessible counter propagating mode for cavity length control [Ou et al(1992) Takeno et al (2007)] It shows no direct back-reflection of incoupledlight which helps reducing parasitic interferences [Stefszky et al (2012)]

48

(a)

(b)

(c)

(d)Squeezingresonator

To inter-ferometer

OPA

SHG

BHD

Mode cleaner

LO

Laser

DBS

DBS

Figure 20 Generation of squeezed light ndash (a) Example of a 2nd-order nonlinearcrystal for the squeezed-light generation at 1064 nm Shown is a bi-convex 65 mm long7MgOLiNbO3 crystal whose polished surfaces also carry the mirror coatings of the res-onator The crystal thus realizes a monolithic squeezing resonator as it was used forthe first demonstration of 10 dB squeezing [Vahlbruch et al (2008)] (b) Optical con-figuration of a half-monolithic (hemilithic) standing-wave squeezing resonator Here thecavity length can be adjusted by displacing the coupling mirror The crystal surface insidethe cavity is anti-reflection coated The photograph shows a 10 mm long PPKTP crystalsqueezing resonator as used for the GEO 600 squeezed-light source [Abadie (2011)] (c)Mechanically stable housing of a standing-wave squeezing resonator The crystalrsquos temper-ature is stabilized at its phase matching condition using Peltier elements (d) Schematicfor the squeezed-light generation After spatial filtering of continuous-wave laser lighttwo hemilithic standing-wave resonators are employed The first generates second har-monic pump light (SHG) The second (OPA) generates a squeezed vacuum field at theinitial wavelength The squeezed states are observed by a balanced homodyne detector(BHD) or alternatively sent and mode-matched to the optical mode of an interferometerbeforehand LO local oscillator DBS dichroic beam splitter

49

Fig 20 (a) and (b) show photographs of typical nonlinear crystals used forsqueezed-light generation at near infra-red wavelengths The crystals shownhere form a monolithic standing-wave squeezing resonator (a) or are partof a half-monolithic standing-wave squeezing cavity (c) shows a tempera-ture stabilized and mechanically stable housing of the squeezing resonator(d) shows a schematic of a full setup for the generation of squeezed vacuumstates of light for an application in a laser interferometer The only brightinput required for the squeezing resonator (OPA) is the second-harmonicpump field The resonator mode at fundamental frequency is thus initiallynot excited by photons ie it is in its ground state characterized by vacuumfluctuations due to the zero point energy see Fig 7 (c) [Gerry and Knight(2005)] The pump field spontaneously decays in the degenerate pair of sig-nal and idler fields The combined down-converted field leaving the resonatorexhibits quantum correlations which give rise to a squeezed photon countingnoise when overlapped with a bright coherent local oscillator beam Thedetection is done either in a balanced homodyne detector (BHD) or with asingle photo diode The squeeze factor increases the closer the pump powerof the squeezing resonator gets to the oscillation threshold and the lower theoptical loss on down-converted photon pairs is

441 High squeeze factors ndash minimizing decoherence

Squeezed states of light have significant impact on the sensitivity of laserinterferometers if large squeeze factors can be produced Squeezing of 3 dBimproves the signal-normalized quantum-noise spectral density by a factorof 2 This factor corresponds to doubling the (coherent state) light powercirculating inside the interferometer Squeezing of 10 dB corresponds to aten-fold power increase The experimentally demonstrated squeeze factorswere considerably improved in recent years [Takeno et al (2007) Vahlbruchet al (2008) Polzik (2008) Eberle et al (2010) Stefszky et al (2012)]culminating in a value of as large as 150 dB [Vahlbruch et al (2016)] Thisvalue corresponds to the same reduction of signal-normalized quantum noisethat is achieved by increasing the light power by a factor of 32 (At this pointit is already noted that squeezing the quantum noise can simultaneouslyreduce quantum measurement noise (shot noise) as well as quantum backaction noise (radiation pressure noise) This is not possible with scaling thelight power of coherent states see Subsec 55)

50

Ideally a parametric squeezed-light source can produce an infinite squeez-ing level see Fig 19 fundamentally just limited by the energy provided bythe pump field In practice the limit is set by decoherence mechanismsThe by far most important one is optical loss Optical loss occurs duringsqueezed-light generation its propagation through the interferometric setupincluding imperfect mode matchings and finally the photo-electric detec-tion Also detector dark noise [Schneider et al (1998)] phase noise [Takenoet al (2007)] and excess noise [Bowen et al (2002)] impair the observablesqueezing strength

Optical loss is usually understood as coupling the squeezed mode to a zerotemperature bath ie overlapping it with a vacuum mode For any amountof loss the resulting state is still squeezed But to be able to directly observesay 10 dB of squeezing the total loss on the state needs to be less than 10in this example cf Eq (16) To minimize optical loss the nonlinear crys-tal as well as lenses and beam splitters in the interferometric path need toshow very low absorption and scattering at the wavelength of the squeezedlight PPKTP shows absorption of about 10minus4cm and below at near-infraredwavelengths Low OH content fused silica is a suitable material for all otheroptics Absorptions of less than 10minus6cm were measured [Hild (2007)] Coat-ings on crystal surfaces and on all other optical components should also showlowest optical loss Total loss of the 10minus6 level are available today Superpol-ished surfaces which show roughnesses with less than 1 A root mean square(integrated over spatial scales from approximately 1 micron to 100 microns)and thus very low scattering are necessary to achieve these low numbersMinimizing the total number of optical components is essential From thisperspective a monolithic squeezing resonator as shown in Fig 20 (a) is theoptimum choice The squeezed mode needs to be matched to the mode ofthe laser interferometer or to the mode of the balanced homodyne detectorVisibilities of up to 998 have been achieved [Eberle et al (2010)] whichcorresponds to a loss of about 04 Of great importance also is the quantumefficiency of the photo-diodes used for detecting the squeezed field (togetherwith the interferometric signal) Recently a quantum efficiency of photo-diodes in a squeezing experiment of (995plusmn 05) was measured [Vahlbruchet al (2016)] To minimize photon loss the photo-diodes had no protectionwindow an anti-reflection coating on the semi-conductor material and theremaining reflection was re-focussed with an external mirror

Also the dark-noise spectral density of the detection electronics reducesthe observable squeezing and needs to be as low as possible Similar to optical

51

noise it also provides a contribution to the observed variance The dark noiseof the detection electronics needs to be much lower than the detected photoncounting noise In [Vahlbruch et al (2016)] it was 28 dB below shot noisebut still reduced the observable squeeze factor from 153 dB to 150 dB

Excess noise emerges if the squeezed mode couples to a nonzero tem-perature bath or to a mode whose excitation is strongly fluctuating (Thecoupling process can always be understood as a beam splitter coupling andis physically described by overlapping electric fields Coupling to a zero tem-perature bath leads to Eq (16)) The captured excess noise variance thenneeds to be added to the initial squeezing variance which deteriorates theobserved squeezing stronger than just mixing in the vacuum mode Excessnoise is less likely to occur at MHz frequencies but can be significant ataudio-band sideband frequencies and below and is thus a serious issue ingravitational-wave detectors [Chua et al (2014)] The reason for that isthat acoustically or thermally excited motions of surfaces produce frequencyshifts of back-scattered light mainly at these low frequencies [Vahlbruch et al(2007)]

Phase noise corresponds to stochastic phase fluctuations between thesqueezed field and the local oscillator within the measuring time It cor-responds to mixing the squeezed mode with itself with a fluctuating squeezeangle [Suzuki et al (2006) Franzen et al (2006)] Phase noise in squeez-ing experiments typically is less of an issue than optical loss [Dwyer et al(2013) Oelker et al (2016) Vahlbruch et al (2016)] The setuprsquos phasenoise can be reduced by making the squeezing resonator more compact andthus mechanically more stable against acoustic and thermal fluctuations ofthe environment and by improving the quality of the servo loops for cavitylength and propagation length controls Operating a squeezed-light resonatorin vacuum might also be beneficial The ability to run a high performancesqueezed-light generator in vacuum was demonstrated in [Wade et al (2015)]

442 Squeezing in the gravitational-wave (GW) detection band

High squeeze factors have been first demonstrated at sideband frequen-cies of a few MHz and above where excess noise is generally negligible whenworking with visible or near-infra-red light Today we know that extendingthe squeezing spectrum towards the audio-band and even below is technicallynot always easy but straight forward once a high squeeze factor is achieved

52

at MHz frequencies In most squeezing experiments the main laser light

Figure 21 Photograph of the GEO 600 squeezed-light source ndash The breadboarddimensions are 135 cm times 113 cm The squeezing resonator is high-lighted by the white ar-row and is set up as a standing-wave hemilithic cavity containing a plano-convex PPKTPcrystal of about 10 mm length (see also Fig 20 b) It is pumped with continuous-wave532 nm light that is produced by frequency doubling of light (at angular frequency ω)from a commercial NdYAG laser Two more laser fields at about 1064 nm having fre-quency offsets of more than 10 MHz with respect to ω(2π) serve as optical control fieldsBoth fields are mode-matched and injected into the squeezing resonator together with thesecond-harmonic pump field

at the squeezing wavelength is accompanied by significant noise up to thelaser relaxation oscillation For this reason laser control fields at the opticalcarrier-frequency in the optical path of the squeezed mode need to be avoided[Bowen et al (2002) Schnabel et al (2004) McKenzie et al (2004)] and thesqueezing resonator length and the optical path stabilized by other means[McKenzie et al (2005) Vahlbruch et al (2006)] Furthermore and mostimportantly excess noise due to back-scattering is an issue Back-scattering(also called lsquoparasitic interferencesrsquo) is produced if DC light scatters out ofthe optical path hits a vibrating surface and re-scatters back into the opticalpath [Vahlbruch et al (2007)] Significant back-scattering can be produced

53

in interferometers for the detection of gravitational waves since light powersof several hundreds of kilowatts are used Even back-scattering from the mil-liwatt local oscillator of balanced homodyne detectors is an issue at acousticsideband frequencies and below The recipe for avoiding parasitic interfer-ences turns out to be threefold (i) avoiding scattering by using ultra-cleansuperpolished optics with close to perfect anti-reflex coatings (ii) avoidingback-scattering by carefully blocking all residual (faint) light fields and (iii)reduce the vibrationally and thermally excited motion of all mechanical andoptical parts that could potentially act as a re-scattering surface with re-spect to the optical path [Vahlbruch et al (2007) McKenzie et al (2007)]The insights described above led to the first demonstration of audio-bandsqueezing at frequencies down to 200 Hz [McKenzie et al (2004)] and laterto the first demonstration of squeezing over the full gravitational-wave de-tection band even from as low as 1 Hz [Vahlbruch et al (2007)] Whilea standing-wave squeezing resonator [Ou et al (1992)] can be built in avery compact way that is rather insensitive against mechanical vibrations[Chelkowski et al (2007)] a traveling-wave bow-tie squeezing resonator [Wuet al (1986)] is more tolerant against back-scattered light [Chua et al (2011)]The strongest squeezing in the audio-band of up to 116 dB was reported inRef [Stefszky et al (2012)]

443 The first squeezed-light source for GW detection

The first squeezed-light source for the continuous operation in GW de-tectors had been designed and completed between 2008 and 2010 [Vahlbruch(2008) Vahlbruch et al (2010)] Since then this source has been producingsqueezed vacuum states in a fully phase controlled way using co-propagatingfrequency-shifted bright control beams [Vahlbruch et al (2006)] as an inte-gral part of the GW detector GEO 600 The source is a turn-key device witha fully automated re-lock system [Vahlbruch et al (2010) Khalaidovski et al(2012)] Re-locking is required if the temperature of the environment changessignificantly which drives the actuators outside their dynamic ranges

Up to 9 dB of squeezing over the entire GW detection band was observedusing a balanced homodyne detector (BHD) located close to the squeezingresonator The squeeze factor has been limited by optical loss due to ab-sorption in the PPKTP crystal transmission of the back-surface and thenon-perfect AR-coating of the crystalsrsquos intra-cavity surface The adjustableair gap has been necessary to allow for an easy way to apply length control

54

-12

-8

-4

0

4

8

12

16

20

10 100 1k 10k

Rel

ativ

e no

ise

pow

er [

dB]

Frequency [Hz]

shot noise (a)

squeezed noise (b)

anti-squeezed noise (c)

~ 9dB

Figure 22 Broadband squeezing spectrum ndash Noise power spectra measured on theoutput of the GEO 600 squeezed-light source shown in Fig 21 with a balanced homodynedetector The traces correspond to the spectra of quadrature amplitude variances normal-ized to vacuum noise The resolution bandwidth used increases towards higher frequenciesto reduce the measurement time (a) Shot noise normalized to unity which serves as thereference level (0 dB) (b) Squeezed noise covering the complete detection band of ground-based GW detectors (c) Anti-squeezed noise Peaks at 50 Hz and 100 Hz are the electricmains frequency and its first harmonic The data was published in Ref [Vahlbruch et al(2010)]

Additional optical loss in the path to the balanced homodyne detector mainlyarose due to a Faraday isolator that eliminated parasitic interferences Fi-nally the mode missmatch to the BHD as well as its non-perfect quantumefficiency provided additional loss Inferring the squeeze factor without theBHD detection loss more than 10 dB of squeezing are provided by the sourceSince 2010 it has been used in basically all observational runs of the GEO 600GW detector see Section 6

444 Generation of two-mode (bi-partite) squeezing

lsquoTwo-mode squeezed lightrsquo or lsquobi-partite squeezed lightrsquo is light that allowsfor joint measurements at two locations A and B These joint quadraturemeasurements reveal correlations and anti-correlations with a remaining un-certainty smaller than the ground-state uncertainty which certifies the pres-

55

ence of entanglement cf Subsec 34 Bi-partite squeezed light has been gen-erated by type I and by type II parametric down-conversion In case of type Ithe squeezed fields from two squeezing resonators as described in Subsec 44are overlapped on a balanced beam splitter with a 90 phase shift The twooutput fields together represent the entangled mode [Furusawa et al (1998)Bowen et al (2003c) Eberle et al (2013)] In case of type II signal and idlerfields are non-degenerate regarding polarisation and a single cavity contain-ing an appropriate crystal and a polarising beam splitter are sufficient for theproduction of bi-partite squeezing Also in this case the measurements ofthe quadrature amplitudes of signal and idler fields show large uncertaintiestogether with bi-partite correlations and anti-correlations that are strongerthan the ground state uncertainty of individual subsystems [Ou et al (1992)Villar et al (2006) Jing et al (2006)]

To date the strongest entanglement of bi-partite squeezed light has beenproduced based on type I parametric down-conversion [Eberle et al (2013)]The requirements of producing strong entanglement are identical to those ofproducing strong squeezing outlined above The strength of bi-partite en-tanglement can be given in decibels in full analogy to the squeeze factorPractically the strength of bi-partite squeezing is always somewhat smallerthan that of single party squeezing since it requires an additional mode-matching that results in additional optical loss

45 Conclusions

The first observation of squeezed light was achieved in 1985 [Slusher et al(1985)] Shortly after cavity-enhanced optical parametric amplification forsqueezed-light generation was demonstrated [Wu et al (1986)] which todayenables the observation of up to 15 dB of squeezing [Vahlbruch et al (2016)]Quite generally the maximum squeezing level that is observed does not de-pend on the strength of the optical nonlinearity Squeezing cavities can easilybe operated at their oscillation threshold where they should provide infinitesqueezing if decoherence is zero The main limiting factor is optical lossincluding that of the photo-electric detection

Dedicated experimental research and development towards a squeezed-light source for applications in gravitational-wave detectors can be tracedback to 2002 [McKenzie et al (2002) Bowen et al (2002)] Since then a sur-prising amount of progress has been made culminating in the first squeezed-light source specifically built for the integration into a gravitational-wave

56

detector For the future squeeze factors above 15 dB will certainly be possi-ble by further reducing optical loss This statement is supported by the highdegree of matching between experimental data and a theoretical loss modelas presented in Fig 3 of [Vahlbruch et al (2016)]

5 Quantum noise in laser interferometers

51 Interferometric measurements

The purpose of a laser interferometer is the precise measurement of smallchanges of an optical path length with respect to a reference path Forthis the interferometer transfers the change of the phase difference betweentwo light fields into an amplitude quadrature change of the interferometerrsquosoutput light The latter can easily be detected by a single photo diode Ofgeneral interest are differential changes of the optical path length that aremuch smaller than the laser wavelength ie differential phase changes thatare much smaller than 2π In this case the differential phase change can bedescribed in very good approximation as a differential change of the phasequadrature instead

In order to transfer the phase quadrature signal with minimum loss ahigh interference contrast at the interferometerrsquos beam splitter is essentialAdditionally instrumental noise in terms of unwanted excitations of the out-putrsquos amplitude quadrature needs to be reduced to a minimum Noise arisesdue to power fluctuations of the input laser light back-scattered laser lightinside the interferometer thermally driven displacements of mirror surfacesand in many more ways The important measure of the sensitivity of aninterferometer obviously is its signal-to-noise-ratio The most useful measureis given in terms of the noise spectral density S(f=Ω2π) that is normalizedto the physical unit of the signal S(f) is in fact a lsquonoise-to-signal-ratiorsquoand can be seen as the signal-normalized variance of the photo diode out-put decomposed into spectral components versus sideband frequency f withthe resolution bandwidth of 1 Hz As an example S(100 Hz) = 10minus39 m2Hzmeans that the instrumental noise in the one hertz band around 100 Hz equalsa signal that would be produced if the mirror of one interferometer arm os-cillates with an amplitude of just

radic10 middot 10minus20 m in the very same band Such

small spectral densities are achieved by gravitational-wave detectors [Abbott(2016)]

57

52 Quantum measurement noise and shot noise

The most fundamental noise source in laser interferometers is due to thequantum noise of light which is in fact two-fold [Caves et al (1980)] Firstof all there is lsquoquantum measurement noisersquo which arises in the processof photo-electric detection For coherent states the quantum measurementnoise is the lsquophoton counting noisersquo of mutually independent photons andusually simply called lsquoshot-noisersquo Fig 2 (b i) shows a time series of suchnoise hiding the actual signal The frequency components of the shot noiseare well described by the quantum uncertainty of the output fieldrsquos amplitudequadratures XΩ∆Ω see Subsec 22 (Recall this quantity corresponds to the

differential phase quadrature YΩ∆Ω of the light beams in the interferometerarms) The photon counting noise has a white Fourier spectrum howeverthe lsquoshot noisersquo of an interferometer is usually normalized to the signal whosetransfer function is usually not white for instance due to the presence of armcavities or a signal-recycling cavity

All current and planned gravitational-wave detectors are Michelson-typelaser interferometers with operating points very close to a dark fringe Thelight power in the output port is just a couple of tens of mW which canbe handled by a single photo diode In this configuration the signal-to-shot-noise-ratio is actually maximized which can be shown in three steps [Bachorand Ralph (2004)] For the first step we use plane waves to describe theelectric field in the output port of a Michelson interferometer For perfectinterference contrast at the balanced beam splitter ie for perfect modematching and for defining φ = 0 as the dark port condition we get

Eout(t φ) =1

2E0 sin(ωt+ φ)minus 1

2E0 sin(ωt) (31)

where E0 is the amplitude of the total internal field whose two parts hasaccumulated a differential phase It directly follows for the squared fields

E2out(t φ) =

(sin

φ

2

)2

(E0 cos(ωt+ φ2))2 (32)

We now turn to a light beam with a localized transversal mode that can befocussed onto a photo-electric detector The photo diode has perfect quantumefficiency ie the rate of photo electrons is not only proportional to the rateof output field photons but also has a unity slope efficiency Since the optical

58

frequency is too high to be resolved we consider the averaged light power

P out(φ) =

(sin

φ

2

)2

P (33)

The next step is a formulation of the signal being the derivative of detectedphoton number versus phase Let n be the average value of the photonnumber per measuring time interval Eq (33) can then be rewritten as

nout(φ) =

(sin

φ

2

)2

n (34)

rArr dnout(φ) = n sinφ

2cos

φ

2dφ (35)

The final step is the calculation of the signal-to-shot-noise-ratio Shot noiserefers to coherent states which have a standard deviation of the photonnumber of σ(n) =

radicn

dnout(φ)

σ(nout)=n sinφ

2cosφ

2dφ

radicn sinφ

2

(36)

and find for a signal-to-noise ratio of unity for coherent states and for anon-zero but still small phase difference ∆φCoh 2π

1 =radicn cos

φ

2∆φCoh with φ 6= 0 (37)

In this equation the smallest measurable phase difference is given for φrarr 0

∆φCohmin =

1radicn (38)

This is the well-known shot-noise limit of high-precision phase sensing ∆φCohmin

is the smallest phase shift that can be measured with a signal-to-noise ra-tio of one when using n mutually independent photons per measuring time(those of a coherent state) when the loss of photons is assumed to be zeroThe typical purpose of a laser interferometer is the continuous sensing (moni-toring) of a continuously changing phase An illustrative example is the phasesignal produced by the black hole merger measured by Advanced LIGO onSept 14 in 2015 (Fig 1 in [Abbott (2016)]) The measuring interval should

59

be short to be able to resolve the time-evolution of the signal Generally themeasurement of an arbitrary signal that lasts for a finite time thus needs tobe understood as l subsequent measurement intervals using n photons eachIt can be shown that Eq (38) is also valid for interferometers operated athalf fringe ie when each output port contains the same light power In thiscase photo diodes need to be placed in both output ports and the actualsignal is provided by their difference voltage

Due to its importance the shot-noise limit deserves some remarksThe phase φ in Eq (31) is the phase difference of two mode-matched fieldsand might be accumulated by a single pass along the length L such as ina Mach-Zehnder interferometer or in a double pass such as in a (simple)Michelson interferometer or in four passes as realized in a Michelson inter-ferometer with folded arms [Grote (2005)] The shot-noise limit in Eq (38)and its scaling therefore holds independent of the number of passes Theclaim in Ref [Higgins et al (2007)] that the scaling according to Eq (38) canbe surpassed by multiple passes is not justifiedThe fact that Eq (38) is derived by approaching φ rarr 0 correctly describesthe actual operation point of gravitational-wave detectors which is close tobut not exactly at a dark port In practice a tiny offset from dark port ischosen at which the shot noise is well above the photo diodersquos electronic darknoiseEq (38) solely depends on the number of quanta but not on the lightrsquos wave-length λ Of course the shot-noise limit for the change of an optical pathlength ∆L does depend on the wavelength and ∆φCoh

min needs to be replacedby ∆φCoh

min = 2π∆LCohminλ

Finally an essential result of the shot-noise limit is that the ideal precisemeasurement should use lsquoas much quanta as possible per measuring inter-valrsquo which translates to lsquoas much light power in the interferometer armsas possiblersquo Eq (38) is indeed the one and only reason why gravitational-wave detectors use high power lasers power-recycling and arm resonatorsExtending the measuring time for a given light power can also improve thesensitivity but only if the signal repeats ie is periodic Let us assume thatone period of the signal is resolved by l intervals using n photons each Inthis case repeating the overall measurement k times improves Eq (38) by1radick The fundamental statement of Eq (38) however does not change

since the actual photon number n may then simply incorporate the factor k

60

For a given average photon number the shot-noise limit in Eq (38) canonly be surpassed by using photons that are quantum correlated ie by usingnonclassical states of light How is the shot-noise limit surpassed with thehelp of squeezed states A Michelson interferometer that is operated closeto a dark fringe acts like an almost perfect mirror for both input ports Allthe input light is back-reflected towards the laser source This also accountsfor the quantum uncertainty of the input light The quantum uncertaintythat impinges onto the photo diode thus (mainly) enters the interferometerthrough its (almost) dark port An interferometer that uses displaced co-herent states entering from one port can thus be improved by replacing theordinary vacuum entering the signal output port by a squeezed vacuum stateThis was the proposal by CM Caves in 1981 [Caves (1981)] which is labeledhere with lsquoCSVrsquo If the differential phase quadrature of the interferometer issqueezed Eq (38) then within the limit of large coherent state displacementα sinh2r improves to

∆φCSVmin asymp

eminusrradicn (39)

(The above expression is an approximation since the squeezing operationproduces a small number of photons that are not accounted for here) Ofcourse the mode of the squeezed vacuum needs to be precisely matchedto the mode of the interferometer The first experimental demonstrations ofsqueezed phase measurements used a Mach-Zehnder [Xiao et al (1987)] and apolarization interferometer [Grangier et al (1987)] Fig 2 shows how spatialdegeneracy between an externally generated squeezed mode and the signalmode in a Michelson interferometer is achieved using a polarizing beam split-ter and a Faraday rotator Again the limit in Eq (39) can only be achievedif optical loss is zero Optical loss not only reduces the signal but here alsoreduces the squeeze parameter see Eqs (15) and (16)

Let us consider an example The sensitivity of a laser interferometer thatuses coherent states with an excitation of 1023 photons per second can beimproved by a factor of

radic10 by either adding 09 middot 1024 photons per second

or by adding about just 2 photons per second and bandwidth in hertz thatbelong to the 10 dB squeezed vacuum confer Eq (30) Since the full sig-nal band of ground-based GW detectors covers sideband frequencies up to10 kHz just 2 middot 104 photons per second are necessary At a wavelength ofλ = 1064 nm these values correspond to a power increase by 168 kW and

61

37 fW respectively

The question arises whether a scaling of the sensitivity better than propradic1n is possible It was theoretically shown that in principle the scaling

can indeed considerably be improved yielding the so-called Heisenberg limitor Heisenberg scaling [Bondurant and Shapiro (1984) Yurke et al (1986)Braunstein (1992) Holland and Burnett (1993)]

∆φHLmin prop

1

n (40)

The Heisenberg scaling requires nonclassical states that have a certain num-ber of quanta similar to Fock states ie n = n The theoretically optimalstates describe a superposition of n (N) indistinguishable photons in oneinterferometer arm while having zero (0) photons in the second arm and viceversa and were named lsquoN00Nrsquo-states [Dowling (2008)] A specific propertyof these states is lsquosuper-resolutionrsquo The output ports of the interferometershow an n-times faster oscillation of the interference fringes when changingthe phase between the two interferometer arms Super-resolution correspondsto an n-times improved signal transfer function and was demonstrated forinstance in Refs [Rarity et al (1990) Kuzmich and Mandel (1998) Mitchellet al (2004) Afek et al (2010)] The presence of this nonclassical phe-nomenon however does not prove a sensitivity better than the semi-classicalbound according to Eq (38) Sensitivity is rather related to the signal-to-noise-ratio and needs to take into account all imperfections as well as theprobability of a successful detection of the sensing state [Thomas-Peter et al(2011)] All experiments so far used post-selection on particular measure-ment outcomes and neglected the typically large probability that nothingwas detected

Super-resolution was demonstrated with up to n = 5 [Afek et al (2010)]In addition to the fact that super-resolution does not prove a sensitivity bet-ter than the semi-classical bound photon numbers in state-of-the-art super-resolution experiments are extremely small compared to the photon numberof about 1023 (within a measuring interval of one second) of coherent statesused in Ref [Abbott (2016)] and of about 1022 using coherent states plussqueezed vacuum states used in Ref [Abadie (2011)]( which did prove a sen-sitivity better than the semi-classical bound)

Another interesting and related question is what the smallest phase is

62

that can be estimated in a single measurement again using a given numberof quanta Taking into account that no prior information about the phaseshift exists still a scaling proportional to 1n is possible In the limit of largen however an additional factor of π is required in the nominator of Eq (40)[Sanders and Milburn (1995) Berry and Wiseman (2000)] yielding

∆φHLmin asymp πn (41)

The state that can actually achieve this bound is different from the N00Nstate and was found in [Summy and Pegg (1990) Luis and Perina (1996)Berry and Wiseman (2000)] A N00N state is not the optimum state forphase estimation (via a single measurement) since it only provides one bit ofinformation A recent review on generell aspects on phase measurements isgiven by Ref [Demkowicz-Dobrzanski et al (2015)]

It is important to note that Eqs (38-40) do not consider photon lossExperiments that demonstrated super-resolution and aimed for achievingthe scaling in Eq (40) were conditioned on zero photon loss Let η gt 0 bethe average efficiency of detecting (all) photons Eq (38) then reads

∆φηCohmin =

radic1

η n (42)

Eq (39) turns into

∆φηCSVmin asymp

radicηeminus2r + 1minus η

η n(43)

and Eq (40) turns into [Demkowicz-Dobrzanski et al (2012 2013)]

∆φηmin =

radic1minus ηη n

for 0 lt η lt 1 (44)

For non-zero photon loss most interestingly the ultimate sensitivity of aphase measurement for a given photon number also shows a 1

radicn -scaling

The difference between the CSV strategy of using bright coherent states incombination with squeezed vacuum states which is bounded by Eq (43) andthe strategy of using the optimal nonclassical state which is bounded byEq (44) is marginal in practice [Demkowicz-Dobrzanski et al (2013)] Forgravitational-wave detectors and for any other laser interferometer using in-tense light there is no need for an alternative to the CSV strategy

63

We now turn back to the shot noise according to Eq (38) Generally noisecan be decomposed into its spectral contributions For a simple Michelsoninterferometer without arm resonators and without a signal-recycling cavitythe square-root of the single-sided shot-noise spectral density normalized tothe differential arm length change x in units of m

radicHz is given by [Saulson

(1994)] radicSMI

SNx =

radic~c2

2ωPprop 1radic

P (45)

where ω is the optical angular frequency of the quasi-monochromatic carrierlight and P the total light power in both arms including the built-ups fromcavities In combination with a squeezed vacuum whose relative phase gen-erates squeezing of the output lightrsquos amplitude quadrature the right handside reduces according to the factor eminusr Note that the single-sided spectraldensity is only defined for positive sideband frequencies and thus twice aslarge as the double-sided spectral density

The spectral density of the measurement of a GW induced strain is givenby the same expression but normalized to h = xL (If the gravitational waveis oriented in an optimal way with respect to the Michelson interferometerone arm is squeezed while the other is expanded by the same amount of ∆L =x2 and h then corresponds to the actual gravitational-wave amplitude)The square-root of the single-sided shot-noise spectral density normalized tostrain in units 1

radicHz is given by

radicSMI

SNh =

radic~c2

2L2ωP (46)

Equations (45) and (46) show that the smallest measurable signal (corre-sponding to unity signal-to-shot-noisendashratio) is inversely proportional to thesquare root of the laser power and has a white spectrum for sideband frequen-cies much smaller than the carrier frequency see horizontal line in Fig 23

All first- and second-generation GW detectors use power-recycling andadditional cavities to improve their sensitivities Fabry-Perot arm resonatorsdo not only increase the light power but additionally also increase the signalfor signal frequencies inside the resonator linewidth For lossless Fabry-Perot

64

arm resonators the spectral densities in Equations (45) and (46) need to bemultiplied by the following factor [Kimble et al (2001)]

HFP =

radicL2(γ2

FP + Ω2)

c2 (47)

where γFP = cTFP(4L) is the Fabry-Perot arm resonatorrsquos half bandwidthand TFP is the light power transmission of the input mirror The end mirrorsare assumed to have perfect reflectivity A similar expression can be derivedfor describing the improvement due to signal-recycling [Buonanno and Chen(2001)]

In summary shorter laser wavelengths higher light powers and squeezingof the amplitude quadrature of the interferometer output reduce shot noise ina broadband way ie for all signal frequencies Fabry-Perot arm resonatorsas well as signal-recycling provide improvements mainly for frequencies insidethe resonator linewidths

53 Quantum back-action and quantum radiation pressure noise

In laser interferometers quantum back-action noise results from the un-certainty of the lightrsquos radiation pressure force on the interferometer mirrorsand is also called lsquo(quantum) radiation pressure noisersquo (RPN) Its origin isthe quantum uncertainty of the differential amplitude quadrature XΩ∆Ω ofthe fields in the interferometer arms It results in an uncertain momentumtransfer to the mirrors and thus in an position uncertainty of the mirrors atfuture times with respect to their differential mode of motion [Caves et al(1980)] The physical mechanism of radiation pressure corresponds to anintensity dependent phase shift [Pace et al (1993)]The higher the light power in the arms of a laser interferometer the loweris its shot-noise spectral density see Eq (45) Unfortunately the spectraldensity of quantum back-action noise increases with light power The single-sided force noise spectral density reads

radicSRPNF =

radic8~ωPc2

(48)

Whereas the force noise of the quantum radiation pressure has a white spec-trum the RPN does not since the mirrorrsquos reaction to external periodicforces depends on frequency The link between the Fourier component of an

65

external force F (Ω) and the Fourier component of the displacement x(Ω) isgiven be the mechanical susceptibility HM It reads for an harmonic oscillatorwith mass M

HM(Ω) =1

M | minus Ω2 + Ω2M + iΩΩMQ|

(49)

where ΩM is the oscillatorrsquos resonance frequency and Q its quality factorThe square root of the single-sided spectral density of the RPN normalized

to the displacement of an harmonic oscillator with mass M is then given by

radicSRPNx = HM(Ω)

radic8~ωPc2

(50)

In GW detectors the test mass mirrors are suspended as pendula with highmechanical Q-factors and their centre of mass motion corresponds to that ofa harmonic oscillator The resonance frequencies of the pendula are lowerthan the detection band of interest The mechanical susceptibility is thereforeoften approximated for the so-called free-mass regime as H fm

M (Ω) = (mΩ2)minus1The square root of the single-sided spectral density of the RPN normalizedto differential displacement of two mirrors with each of mass M in a simpleMichelson interferometer is given by [Saulson (1994)]radic

SfmMIRPNx =

radic2~ωPc2m2Ω4

propradicP (Ω ΩM) (51)

where m = M2 is the mirrorsrsquo reduced mass In case of a simple Michel-son interferometer that is enhanced with arm cavities the spectral density inEq (51) needs to be multiplied with the expression given in Eq (47) In com-bination with a squeezed vacuum whose relative phase generates squeezing ofthe output lightrsquos phase quadrature the right hand side reduces according tothe factor eminusr Note if the radiation pressure noise is squeezed the shot noisemust be anti-squeezed or vice versa The radiation pressure noise calibratedto strain of space time is given by the right side of Eq (51) divided by theinterferometer arm length LIn summary heavier masses longer laser wavelengths lower light powersand squeezing of the amplitude quadrature in the interferometer arms re-duce radiation pressure noise

66

1 10 100 1000Frequency [Hz]

Radiation pressure noise

Shot noise

SQL

10-21

10-18

10-15

10-12

radicSx

[mradic

Hz]

ndashndash

ndashndash

Figure 23 Displacement-normalized quantum noise spectral densities ndash Consid-ered is a simple Michelson interferometer with neither arm cavities nor signal recyclingThe two end mirrors (m = 100 g) of the interferometer arms are suspended as pendulahaving a resonance frequency of ΩM2π = 1 Hz and a Q-factor of 107 The interferometeruses quasi-monochromatic light (in coherent states) with a total power of 4 kW Opticalloss and the offset from a dark output fringe is assumed to be negligible Wavelengthλ = 1550 nm The standard quantum limit (SQL) corresponds to the lowest noise achiev-able at a given sideband frequency when varying the light power without using quantumcorrelations

54 Interferometer total quantum noise and the standard quantum limit

Both shot noise and radiation pressure noise contribute to the total quan-tum noise of a given interferometer If they are not quantum correlatedwhich is the case for a conventional Michelson interferometer when detectingthe output lightrsquos amplitude quadrature their variances add up (The resultis not shown in Fig 23) It can easily be deduced from the previous sectionsthat changing the laser power will shift the two quantum noise contributionsHowever the total quantum noise never goes below the standard quantumlimit (SQL) [Braginsky and Manukin (1967)]

Let us consider Fig 23 for sideband frequencies much greater than thependulum resonance Here the test mass mirrors react as free masses whenexerted to external forces The SQL in this free-mass regime is calculated

67

by minimizing the sum of the squares of Eqs (45) and (51) [Saulson (1994)]

Sfmtotx =

~c2

2ω

[1

P+

4ω2

c4m2Ω4P

] (52)

Its derivative reads

dSfmtotx

dP=minus1

P 2+

4ω2

c4m2Ω4 (53)

Setting the above equation to zero provides the optimum laser power versussideband frequency in order to achieve the lowest total quantum noise

P fmopt =

c2mΩ2

2ω (54)

Inserting the optimal light power into Eq (57) provides the square root ofthe single-sided noise spectral density of the free-mass SQL in m

radicHzradic

SfmSQLx =

radic2~mΩ2

(55)

Again m is the reduced mass and dividing by the interferometer arm lengthL yields normalization to the GW-induced strain h Eq (55) shows that theSQL falls off with sideband frequency The corresponding equation for aMichelson interferometer that uses arm cavities readsradic

SfmFPSQLx =

radic~

mΩ2

(1

HFP

+HFP

) (56)

with HFP according to Eq (47)Using the expression for the SQL the square root of the total quantum

noise spectral density of a Michelson interferometer in the free-mass approx-imation can be written asradic

SfmFPtotx =

radicSfmFP

SQLx

2

[1

k+ k

] (57)

with the radiation pressure coupling parameter

k(Ω) =2ωP

mc2Ω2 (58)

68

For a fixed light power and fixed reduced mass of the mirrors the quantumnoise limited interferometer reaches the SQL when k = 1 which is realizedat the angular sideband frequency ΩSQL =

radic2ωP(mc2)

Note that neither squeezing the phase quadrature nor squeezing the am-plitude quadrature of the interferometer light leads to sub-SQL performance[Caves (1981)] also confer Ref [Schnabel (2005)] Fig 3 (left) As we willsee in the next sections the standard quantum limit can be surpassed if shotand radiation pressure noise are correlated Then the total quantum noise isnot given by the sum of the variances ie the sum of the squares in Eqs (45)and (51)

55 Squeezed light for surpassing the standard quantum limit

A measurement with sensitivity better than the standard quantum limit(SQL) is also called a lsquoquantum non-demolition (QND)rsquo measurement [Bra-ginsky and Khalili (1995 1996) Kimble et al (2001)] Several QND tech-niques for laser interferometers were proposed in recent decades [Jaekel andReynaud (1990) Kimble et al (2001) Purdue and Chen (2002) Chen (2003)McClelland et al (2011) Danilishin and Khalili (2012) Graf et al (2014)]What they have all in common is they exploit quantum correlations betweenobservable uncertainties

Arguably the most extensive way of introducing quantum correlationsand surpassing the SQL is the injection of squeezed states of light [Jaekeland Reynaud (1990)] If the squeezed quadrature angle of the injected statesis neither 0 nor 90 the quantum uncertainties of the amplitude and phasequadrature amplitudes that describe the differential field in the two interfe-rometer arms become correlatedLet us consider a very simplified setup that just consists of a quasi-mono-chromatic light field that is back-reflected from a quasi-free mirror Thelight power and the mass be such that reflected light in a coherent stateresults in a measurement of the mirror position with a noise spectral den-sity at the SQL at sideband angular frequency ΩSQL At this frequencyquantum measurement noise and back-action noise are of the same sizeie the uncertainty in X produces an equally large additional uncertaintyin Y Upon reflection the quadrature amplitude variances change from∆2X = ∆2Y = 14 to 2∆2X = ∆2Y = 12 This result corresponds tothe situation in Fig 23 at the crossing frequency of shot noise and radiation

69

pressure noise The coupling of the uncertainty variances can be describedby the matrix K = (1 minusk 0 1) where k = 1 at the SQL If the modulationstate at ΩSQL is the ground state its variances are transferred according to

KT

(1 00 1

)K =

(1 0minus1 1

)(1 00 1

)(1 minus10 1

)=

(1 minus1minus1 2

) (59)

In accordance with Fig 23 the variance of YΩSQLis twice as large as the vac-

uum noise varianceNow let the quantum noise of the light field be 10 dB squeezed at 45

(Eq (19)) The projection of the quantum uncertainty onto the X-observableproduces the radiation pressure noise by being transferred with the couplingfactor k = 1 at the SQL into the Y -observable in fact with negative signsince a larger value of X produces a larger optical path length and thus aretardation of the phase Due to the squeezing at 45 the initial uncertaintyin Y cancels with the additional uncertainty that originates from the onein X The following calculation shows that the strength of the cancellationcorresponds to the initial squeezing strength Upon reflection the quantumuncertainties transform in the following way(

1 0minus1 1

)(505 495495 505

)(1 minus10 1

)=

(505 minus01minus01 02

) (60)

The state of light after reflection has a squeezed phase quadrature amplitudeThe improvement in comparison to Eq (59) is exactly 10 dB The quantumnoise improvement corresponds to the input squeeze factor and is also a mea-sure by what factor the SQL is surpassed Squeezed vacuum injection thusallows surpassing the SQL upon measuring the conventional Y -quadrature(which is realized by a single photo diode in the interferometerrsquos output port)as first realized by [Unruh (1983) Yuen (1983) Jaekel and Reynaud (1990)]

In the example above the input squeeze angle is optimized for a sin-gle sideband frequency Injecting a broadband squeezed vacuum field withfrequency-independent squeeze angle of 45 would result in a rather bad inter-ferometer quantum-noise performance at frequencies far smaller or largerthan ΩSQL Fig 24 shows the quantum-noise performance if the input fieldhas squeeze angles that are optimized for every k(Ω) as given in Eq (58)Injected squeezing can thus lead to a broadband sub-SQL performance ifthe quantum measurement noise (shot noise) and the quantum back-actionnoise (radiation pressure noise) are correlated in an optimal way Due to the

70

10 50 100 500 1000 5000Frequency [Hz]

Dis

plac

emen

t noi

se sp

ectra

l den

sity

[mradic

Hz]

10-20

10-19

10-18

Total quantum noise 0dB

Total quantum noise -10dB

SQL

Y

X

Y

X

Y

X

Figure 24 Surpassing the SQL with squeezed-light injection ndash At shot-noise lim-ited sideband frequencies squeezing of the Y -quadrature amplitude improves the noisespectral density of the interferometer At radiation-pressure-noise limited sideband fre-quencies squeezing of the X-quadrature amplitude improves the noise spectral density ofthe interferometer If both kinds of quantum noise contribute equally (at the SQL markedwith a dot) a squeeze angle of 45 results in surpassing the SQL by the full squeeze factorsee Eq (60) In the graph here the squeeze angle is optimized for all frequencies result-ing in a broadband quantum noise reduction [Jaekel and Reynaud (1990)] Measurementsensitivities beyond the SQL (shaded area) are in the so-called quantum non-demolition(QND) regime [Kimble et al (2001)] Dashed horizontal lines represent the (squeezed)shot noise Dashed straight lines with negative slope represent the (squeezed) radiationpressure noise The calculations use 10 dB of squeezing a conventional Michelson inter-ferometer with neither arm resonators nor signal recycling a light power at the beamsplitter of 1 MW at a wavelength of λ = 1550 nm and mirror masses of 1 kg

correlation shot noise and radiation pressure noise can be squeezed simulta-neously

71

Light with a frequency-dependent squeeze angle

The discovery that shot noise and radiation pressure noise can be squeezedsimultaneously and thus a broadband reduction of quantum noise beyond theSQL be achieved required the insight that the spectral analysis of light definesa spectrum of many lsquosideband modulation modesrsquo that all can be in differ-ent quantum states An ordinary squeezing resonator which is on resonancefor light at twice the pump wavelength produces a spectrum of modulationmodes that all have the same squeeze angle A frequency-dependent squeezeangle can be introduced by reflecting such a field from a detuned single-ended filter cavity which was suggested by Kimble and coworkers [Kimbleet al (2001)] They showed that the optimal frequency dependence thatleads to the broadband improvement shown in Fig 24 can be realized byusing altogether two filter cavities as shown in Fig 25 Motivated by this re-sult research and development on filter cavities for optimizing the frequencydependence of broadband squeezed fields has been very active in recent years[Corbitt et al (2004) Chelkowski et al (2005) Dwyer et al (2013) Kweeet al (2014) Straniero et al (2015) Oelker et al (2016)]

Photo diode Squeezed vacuum

Faraday Rotator

Coherent light

YΩΔΩ

Figure 25 Frequency dependent squeezing injection ndash A broadband squeezed fieldwith a frequency-dependent squeeze angle that is optimal for gravitational-wave detectorsis produced by reflecting off an ordinary broadband squeezed field from two detuned opticalfilters [Kimble et al (2001)]

A light field with a frequency-dependent squeeze angle was first demon-strated in Ref [Chelkowski et al (2005)] see Figs 26 and 27 The experi-

72

Detuned filter cavityLockingphoto diode

Isolator

EOM R asymp 1

SHG

EOM

DBSOPA

LaserIsolator

Mode cleaner

LO

Homodyne detector

Figure 26 Generation of a frequency-dependent squeezing ndash A frequency-dependent orientation of the squeeze ellipse was first demonstrated in Ref [Chelkowskiet al (2005)] Initially a conventional spectrum of squeezed vacuum states of light was gen-erated in a squeezing resonator (lsquoOPArsquo) The squeezed vacuum was transmitted throughan optical isolator to a detuned filter cavity After reflection the squeezed vacuum stateswere absorbed in a balanced homodyne detector (BHD) The phase of the BHDrsquos localoscillator (LO) was changed for quantum state tomography of the squeezed states in diffe-rent regions of the spectrum The result showed a frequency-dependent orientation of thesqueeze ellipse see Fig 27 SHG second harmonic generation EOM electro-optical mod-ulator for applying phase modulation sidebands for cavity length control DBS dichroicbeam splitter R mirror reflectivity λ4 quarter wave plate for turning linear polarizedlight into circular polarised light and vice versa

ment consisted of a standing-wave squeezing resonator which produced ans-polarized broadband amplitude quadrature squeezed field accompanied bya dim continuous-wave DC control field with a wavelength of λ = 2πcω =1064 nm The squeeze bandwidth covered sideband frequencies up to aboutΩ(2π) = 30 MHz which corresponded to the linewidth of the squeezing res-onator The optical cavity for producing the frequency dependence of thesqueeze angle was a standing-wave cavity composed of a plane incouplingmirror of reflectivity r1 =

radic097 and a concave end mirror of reflectivity

r2 =radic

09995 The cavity length was L = 50 cm resulting in a linewidthof 147 MHz The squeezed field first passed a Faraday isolator to preventinterference effects between the filter cavity and the squeezing resonator A

73

Phase quadrature

min

max

-2 -1 0 1 2

141 MHz

-2

-1

0

1

2

Am

plit

ud

e q

uad

ratu

re

Figure 27 Frequency-dependent squeezing ndash Picture top right Reconstructed con-tour plot of the Wigner function of the sideband modulation at Ω(2π) = 141 MHz afterreflection from a 1515 MHz detuned filter cavity The state shows quantum correlationsbetween phase and amplitude quadratures ie squeezing at an angle of here about 40The white circle visualizes the standard deviation of the vacuum state uncertainty Thewhite ellipse represents the standard deviation of the squeezed uncertainty Small picturesMeasurement results on the same continuous-wave laser beam at various sideband frequen-cies around 15 MHz For each tomographic picture noise histograms of 100 equidistantquadrature angles were measured In each case the laser beam was phase locked to a ref-erence beam and the quadrature angle stably controlled and stepwise rotated The phasereference was given by a phase modulation at 198 MHz [Chelkowski et al (2005)] Thepicture was first published in Ref [Schnabel (2005)] (copyright ccopy2007 by Imperial CollegePress)

λ4-waveplate turned the s-polarized field into a circularly polarized beamwhich was then mode matched into the detuned cavity The retro-reflectedfield was analyzed by a balanced homodyne detector (BHD) for quantumstate tomography The filter cavity was electro-optically controlled to be de-tuned by 1515 MHz with respect to the DC control field The cavity length

74

control was achieved by the Pound-Drever-Hall (PDH) locking techniqueutilizing a circularly polarized laser beam that carried 15 MHz phase mod-ulation sidebands and was coupled into the filter cavity from the back Theoutput voltage of the BHD was characterized by a spectrum analyser as wellas used to perform quantum state tomography In the latter case the BHDoutput voltage was mixed down with an electronic local oscillator at differentradio-frequencies around 15 MHz and low-pass filtered to set the resolutionbandwidth to ∆Ω(2π) = 100 kHz The final electric signal corresponds to atime series of quadrature amplitude measurements XθΩi∆Ω Quantum statetomography is a method to reconstruct the phase space quasi-probability dis-tribution (Wigner function) of quadrature amplitudes from sets of measuredXθΩ∆Ω distributions when varying the angle θ [U Leonhardt (1997)] Forevery sideband frequency Ωi 100000 quadrature values were measured di-vided up on 100 equidistant quadrature angles Each quadrature angle wasstably controlled with a precision of plusmn1 Fig 27 shows the reconstructedWigner functions which were all measured on the same laser beam but atdifferent sideband frequencies For these measurements the detuned filtercavity was locked to the lower sideband at minus1515 MHz The result clearlyshows the frequency-dependent orientation of the squeeze ellipse In a morerecent experiment a frequency-dependent squeeze angle was also realized inthe kHz regime [Oelker et al (2016)]

56 Optomechanically induced (ponderomotive) squeezing

The radiation pressure of light when acting on a movable mirror re-sults in an intensity dependent phase shift [Pace et al (1993)] The cou-pling produces a so-called lsquoponderomotive effectrsquo [Braginsky and Manukin(1967)] which is of third order optical nonlinearity and which transforms abright coherent state inside an interferometer into a squeezed state of light[Vyatchanin and Matsko (1993)] This type of squeezed-light generationis usually called lsquoponderomotive squeezingrsquo or lsquooptomechanical squeezingrsquoConsequently even if no squeezed field is injected into the interferometercorrelations between the quadrature amplitudes are generated that allow forsurpassing the SQLPonderomotive squeezing as produced by the interferometer itself can onlybe exploited for evading back-action (radiation pressure noise) It can notbe used to squeeze the interferometer shot noise This is why ponderomotivesqueezing is fundamentally less extensive than injecting externally producedsqueezed states of light [Corbitt et al (2006)] suggested an external pon-

75

deromotive squeezing source for gravitational-wave detectors In this casedue to its external generation also the interferometerrsquos shot noise can besqueezed Recently ponderomotive squeezing was observed for the first time[Brooks et al (2012) Purdy et al (2013)] The achieved squeeze factors aremuch smaller than those produced by optical-parametric down-conversion[Vahlbruch et al (2016)]

Let us have a look again at Eq (59) Rotating the covariance matrix onthe right by arctan(minus

radic54minus 12) asymp minus58 indeed reveals squeezing

(cos 58 minussin 58

sin 58 cos 58

)(1 minus1minus1 2

)(cos 58 sin 58

minussin 58 cos 58

)asymp(

262 00 038

) (61)

The vacuum-noise normalized variance of 038 corresponds to about 42 dBof ponderomotive squeezing This is the general value that is produced atthe angular sideband frequency ΩSQL At higher frequencies the squeezefactor gets smaller at lower frequencies higher The squeezing strength of42 dB can be observed if the photo diode in the interferometer output portis replaced by a balanced homodyne detector using a local oscillator phase ofabout minus58 It can be shown however that the optimal signal-to-quantum-noise-ratio at the SQL is achieved for a local oscillator phase of exactly 45At this angle back-action is fully evaded

Full evasion of radiation pressure noise at all frequencies requires an op-timized frequency dependence of the relative local oscillator phase This canbe achieved by reflecting off the interferometer output field from two detunedfilter cavities [Kimble et al (2001)] The scheme was called lsquovariational out-putrsquo In the case of zero optical loss this scheme can fully evade radiationpressure noise just leaving the shot noise as the only quantum noise contri-bution

The variational-output scheme can be used to enhance the frequency-dependent squeezed input scheme The right site of Eq (60) shows thatthe output statersquos squeezing is not optimally detected in the Y -quadratureRather than with a single photo diode the detection should be done witha balanced homodyne detector with optimized phase of its local oscillatorIn this case the output lightrsquos quantum noise is solely given by squeezedshot noise The total quantum noise in Fig 24 would then be given by thelowest (dashed) horizontal line This combined scheme was called lsquosqueezed

76

variationalrsquo [Kimble et al (2001)] It can be realized by reflecting off theinterferometer output light from in total two optical filter cavities placed infront of the balanced homodyne detector

57 Conclusions

The highest quantum-noise-limited sensitivities of high-precision laserinterferometers are achieved by employing a large number of quanta tomaximize the signal strength in combination with strongly squeezed statesto minimize the quantum noise From this perspective it is clear that thequantum-noise-limited sensitivity of future gravitational-wave detectors willbe further improved ndash by increasing the light power and the squeeze factorTo be able to do so the optical loss in these devices needs to be reduced

In principle the optical loss in laser interferometers can be made smallbut never zero Recent theoretical research has shown that for any non-zero loss the sensitivity scales proportional to 1

radicn at best where n is the

average photon number per measurement This scaling is efficiently achievedby combining strongly displaced coherent states with squeezed vacuum statesof light

If a repeated measurement is not only limited by quantum measurementnoise but also by quantum back-action noise squeezed states of light can beused to simultaneously reduce both ie in the case of an interferometer shotnoise and radiation pressure noise

6 The first application of squeezed light in an operating gravita-tional-wave detector

Squeezed states of light have been successfully used to improve the sensi-tivity of the gravitational-wave detector GEO 600 from 2010 up to the pointwhen this Review was written [Abadie (2011) Grote et al (2013)] Afterdecades of proof-of-principle experiments [Xiao et al (1987) Grangier et al(1987) McKenzie et al (2002 2004) Vahlbruch et al (2005 2006 20072008) Goda et al (2008)] the implementation of a squeezed-light source inGEO 600 has resulted in the first sensitivity improvement beyond shot noiseof a measurement device that targets new observations in nature The im-plementation of squeezed states in GEO 600 was not done to provide anotherproof-of-principle demonstration but was realized because it offered a rela-tively cheap way of further improving the measurement sensitivity Of course

77

the sensitivity of GEO 600 can also be further increased by purely classicalmeans however the implementation of arm resonators to enable higher lightpowers without increasing the thermal load on the beam splitter or eventhe realization of longer interferometer arms are much more expensive Inthis respect the sensitivity improvement of GEO 600 with squeezed light canarguably be regarded as the first lsquotruersquo application that developed out of thefield of lsquononclassical (quantum) metrologyrsquo (Note that the term rsquoquantummetrologyrsquo is currently defined in different ways [Giovannetti et al (2006)Gobel and Siegner (2015)] and the term rsquononclassicalrsquo referring to a non-classical P-function gives a distinct description)

61 Gravitational waves

Einsteinrsquos General Theory of Relativity [Einstein (1916)] or simply lsquoGen-eral Relativityrsquo (GR) predicts that accelerating mass distributions producegravitational radiation analogous to electromagnetic radiation from accel-erating charges Experimental evidence of their existence is given by theobservation of the slow spiraling together of two neutron stars caused bythe loss of orbital energy to gravitational waves The inspiral rate exactlymatches the predictions of Einsteinrsquos theory [Weisberg and Taylor (2005)]Recently Advanced LIGO observed gravitational waves for the first time[Abbott (2016)] thereby giving the go-ahead for gravitational-wave astron-omy The gravitational-wave source was the final inspiraling and the mergerof two black holes 13 billion light years away from earth

Gravitational-waves evolve in the far field of the source propagate withthe speed of light and are measurable on earth with laser interferometersFig 28 displays a gravitational wave propagating along a certain directionGravitational waves are dynamical changes of space-time They are transver-sal and quadrupolar in nature and have two polarization states

A variety of known astrophysical and cosmological sources are predictedto emit gravitational radiation that should reach the Earth with a measur-able strength [Sathyaprakash and Schutz (2009)] The first gravitational waveevent detected was produced by two black holes of 36 and 29 solar massesDuring the final 02 seconds of their inspiraling they produced a peak gravi-tational strain in our solar system of 10minus21 covering frequencies up to 250 Hz[Abbott (2016)] Other predicted sources are mergers of neutron stars supernovae and background signals from the Big Bang According to GR GWsfrom complex astrophysical sources carry a plethora of information that willhave a major impact on gravitational physics astrophysics and cosmology

78

L +Δ L

Binary system

fBS

Figure 28 Space-time oscillation ndash Gravitational waves are dynamical deformations ofspace-time that form in the plane perpendicular to the direction of wave propagation Asa result distances between free-falling test masses in a transverse plane will change witha strain h = ∆LL For black hole or neutron star binary systems with orbital frequencyfBS distances will oscillate at frequency fGW = 2fBS The wavelength of this oscillation isgiven by λGW = cfGW where c is the speed of light The wave of orthogonal polarizationwith respect to the one shown is rotated by 45 around the propagation axis

62 Interferometric detection of gravitational waves

Current gravitational wave detectors are kilometre-scale laser interfero-meters [Dooley et al (2016) Aasi (2015) Acernese (2015) Aso et al (2013)]Continuous-wave laser light is split into two beams traveling in orthogonaldirections Both beams are reflected back towards the central beam splitterwhere they interfere Gravitational waves change the optical path lengthdifference and thus the light power directed towards the photo-diode thatis positioned in the signal output port of the beam splitter A gravitationalwave at frequency fGW = ΩGW(2π) reveals itself as a light-power modula-tion at the same frequency The spectral decomposition of the output signalis described by a spectrum of the quadrature amplitude YΩ∆Ω introduced inSec 3 It corresponds to the amplitude quadrature amplitude of the outputlight and relates to the differential phase quadrature of the interferometerarms

79

Photo diode

Squeezed vacuum

Faraday Rotator

Signal

Quantum noise

Coherent light XΩΔΩ

YΩΔΩ

(a) (b)

YΩΔΩFigure 29 Squeezed-light-enhanced interferometric measurement ndash (a) Michelsoninterferometer with arm cavities power recycling and signal recycling (see main text forexplanation) The interferometer is operated close to a dark fringe such that the quantumnoise entering from the dark port is back-reflected The squeezed field is mode-matched tothe signal output field (b) Phase space diagram of the gravitational-wave signal outputat sideband frequency Ω(2π) The quantum noise is squeezed below the ground stateuncertainty and thus the signal to quantum noise ratio improved

The first key ingredient of an interferometric gravitational-wave detectorare suspended heavy mirrors that can be regarded as quasi-free in the di-rection of laser light propagation thereby acting as test masses that probespacetime Being on ground current detectors are located in rather noisy en-vironments that allow the realization of undisturbed quasi-free mirrors onlyabove a sideband frequency of the order of 10 Hz Since sufficiently strongGW signals are expected up to a frequency of 10 kHz todayrsquos gravitationalwave detectors target at signals in the acoustic band from 10 Hz to 10 kHzThe quasi-free motion of the test mass mirrors in this frequency regime isachieved by suspending the mirrors as sophisticated multiple-stage pendulain vacuum chambers [Aasi (2015)] Far above the pendularsquos resonant frequen-cies which are typically around 1 Hz the centre of masses of the mirrors areisolated from vibrations of the ground and they react on frequency compo-nents of small external forces approximately as free masses The mirrors andtheir suspensions are built from materials having exquisitely high mechani-cal quality factors This helps to concentrate the thermal energy that causes

80

displacements of the mirror surface into well-defined vibrational frequencymodes At these particular very sharp frequencies no gravitational wavescan be detected

The second key ingredient of an interferometric gravitational-wave de-tector is laser light with a power of up to hundreds of kilowatts or evenmegawatts The light is quasi-monochromatic and needs to show very lowamplitude (quadrature) noise and phase (quadrature) noise at sideband fre-quencies within the detection band Low amplitude noise is necessary toprovide a shot noise limited output field It is also necessary to avoid classi-cal radiation pressure noise which becomes an issue if the light power or themirror masses in the two arms are not identical Low phase noise is requiredif the storage time of the light in the two arms is not identical This mightaccidentally occur due to different linewidths of the arm cavities or mightbe part of the interferometer design to allow for the length control schemeproposed by Schnupp [Heinzel et al (1998)] To maximize the light power in-side the interferometerrsquos cavities it should be produced in an almost perfecttransversal spatial distribution of a Gaussian TEM00 mode

Light sources of gravitational-wave detectors are ultra-stable NdYAGmaster-slave systems that provide up to 200 W of light at 1064 nm [Winkel-mann et al (2011) Kwee et al (2012)] The high power in the interferometerarms is achieved by cavity built-ups in the so-called power-recycling cavityand in the arm cavities Power recycling uses a partially reflective mirror thatis located between the light source and the interferometer beam splitter Itssurface is matched to the lightrsquos wave front and forms an optical cavity to-gether with the rest of the interferometer Since gravitational-wave detectorsare operated close to a dark fringe large power built-ups can be achievedThe highest power built-up is achieved for a mirror transmission equal to the(given) interferometer round trip loss In this case an impedance-matchedcavity is achieved The power-recycling cavity as well as the arm cavitiesare stabilized on resonance for the input light The difference between theirfunctionality is that the power-recycling cavity does not limit the detectionbandwidth of the interferometer GEO 600 as well as Advanced LIGO em-ploy a third type of cavity the so-called signal-recycling cavity Similarlyto power recycling a partially reflecting mirror that is placed between theoutput port of the beam splitter and the photodiode is used to resonantly en-hance the GW signal [Meers (1988)] The signal-recycling cavity resonantlyenhances the signal modulation fields within its linewidth without furtherenhancing the carrier light power In combination with low linewidth Fabry-

81

Perot arm resonators it can also be used to extract the signal by reducing theeffective finesse of the arm resonators for the signal sidebands This schemeis called resonant sideband extraction [Heinzel et al (1996)] The signal-recycling cavity has also been tested in a detuned setting in which just theupper or lower sideband is extracted or resonantly enhanced respectively[Heinzel et al (2002)] Current gravitational-wave detectors however usecarrier-tuned signal recycling

All these techniques are lsquoclassicalrsquo approaches for maximizing the signal-to-shot-noise ratio At frequencies above a few hundred Hertz howevershot-noise is still the limiting noise source in gravitational-wave detectorsFuture gravitational-wave detectors will therefore use even higher light pow-ers but further increasing the light power becomes more and more challeng-ing Optical materials with less light absorption need to be found to coun-teract an increasing thermal load inside the mirrors Mirror masses need tobe further increased to counteract the increasing radiation pressure noiseNonclassical approaches are superior and become more and more attractivethe farther classical approaches are pushed to the extremes Nonclassicalapproaches allow for simultaneously increasing the signal-to-shot-noise ratioand the signal-to-radiation-pressure-noise ratio without changing light poweror mirror masses see Fig 24 They also allow for a complete evasion of radia-tion pressure noise [Braginsky and Khalili (1995 1996) Kimble et al (2001)]see Subsec 56

63 Squeezed-light enhancement of the gravitational-wave detector GEO 600

In 2010 GEO 600 was equipped with the squeezed-light source shownin Fig 18 The location of the squeezed-light source close to the outputport is shown in Fig 30 It was known that GEO 600 was shot-noise limitedat sideband frequencies above about 700 Hz In this frequency regime thereplacement of the ordinary vacuum states that entered the interferometerfrom the output port by a spectrum of squeezed vacuum states was expectedto reduce the noise spectral density into the nonclassical regime It was notprecisely clear what squeezing factors could be expected since the opticalloss upon mode-matching an external field into the output port propagationalong the arms and the final photo-electric detection was not determined

Critical components were the quantum efficiency of the photo-diode aswell as the optical loss of Faraday rotator for overlapping the squeezed fieldwith the output mode Also the transversal output mode of GEO 600 was an

82

issue since it contained significant contributions from higher order modesto which a good mode matching of the squeezed mode was not possible Theimplementation of the squeezed-light source thus was accompanied with anew custom made InGaAs photo-diode with 3 mm diameter The goal was aquantum efficiency of greater 99 [Vahlbruch et al (2016)] Also the Fara-day rotator was custom-made and optimized for lowest optical loss which

Figure 30 GEO 600 ndash View into the central building of the British-German GW detectorlocated close to Hannover Germany The vacuum chambers contain the suspended beamsplitter power- and signal recycling mirrors additional input and output optics as well asmirrors to realize a double pass of the laser light through the 600 m long interferometerarms By courtesy of the AEI

83

involved a rather precise rotation of the polarisation of 45 plusmn 05 over anaperture of more than 15 mm Finally a ring cavity (output mode cleaner)was placed in front of the photo-diode which acted as a passive filter forhigher transversal modes Since GEO 600 was not limited by radiation pres-sure noise and since it used a carrier-tuned signal-recycling cavity a frequencyindependent orientation of the squeezing angle was optimum After several

100 200 300 500 700 1000 2000 3000 500010-22

10-21

10-20

Obs

erva

tory

noi

se c

alib

rate

d to

str

ain

[1radic

Hz]

10-19

10-18

10-17

Noise calibrated to test m

ass displacement [m

radicHz]

Sideband frequency f

Figure 31 Nonclassical reduction of the GEO 600 instrumental noise ndash Shownare the square roots of the noise spectral densities without squeezed vacuum states (top)and with squeezed vacuum states (bottom) [Abadie (2011)] Their calibrations [Affeldtet al (2014)] to gravitational-wave strain and differential arm length change are shownon the left and right y-axes respectively Note that both traces increase towards higherfrequencies due to the frequency-dependent signal enhancement of the signal-recyclingcavity The injection of squeezed vacuum states leads to a broadband noise reductionof up to 35 dB at shot-noise limited frequencies The spectral features are for instancecaused by excited violin modes of the mirror suspensions (600ndash700 Hz and harmonics)Data by courtesy of the AEI

months the combination of the squeezed-light source and the gravitational-wave detector succeeded The main laser of the squeezed-light source wasphase locked to the main laser of GEO 600 and a stable mode matching be-

84

tween the squeezed field and the interferometer output field was achieved[Abadie (2011)] The effect on GEO 600rsquos (strain normalized) noise spectraldensity is shown in Fig 31 At frequencies above about 700 Hz the squareroot of noise spectral density was reduced by up to one third This value cor-responds to a quantum noise variance of 045 (minus35 dB) with the shot noisevariance normalized to unity For isotropically distributed gravitational-wavesources this factor produces a detection rate increase by a factor of 153 asymp 34After its integration into GEO 600 the squeezed-light source was used in allscientific runs seeking for gravitational waves for instance in the observa-tional run S6eVSR4 that was undertaken from June 3rd to September 5th

in 2011 [Grote et al (2013)]Towards the end of 2011 right before the start of the detector upgrade

to Advanced LIGO a nonclassical sensitivity improvement was also demon-strated in one of the LIGO detectors [Aasi (2013)] In this experiment anonclassical sensitivity improvement corresponding to up to 215 dB abovefrequencies of about 150 Hz was achieved The successful test is a strongmotivation for a squeezed-light upgrade of Advanced LIGO Note that thedesign of Advanced LIGO was completed in 1999 and squeezed-light sourceswere not mature at those times

In the past years the squeezing enhanced GEO 600 detector was not onlyused for observations but also was the control of the injected squeezed modefurther improved Stabilizing the overlap between squeezed mode and brightmode of the interferometer to close to perfect is necessary to reduce theeffective optical loss and to maximize the measurable squeezing factor Re-cently superior methods for stabilizing the longitudinal phase of squeezedvacuum mode were found [Dooley et al (2015)] and the first automatic align-ment system for stabilizing and optimizing the transversal mode overlap wasdemonstrated [Schreiber et al (2016)]

64 Are squeezed states the optimal nonclassical resource ingravitational-wave detectors

For a given number of photons Eq (41) quotes the ultimately smallestphase change that can be measured with a signal-to-noise-ratio of one Thescaling with number of photons per measuring time of this Heisenberg limitseems appealing compared to the scaling achievable with coherent states orsqueezed states according to Eqs (38) and (39) respectively The Heisenberglimit however is only valid for precisely zero photon loss Since the non-classical states required to achieve Eqs (40) and (41) show an exponentially

85

increasing sensitiveness to loss when increasing the photon number the ac-tual scaling can not be deduced from Eq (40) Proposals to use Fock statesand the so-called N00N states for optimizing interferometer sensitivities [Hol-land and Burnett (1993) Dowling (1998) Mitchell et al (2004) Afek et al(2010)] are thus only applicable when the experiment is conditioned on zerophoton loss As discussed in recent publications the correct expression forthe fundamental sensitivity limit needs to consider not only the total photonnumber inside the interferometer but also the total photon loss [Dorner et al(2009) Ko lodynski and Demkowicz-Dobrzanski (2010) Knysh et al (2011)Escher et al (2011) Demkowicz-Dobrzanski et al (2012)]

Based on these earlier works Ref [Demkowicz-Dobrzanski et al (2013)]proved that the nonclassical sensitivity enhancement of GEO 600 reportedin Ref [Abadie (2011)] has been exceedingly close to fundamental quantuminterferometry bound under given energy constraints and photon loss lev-els More than that it was generally proven that the approach of com-bining displaced coherent states and squeezed vacuum states is optimal forgravitational-wave detectors

In Ref [Abadie (2011)] the gravitational-wave detector GEO 600 used aneffective number of photons per second of approximately n = 2 middot 1022 whichcorresponded to a total optical power inside the interferometer arms of P asymp37 kW at a wavelength of 1064 nm The total optical loss was 1minus η asymp 038The injected squeezing factor was eminus2r asymp 01 For these numbers the ratioof Eqs (43) and (44) is calculated to

∆φCSVmin

∆φgenmin

asymp

radicηeminus2r + 1minus η

1minus ηasymp 108 (62)

which is a good approximation within the limit of large coherent state dis-placements α sinh2r The quantum noise of GEO 600 including thesqueezed-light source was just 8 above the fundamental quantum inter-ferometry bound An increased squeezing strength of 16 dB (eminus2r asymp 0025)which is in reach would bring the approach based on coherent states andsqueezed vacuum states to within just 2 above the fundamental bound

Future GW detectors will have significantly reduced optical loss values(1 minus η) lsquoLossrsquo includes scattering and absorption at mirrors non-perfectfringe contrasts and the non-perfect quantum efficiency of the photo detec-tor Optical loss reduction is important for at least four reasons First itleads to an increased signal second it leads to a reduced quantum noise when

86

employing squeezed states third less absorption reduces the thermal load onthe test mass mirrors and fourth less scattering reduces the probability ofback-scattered light which produces disturbance signals [Billing et al (1979)Vahlbruch et al (2007) Punturo et al (2014)] The higher the finesse valuesof the arm and signal-recycling cavities are the more significant is opticalloss at mirror test masses the beam splitter and the signal-recycling mirrorThe finesse value of the power-recycling cavity and the loss of mirrors andlenses that guide the output field to the photo-diode are less critical Suitablephoto detectors of 995 quantum efficiency are available today [Vahlbruchet al (2016)] but achieving a total optical loss of 10 is still challengingThe reason for that is that first of all a measurement device aiming for bestabsolute sensitivity should use as much quanta (photons) as possible Highfinesse values for the enhancement cavities are thus essential but results inan unavoidable scaling-up of the effect of mirror losses A realistic exampleof future gravitational wave detectors thus considers η = 09 with a squeezingfactor of 20 dB (eminus2r = 001) In this case the quantum noise will be about4 above the ultimate fundamental bound for a given photon number

From Eq (62) it can be concluded that there is no need for any moresophisticated nonclassical states than squeezed states In particular non-classical states with a defined photon number such as N00N states are notrequired Within the approximation quoted this result is independent ofthe photon number This result is also independent of the physical systemused for interferometric phase estimation and can also be made for quantum-enhanced atomic clock calibration in the presence of dephasing Here the-oretical results indicate that the precision of Ramsey interferometry withspin-squeezed states is close to the optimal one in the asymptotic regime ofa large number of atoms [Huelga et al (1997) Ulam-Orgikh and Kitagawa(2001) Escher et al (2011)] as already stated in Ref [Demkowicz-Dobrzanskiet al (2013)] More sophisticated nonclassical states with fixed number ofn quanta might still be useful for the exceptional case when the absorptionof one quantum already results in zero measurement sensitivity anyway Anexample is an ensemble measurement where the absorption of a single photondemolishes the source of the phase change to be characterized A typicallyused approach of conditioning the measurement result on n clicks of n singlephoton counters conditions on precisely zero loss and is thus able to use theadvantage of Eq (40) over Eq (39)

87

65 Conclusions

Squeezed states of light will contribute to realizing gravitational-wave ob-servatories with much higher sensitivities than existing or planned ones Tobenefit from squeezed states in a most efficient way optical loss in terms ofabsorption and scattering must be minimized In particular the optical lossof mirror coatings and mirror substrates need to be minimized The rele-vant mirrors include the test masses the balanced beam splitter the signalrecyclingextraction mirror and all optical components between the latterand the photo diode Excellent spatial mode matching between the brightinterferometer field and the squeezed vacuum field is also of great impor-tance Achieving this requires further improvement of the surface figures ofall reflective optical components of the interferometer as well as improvedhomogeneity of all optical components that the light passes through

The quantum noise reduction achieved in a gravitational-wave detectoris of course always smaller than the highest squeeze factor provided by thesqueezed-light source As an example let us consider the observation of 15 dBof nonclassical noise suppression directly at the source If the squeezed fieldsenses an additional loss of 5 when propagating through the interferometerwhich is a very challenging number from todayrsquos point of view the remainingsqueezing level is about 11 dB see Eq (16)

7 The application of 2-mode-squeezed light in laser interferome-ters

71 Quantum Dense Metrology

At first glance the application of bi-partite (two-mode) squeezed statesto a device whose goal is measuring a single observable seems meaninglessSqueezing the uncertainty of that observable should be the optimum one cando This is indeed true when concerning just quantum noise but recentlyit was discovered that in the presence of classical disturbances bi-partitesqueezing can improve such measuring devices [Steinlechner et al (2013)]The concept was named quantum dense metrology (QDM) The potentialimprovement of a gravitational-wave detector with bi-partite squeezed statesis shown in Fig 32 (a) A description is given in the caption The pre-condition for a potential improvement can be best understood within a phasespace diagram Fig 32 (b) contains two different kinds of lsquosignalsrsquo The firstis the actual signal which always shows up as a phase space displacement

88

along the Y axis The second is a disturbance signal that can produce adisplacement in arbitrary direction in phase space A prominent example

Faraday Rotator

Coherent light

Photo detectors Balanced homodyne detectors

Squeezed vacuum

Squeezed vacuum

Quantum noise in

Y (A) and X (B)

XΩΔΩ(B)

YΩΔΩ(A)

(a) (b)

Disturbance projected

onto Y

and X

XΩΔΩ(B)

YΩΔΩ(A)

Example of (unknown) disturbance

Signal

Figure 32 Bi-partite-squeezed-light-enhanced measurement ndash (a) Setup for theapplication of bi-partite (two-mode) squeezed light in a laser interferometer on the basisof QDM Two squeezed vacuum fields are overlapped on a balanced beam splitter withprogrammable squeeze angles for instance with a relative angle of 90 which produces abi-partite state as shown in Fig 12 The beam splitter outputs are entangled for any rela-tive angle greater than zero One part is matched to the interferometer mode The secondpart is kept outside as a reference beam The interference of the interferometer outputand the reference beam is arranged with such a phase difference that it reproduces the twosqueezed inputs on the photo detectors The two squeezed beams are photo-electricallydetected measuring the respective squeezed quadrature (using balanced homodyne detec-tors) Both beams carry half of all interferometer induced modulations which includesignals as well as disturbances A single readout as shown in Fig 21 cannot distinguishbetween the two kinds The double readout shown here provides additional informationand allows for recognition of the disturbance [Steinlechner et al (2013)] as well as in prin-ciple a modeling of the disturbance and with a correct model an improvement of thenoise spectral density of the interferometer [Ast et al (2016)] (b) Phase space diagramdescribing phase quadrature readout A as well as amplitude quadrature readout B Bothshow squeezed quantum noise The amplitude quadrature readout does not contain anygravitational-wave signal ie any feature in this channel must be due to disturbancesThis information can be used to improve the interferometer

89

for such a disturbance is parasitic interference due to back-scattered laserlight [Vahlbruch et al (2007)] Back-scattering is a limiting noise at lowsignal frequencies of gravitational-wave detectors [Billing et al (1979) Vinetet al (1997) Hild (2007) Ottaway et al (2012) Punturo et al (2014)] Notethat all noise that couples in via unwanted motions of the test mass mirrorsso-called lsquodisplacement noisersquo always produces a phase space displacementalong the Y axis and cannot be tackled with QDM

Fig 33 shows measurement results obtained in Ref [Steinlechner et al(2013)] In a table-top experiment one part of a bi-partite squeezed state ofa continuous-wave mode at 1064 nm was mode-matched into the output portof a Michelson laser interferometer operated at its dark fringe in full analogyto Fig 32(a) A lsquosignalrsquo was produced by driving the piezo behind one of theend mirrors at a frequency of 555 MHz The lsquodisturbancersquo was introducedby re-injecting a small amount of light that leaked through the second endmirror with an additional piezo-mounted mirror The piezo was driven ata frequency of 517 MHz to produce a phase modulation An additionalDC voltage defined an arbitrary and unknown optical path length of thelight before being re-injected and as such the phase space orientation of thedisturbance signal This mechanism of a parasitic interference is realizednaturally in any interferometric device due to back-scattering of quanta frommoving surfaces in the environment

The interferometer output consisted of the signal as well as the distur-bance with a quantum uncertainty given by one subsystem of the bi-partiteentanglement It was overlapped with the second subsystem of the entan-gled state on a balanced beam splitter and the two outputs were analysedwith balanced homodyne detectors The phases of the bi-partite entangle-ment and the BHD local oscillators were controlled to resemble Fig 32(b)ie both BHDs measured a squeezed uncertainty regardless of the phase ofthe (generally unknown) disturbance

The beam splitter that combines interferometer output and the entangledreference beam unavoidably splits the signal as well as the disturbance intotwo paths For a balanced beam splitter this generally reduces the signaland disturbance power by 3 dB for both quadrature measurements Fig 33shows however that both BHDs performed about 6 dB below shot noisewhich demonstrates the usefulness of the scheme The squeeze factor can inprinciple be infinite which thus qualifies the lsquo3 dB penaltyrsquo In the abovefigure the additional information from the second BHD output was used torecognize the parasitic interference in the first BHD output providing a lsquovetorsquo

90

50 51 52 53 54 55 56 57

50 51 52 53 54 55 56 57-70

-75

-80

-85

-90

-70

-75

-80

-85

-90

Y (A) (Ω2π)Ω∆Ω

Frequency ( f )

Noi

se p

ower

(dB

m)

Noi

se p

ower

(dB

m)

X (B) (Ω2π)Ω∆Ω

~ ndash6 dB

~ ndash6 dB

Shot noise reference

Shot noise reference

Signal Projected disturbance

Projected disturbance

Vet

o

No signal

Figure 33 Bi-partite-squeezed-light-enhanced measurement ndash The result wasachieved in a table-top setup [Steinlechner et al (2013)] In the two panels the lower

(blue) traces show the squeezed quadrature noise-power spectra ∆2Y(A)Ω∆Ω(Ω2π) (top) and

∆2X(B)Ω∆Ω(Ω2π) (bottom) as simultaneously measured with balanced homodyne detectors

lsquoArsquo and lsquoBrsquo respectively The conventional Y -measurement (top) cannot distinguish be-tween signal and disturbances The additional X-measurement (bottom) does not detectany phase quadrature signal thus any feature in this measurement is a parasitic signal dueto a disturbance The respective projection onto the Y -measurement can thus be lsquovetoedrsquoIn a more sophisticated approach the X-data might be used to model and then to elim-inate the disturbance as well as its projections on both quadrature measurements Theresult is a reduced spectral density of the actual phase quadrature measurement [Ast et al(2016)] Traces shown here are slightly sloped due to the decreasing transfer functions ofthe balanced homodyne detectors The resolution bandwidth was ∆Ω(2π) = 10 kHz thevideo bandwidth was 100 Hz All traces were averaged three times

signal to trigger its removal from the data streamThe question arose whether the additional information can be used to

reduce the actual noise spectral density of the first measurement ie to re-cover signals that were buried by parasitic interferences Very recently it wasshown that this is indeed possible Ref [Ast et al (2016)] reports a table-top proof-of-principle experiment in which the additional information in the

91

QDM approach could be used for improving the sensitivity of an interfero-meter The measurement sensitivity was improved from above-shot-noise tosub-shot-noise (sub-Poissonian) performance This result was possible notbecause the way the parasitic interference arose was known but because theadditional information provided by QDM allowed for fitting a model of theexcess noise to the readout data

Quantum dense metrology (QDM) as shown in Fig 32 improves a mea-surement by simultaneously reading out two conjugate observables Bothreadout observables show a squeezed quantum noise and act as estimatorsof independent physical quantities This situation was recently described aslsquoquantum-mechanics freersquo [Tsang and Caves (2012)] QDM is based on anEinstein-Podolsky-Rosen (EPR) entangled [Einstein et al (1935)] bi-partitesystem as described in Subsec 34 EPR entanglement was previously con-sidered for the quantum-informational task of dense coding which doublesthe capacity of quantum communication channels [Bennett et al (1992)Braunstein and Kimble (2000)] The application of EPR entanglement inmetrology was first proposed by DrsquoAriano et al [DrsquoAriano et al (2001)]

72 Conclusions

A single beam that carries an optimized spectrum of squeezed vacuumstates and that is injected into the interferometerrsquos dark port provides themost efficient and practically optimal approach to reduce the quantum noisein laser interferometers by means of nonclassical states (see Section 5) Theconclusion of the section here is that two entangled beams provide a superiorapproach if the interferometerrsquos sensitivity is limited by classical noise thatis not exclusively restricted to the actual observable which is the phasequadrature amplitude Y Parasitic interferences due to laser light that isbackscattered from vibrating surfaces are an example Current gravitational-wave detectors use light fluxes of about 1024 photons per second [Abbott(2016)] Just a single photon per second and hertz which leaves the mainlight beam and is backscattered from a vibrating surface and in this way getsfrequency shifted into the detection band produces a significant disturbancesignal The lsquoquantum-dense metrologyrsquo approach might provide a powerfultechnique to tackle this problem

Very recently it turned out that QDM is not the only technique that mayexploit EPR entanglement to improve phase measurements Ref [Ma et al(2017)] proposes to use EPR entanglement to simultaneously suppress shotnoise and radiation pressure noise in a gravitational-wave detector without

92

the need for an additional filter cavity (confer subsection 55) In this caseEPR entanglement is exploited that is carried by one broadband squeezedbeam and that is present between quadrature amplitudes defined with re-spect to different optical frequencies ω and ωprime as investigated in Ref [Hageet al (2010)] Such lsquofrequency multiplexedrsquo EPR entanglement might resultin considerably lower costs of building a gravitational-wave detector with abroadband simultaneous squeezing of shot noise and radiation pressure noiseAlso this proposal does not lead to a fundamentally lower quantum noise butrather improves on classical aspects of an interferometer

8 Summary and Outlook

In many cases experiments that involve interference of quantum statescan be described in a semi-classical way This description uses the classicalwave picture for the interference part of the experiment and subsequently theclassical particle picture when the states transfer their energy to a detectoror more generally to a thermal bath This semi-classical description is notpossible when using the specific class of lsquononclassicalrsquo states Squeezed statesof light are a prominent example of these Squeezed states and other nonclas-sical states allow for observations that made Einstein Podolsky and Rosenformulate their critical and seminal paper on quantum theory [Einstein et al(1935)]

In the review here it is argued that after many successful proof-of-principle experiments with nonclassical states in the past decades the routineuse of squeezed-light in observational runs of the gravitational-wave detectorGEO 600 goes beyond proof-of-principle and is a true application of nonclassi-cal light Since 2010 the squeezed-light source has improved the measurementsensitivity of GEO 600 in basically every observational run [Abadie (2011)Grote et al (2013) Dooley et al (2016)] At quantum noise limited frequen-cies ie above a few hundreds of hertz the sensitivity has been improvedcorresponding to a squeezing strength in the noise spectral density of up to37 dB which corresponds to an increase of the average gravitational-wavedetection rate by a factor of 043minus32 = 36 This success is a strong motiva-tion to also equip the Advanced LIGO Virgo and Kagra gravitational-wavedetectors with squeezed light Similar improvement factors even down tolower signal frequencies are expected [LSC (2013)] The achievable improve-ment factors are mainly limited by the optical loss on the squeezed states

93

and much higher factors are achievable in principleUp to now squeezed states have not been used to reduce the radiation-

pressure noise in gravitational-wave detectors The reason is that so far othernoise sources are larger than radiation pressure noise and such an effect can-not be observed It is expected however that future gravitational-wavedetectors will eventually be partly limited by radiation pressure noise Fromthis point on squeezed light will be used to simultaneously reduce shot noiseand radiation pressure noiseSqueezed states are the optimum nonclassical states for gravitational-wavedetectors or more generally for all laser interferometers operating with largeaverage photon numbers per measuring interval [Demkowicz-Dobrzanski et al(2013)] In addition to using higher light powers and heavier test mass mir-rors higher squeeze factors will thus contribute to mitigate the lightrsquos quan-tum noise in laser interferometers

Two-mode (bi-partite) squeezed light has not been used in gravitational-wave detectors so far They are not capable of further reducing the quantumnoise in laser interferometers but they can be used to mitigate classicalnoise that originates from fluctuating phase space displacements A well-known such noise source is back-scattered light Proof-of-principle experi-ments were performed recently [Steinlechner et al (2013) Ast et al (2016)]This new technique could turn out to be valuable in next generations ofgravitational-wave detectors in particular in those targeting high sensitivi-ties at low sub-audio signal frequencies and using high light powers Suchan implementation in gravitational-wave detectors does not require any newtechnology Compared to a squeezed-light enhanced interferometer just asecond squeezed-light source is required

It is certainly remarkable that those quantum states that made EinsteinPodolsky and Rosen falsely think quantum theory incomplete are now ex-ploited as new technologies in measurement devices targeting new observa-tions in nature

Acknowledgements

RS thanks M Ast J Bauchrowitz C Baune S Chelkowski J DiGugliel-mo A Franzen B Hage J Harms A Khalaidovski L Kleybolte NLastzka M Mehmet S Steinlechner and H Vahlbruch for their contri-butions many fruitful discussions and their support with the figures and JFiurasek for many valuable comments on the manuscript Thanks are also

94

due to Y Chen F Khalili and H Miao for fruitful discussions within thequantum noise working group of the LIGO Scientific Collaboration (LSC)Special thanks are due to H Vahlbruch and H Grote together with theGEO 600 team for their pioneering work on the squeezed-light implementa-tion in GEO 600 RS is supported by the Deutsche Forschungsgemeinschaft(Grant No SCHN 757-6) and by the European Research Council (ERC)project lsquoMassQrsquo (Grant No 339897)

References

References

Aasi J et al Jul 2013 Enhanced sensitivity of the LIGO gravitationalwave detector by using squeezed states of light Nature Photonics 7 (8)613ndash619URL httpwwwnaturecomdoifinder101038nphoton2013177

Aasi J et al Apr 2015 Advanced LIGO Classical and Quantum Gravity32 (7) 074001URL httparxivorgabs14114547httpstacks

ioporg0264-938132i=7a=074001key=crossref

20895763c84bce3f8929251031b2475c

Abadie J et al Sep 2011 A gravitational wave observatory operatingbeyond the quantum shot-noise limit Nature Physics 7 (12) 962ndash965URL httparxivorgabs11092295httpwwwnaturecom

doifinder101038nphys2083

Abbott B P et al Feb 2016 Observation of gravitational waves from abinary black hole merger Phys Rev Lett 116 061102URL httplinkapsorgdoi101103PhysRevLett116061102

Acernese F et al 2015 Advanced virgo a second-generation interferome-tric gravitational wave detector Classical and Quantum Gravity 32 (2)024001URL httpstacksioporg0264-938132i=2a=024001

95

(2015)] and KAGRA [Aso et al (2013)] conventional laser technology hasbeen pushed to extremes over the past decades Noise spectral densities nor-malized to space-time strain of less than 10minus23 Hzminus12 have been measured[Abbott (2016)] Progress will continue and based on the successful appli-cation in GEO 600 squeezed light is now widely accepted to provide a newadditional technology to contribute to the new field of gravitational-waveastronomy It was also successfully tested in one of the LIGO detectors in2013 [LSC (2013)] and is an integral part of the European design study forthe 10 km Einstein-Telescope [Punturo et al (2010)]

GEO 600 has already taken several years of lsquosqueezedrsquo observational datawhich has increased its sensitivity at signal frequencies above 500 Hz Withthe implementation of a squeezed light source in GEO 600 the applicationof nonclassical states in metrology has been pushed beyond merely proof-of-principle

lsquoTwo-mode squeezed statesrsquo show a squeezed uncertainty in at least onejoint continuous variable of two subsystems lsquoArsquo and lsquoBrsquo Examples of jointvariables are differences and sums of phase-space observables of A and BTwo-mode squeezed states not only belong to the class of nonclassical statesbut due to their bi-partite character also to the class of lsquoinseparablersquo orlsquoentangledrsquo states They are the ideal states to demonstrate the Einstein-Podolsky-Rosen paradox [Einstein et al (1935)] as first achieved in [Ouet al (1992)] Apart from fundamental research on quantum mechanicsrecent proof-of-principle experiments demonstrated their usefulness in inter-ferometric measurements that go beyond the application of simple squeezedstates [Steinlechner et al (2013) Ast et al (2016)] This experiment is thefinal topic of this review

2 Observations on light fields in squeezed states

Generally there are two different kinds of observables that can be subjectof a measurement performed on a quantum system The first kind is associ-ated with the systemrsquos wave property In optics it corresponds to the electricfield strength at a given phase angle ϑ The according (dimensionless) opera-tors are called the quadrature amplitudes Xϑ and have a continuous spectrumof eigenvalues Quadrature amplitudes are measured in very good approx-imation with a balanced homodyne detector using the interference with a

9

bright local oscillator beam see Fig 3 (a) In practice any measurement ofXϑ integrates over some sideband (Fourier) spectrum within the angular fre-quencies Ω plusmn ∆Ω2 The sideband information always needs to be quotedA straight forward but rather untypical way is by adding subscripts whichleads to Xϑ

Ω∆Ω The classical analogue of the quadrature amplitude operator

XϑΩ∆Ω is the modulation depth of the optical field at modulation phase angle

ϑ and at angular modulation frequency Ω measured over the band ∆Ω lt 2ΩThe uncertainties of the statersquos quadrature amplitudes at different phases ϑare limited by a Heisenberg uncertainty relation see section 3 The secondkind of measurement is associated with the systemrsquos particle property andis given by the photon number operator n associated with a measuring timeinterval ∆t Its precise measurement requires a photon counter ideally withsingle photon resolution The measurement result obviously has a discretespectrum Continuous as well as discrete observables are usually subject toquantum uncertainties and thus quantum noiseUsually the measurementrsquos integration time and frequency band actuallydefine the physical system that is characterized In quantum optics experi-ments the interrogated physical system is called a lsquomodersquo

21 Definition of a lsquosingle modersquo

Let us define a light field or generally any quantum system to be asingle mode if it corresponds to the lsquosmallest entity of a waversquo In this caseits spectral and temporal distributions as well as waist size and divergenceare at their Fourier limits and all other properties such as optical axis waistposition and polarization are well defined For instance a linearly polarizedlongitudinal resonance of an optical standing-wave cavity defines such a singlemode if the cavity finesse is high and transversal modes are non-degenerateThe complete photo-electrical detection of a cavity mode however is notstraight forward Most quantum optical experiments are instead performedon propagating light In this case single modes are defined by spatial filtersand by temporal-spectral measurement windows both being at the Fourierlimit Examples for single modes are a laser pulse and a spectraltemporalcutout from a continuous observation of a quasi-monochromatic continuous-wave light beam in the spatial TEM00 mode both at the Fourier limits

In classical physics the only remaining free parameter of a given singlemode is its excitation energy In quantum physics the situation is differentFor a given energy a single mode can be in many different quantum states

10

which differ in their quantum statistics Examples are coherent states num-ber (Fock) states and squeezed states

22 Observations on squeezed states using a single PIN photo-diode

An ideal PIN photo-diode absorbs the full energy of a light mode andproduces one photo electron for every absorbed photon energy It uses theinternal photo-electric effect inside a semiconductor such as silicon or In-GaAs In contrast to avalanche photo-diodes PIN photo-diodes operatewith unity gain lsquoPINrsquo stands for lsquopositiversquo lsquointrinsicrsquo and lsquonegativersquo and isdescribing the doping of the semiconductor layers A PIN photo-diode is op-timally suited for the continuous monitoring of a rather bright light field ofup to several tens of milliwatts An example is the photo-diode in the outputport of a gravitational-wave detector as shown in Fig 2 (a) The prominentwavelength of 1064 nm which is emitted by NdYAG lasers has an opticalfrequency of ν = ω(2π) = 282 middot 1014 Hz The period of the field oscillationis a few femtoseconds and cannot be directly resolved with photo-electric de-tectors However variations of the electric field around the averaged opticalfield oscillation on longer time-scales can be resolved Applying an electronicbandpass filter at the sideband angular frequency Ω plusmn ∆Ω2 to the photovoltage provides information about the lsquodepth of the lightrsquos amplitude mod-ulationrsquo which is also called the lsquoamplitude of the amplitude quadraturersquo Itcan also slowly vary in time and reads

Xϑ=0

Ω∆Ω(t) equiv XΩ∆Ω(t) equiv X (1)

The subscript is usually skipped as it is done with the time dependenceas indicated on the right Applying the electronic bandpass filter in factdefines the mode of the light being detected The structure of the defini-tion in Eq (1) forms the basis of interferometric signals and quantum noisealso in the semi-classical case of coherent states Lets take an example Inthe recent observation of gravitational waves [Fig 1 bottom row in Abbott(2016)] the time-frequency representation of the gravitational-wave signalcorresponded to the amplitude quadrature amplitude XΩ∆Ω(t) of the inter-ferometer output light Note that a larger value of ∆Ω allows for changes ofthe quadrature amplitude on shorter time scales

If the light fieldrsquos lsquomodulation modersquo does not contain any quanta simplybecause there are no photons that have a frequency difference of plusmnΩ with

11

respect to the carrier it is in its ground state In this case lsquovacuum noisersquois observed which originates from the ground state uncertainty Since thevacuum noise only becomes measurable as a beat with a bright light field itcan also be seen as the carrierrsquos band-path filtered shot noise A modulationmode in a displaced vacuum state (a coherent state) corresponds to nonzerocoherent modulationThe measured level of the vacuum noise generally depends on the power ofthe bright carrier light and on the electronic amplification In any case itprovides the reference for certifying lsquosqueezingrsquo Observations using a singlePIN photo-diode require an independent measurement to quantify vacuumnoise A necessary condition is that attenuating the total fieldrsquos light powerresults in the same attenuation of the measured XΩ∆Ω values If they showa stronger attenuation a coherent modulation or thermal noise might bepresent If they show a weaker attenuation the photo-diode and its electron-ics might be saturated

Fig 2 (b) illustrates how a broadband squeezed field improves the mea-surement of an amplitude modulation in time domain based on a PIN photo-diode Shown is a simulated time sequence of XΩ∆Ω-data sampled from thephotoelectric voltage In this simulation all sideband frequencies from zero(DC) to the cutoff frequency of the detector electronics (Ωcut) are included(Ω = ∆Ω2 = Ωcut2) No additional band pass filter is applied making it amaximally broadband detection Although the data in Fig 2 (bi) contains aclassical amplitude modulation of the detected light this signal is not visibledue to random noise here representing shot noise Fig 2 (bii) shows thesame situation but with shot noise that is squeezed over the full detectionband The quantum uncertainty of the modulation depth is squeezed andthe classical signal becomes visible

It needs to be noted that a single PIN photo-diode can only measure theamplitude of the amplitude quadrature XΩ∆Ω(t) but not the non-commutingobservable the lsquoamplitude of the phase quadraturersquo

Xϑ=90

Ω∆Ω (t) equiv YΩ∆Ω(t) equiv Y (2)

For values that are small compared to the field strength of the bright fieldthe quantity Y approximately describes the bright fieldrsquos lsquophase modulationdepthrsquo

12

23 Observations on squeezed states using a balanced homodyne detector

-72

-69

-66

-63(i)

(ii)

(iii)

(a)

LO

(Squeezed)signal input

Phaseshifter

PD1

PD2

5050

01 0200 03

-75

-60

-57Balanced homodyne detector

(b)

Time [s]

Noi

se p

ower

[dB

m]

Figure 3 Balanced homodyne detection (BHD) ndash (a) Setup The quadrature atchoosable angle ϑ of the signal field is measured by overlapping the latter with a localoscillator (LO) field of the same mode parameters on a balanced beam splitter and record-ing the difference voltage from two PIN photo-diodes as shown In order to meet theBHD approximation the LO needs to be much more intense than the signal field A closeto perfect mode overlap between LO and signal input field is crucial For a non-perfectoverlap the detector measures the input state with unwanted contributions of the vacuumstate (b) Noise power measurements (i) on an electronically amplified and band-passfiltered quadrature amplitude of the vacuum field (Xvac

Ω∆Ω) (signal input blocked) (ii) on

a squeezed quadrature (XsqzΩ∆Ω) of a squeezed vacuum state (ϑ = 0) and (iii) on re-

spective quadratures of the same squeezed state where the phase angle ϑ was continuouslyshifted by changing the optical path length of the LO The measurement data shows about5 dB of squeezing and was first published in [Chelkowski et al (2007)] Ω2π = 5 MHz∆Ω2π = 100 kHz

In contrast to a single PIN photo diode a balanced homodyne detector(BHD) is suitable to measure the quantum statistic of all types of modu-lations ie for all angles ϑ Such a detector consists of two identical PINphoto-diodes a balanced beam splitter and an external homodyne local os-cillator field that is much brighter than the signal beam and that has anadjustable phase The signal beam corresponds to the squeezed field whichin many experiments is in a squeezed vacuum field having an optical powerthat usually corresponds to just a few photons per mode The two beams areoverlapped on the balanced beam splitter with close to perfect mode match-ing and the two interference outputs are focussed onto the photo diodes seeFig 3 (left) The electric output signal of the BHD is the difference of the

13

photo diode voltages The LO takes over the role of the carrier light fieldbut with the possibility to choose the phase shift ϑ This way eigenvaluesof X Y or Xϑ can be measured where the latter is given by the followinglinear combination of the first two

Xϑ(t) = cos(ϑ) X(t) + sin(ϑ) Y(t) (3)

If the modulation depths of signal and local oscillator beams are weak com-pared to their coherent amplitudes |α| and |αLO| the output voltage of aBHD corresponds to eigenvalues of the following operator

V (t) prop 2cos(ϑ) |αLO||α|+ |αLO| Xϑ(t) + |α| XϑLO(t) (4)

The lsquohomodyne approximationrsquo further involves |αLO| |α| such that theterm on the right can be neglected even if the local oscillator shows someclassical quadrature excitation The output voltage of a BHD is usually spec-trally analysed or at least spectrally filtered which removes the DC part infull analogy to a single photo diode (see previous subsection) Sampling thefiltered voltage provides eigenvalues proportional to the generalized quadra-ture amplitude in Eq (3)

V BHDΩ∆Ω(t) prop |αLO| Xϑ

Ω∆Ω(t) (5)

Fig 3 (a) shows the setup of a balanced homodyne detector for the char-acterization of squeezed states Setting ϑ = 0 eigenvalues of the ampli-tude modulation depths can be sampled from the photo voltage accordingto Eq (5) Setting ϑ = 90 eigenvalues of the phase modulation depths aremeasured The datarsquos expectation values 〈Xϑ〉 provide the coherent displace-ment of the squeezed state The datarsquos variances

∆2Xϑ equiv 〈(Xϑ)2〉 minus 〈Xϑ〉2 (6)

provide the statersquos (quantum) noise A pure squeezed state as well as asqueezed state that experienced photon loss have Gaussian quantum statisticsand are thus fully described by the expectation values and variances (first andsecond moments) of two orthogonal quadratures but only if one quadraturereflects the lowest quadrature variance

14

In most experiments with squeezed light the photo electric voltage ac-cording to Eq (5) is not sampled with a data aquisition system but the signalis directly fed into a spectrum analyser measuring the noise power of the volt-age If the expectation value 〈Xϑ〉 is zero the noise power is proportionalto the variance ∆2Xϑ in Eq (6) The reference for quantifying the squeezefactor is measured by blocking the (squeezed) signal field in Fig 3 (a) Themeasured vacuum noise level corresponds to the LOrsquos (electronically ampli-fied) shot noise level

Traces (ii) and (iii) in Fig 3 (b) show measured noise powers of the mod-ulation mode (Ω2π = 5 MHz ∆Ω2π = 100 kHz) being in a squeezed vac-uum state (i) is proportional to the variance of the ground state uncertainty∆2Xvac

Ω∆Ω (ii) is proportional to the quantum noise variance of the squeezed

quadrature amplitude ∆2XsqzΩ∆Ω (iii) is proportional to the quantum noise

variance of the quadrature amplitude with scanned phase ∆2XΩ∆Ω(ϑ(t))

To fully characterize a quantum state ie to do quantum state tomogra-phy [Vogel and Risken (1989)] a BHD is a prerequisite But also interfero-metric measurements with balanced homodyne detectors instead of singlePIN photo-diodes have several advantages A correctly implemented BHDreadily provides the vacuum noise level when the signal beam is blockedWith a BHD the optimum operating point of the interferometer is preciselyat a dark fringe If a perfect dark fringe can practically be achieved ampli-tude noise of the laser does not couple into the signal port If the interfero-meter has balanced arm length also frequency noise of the laser then does notcouple into the signal port Some quantum non-demolition schemes with theprospect of evading quantum radiation pressure noise require the detection ofa non-canonical quadrature angle [Jaekel and Reynaud (1990) Kimble et al(2001)] Here the adjustable phase of a BHD provides a straight forwardapproach The experimental exploration of BHDs for gravitational-wave de-tectors only has started recently [Steinlechner et al (2015)]

A light field can be analysed with respect to many different modulationfrequencies Ω The result constitutes a spectrum [Breitenbach et al (1998)]where in principle every modulation mode can be in a different quantumstate Fig 4 shows spectra of squeezed states from 5 MHz to 100 MHz with∆Ω2π = 1 MHz The lower curve shows the spectrum of the most strongly

15

-14-12-10-8-6- 4-20 2 4 6 8

10 12 14 16 18

6 8 10 20 40 60 80 1005

Squeezed noise

Anti-squeezed noise

Vacuum noise

Frequency [MHz]

Nor

mal

ized

noi

se v

aria

nce

[dB

]

Figure 4 Spectrum of quadrature amplitude variances ndash Shown are the quantumnoise properties of a large number of modulation modes having a resolution bandwidth of∆Ω(2π) = 1 MHz For all traces the balanced homodyne detector output was analysedwith a spectrum analyser Squeezing of XΩ∆Ω (bottom trace) and anti-squeezing of

YΩ∆Ω (top trace) versus f = Ω(2π) are shown relative to the vacuum noise variance Thespectrum below 5 MHz is not shown since it contained less squeezing due to laser relaxationoscillation in the carrier field as well as disturbances from back-scattered light [Vahlbruchet al (2007)] Disturbances at frequencies above 70 MHz originated from relatively largedetector dark noise which was subtracted from all traces shown here The thin linerepresents a theoretical model that takes into account for the linewidth of the squeezingcavity The data was first presented in in Ref [Mehmet et al (2010)]

squeezed variances in this case the variances of XΩ∆Ω The upper spectrum

shows the variance in the orthogonal quadrature amplitude (YΩ∆Ω) Allvariances are normalized to those of the corresponding vacuum state Thesqueeze factor reduces towards higher frequencies due to the linewidth ofthe squeezing cavity The anti-squeezing is always higher than the absolutevalue of the squeezing due to Heisenbergrsquos uncertainty relation and due to thepresence of optical loss The curves do not represent pure squeezed states butmixed squeezed states with a significant contribution from vacuum statesdue to optical loss Pure squeezed states can only be produced by making

16

the influence of all decoherence processes negligibleThe choice of the resolution bandwidth (RBW ∆Ω) during data taking

and processing defines the spectral-temporal modulation modes includingtheir number within the detected spectrum For any setting of the RBWthe quantum mechanical properties of the quadrature amplitudes XΩ∆Ω and

YΩ∆Ω [Caves (1985)] fully correspond to those introduced for quadratures instandard text books and which are reviewed in Sec 3

24 Observations on two-mode squeezed states using balanced homodyne de-tectors

Two-mode squeezed states are composed of two subsystems lsquoArsquo and lsquoBrsquoand are bi-partite entangled states with a Gaussian quantum statistic Toavoid conflicts with different usage of the term lsquomodersquo they can synony-mously be named lsquobipartite Gaussian entangled statesrsquo or lsquobipartite squeezedstatesrsquo which will be mainly used in this Review In the same way multi-partite Gaussian entangled states correspond to multi-partite squeezed states

The measurement observables that prove or disprove the bi-partite squeez-ing property are XA

Ω∆Ω minus XBΩ∆Ω and Y A

Ω∆Ω + Y BΩ∆Ω where the minus and

plus signs may be swapped Bi-partite squeezed states are precisely thosestates that were discussed by Einstein Podolsky and Rosen (EPR) in theirseminal paper [Einstein et al (1935)] Fig 5 shows a measurement resulton bi-partite squeezed light [Eberle et al (2013)] The variances of bothjoined observables are squeezed as shown in the two lower traces They wererecorded consecutively by adding or subtracting the outputs of two balancedhomodyne detectors But by interfering the subsystems on a beam splitterone could even measure both joined observables simultaneously This pos-sibility is correctly described in quantum theory since their commutator iszero

The so-called EPR paradox arises as follows If we either measure XAΩ∆Ω

and XBΩ∆Ω or Y A

Ω∆Ω and Y BΩ∆Ω it is obvious from the data in Fig 5 that

we can always predict the measurement result at subsystem lsquoBrsquo when know-ing the result at subsystem lsquoArsquo This seems to suggest that both quantitiesat lsquoBrsquo are precisely defined simultaneously before the measurement on lsquoArsquowhich contradicts the rigorous (and correct) interpretation of their non-zerocommutator that they are not precisely defined simultaneously

To solve this paradox EPR conjectured that the wavefunction as definedby quantum theory does not provide the full information This led to a

17

discussion of whether hidden variables existed that needed to be included ina complete theory of quantum mechanics (see also Bell [Bell (1966)]) Theexperimentally observed violation of Bellrsquos inequality [Bell (1964) Aspectet al (1981) Giustina et al (2013) Hensen et al (2015)] however ruled outthe existence of (local) hidden variables

Based on that the EPR paradox needs to be solved in a different wayContrary to what EPR assumed it is in fact possible to predict the valueof an arbitrary observable of a physical system A with certainty via a mea-surement on system B although this observable was not defined before themeasurement Without any interaction a measurement on subsystem lsquoArsquo notonly creates lsquorealityrsquo of eg XA

Ω∆Ω simultaneously lsquorealityrsquo is also created

regarding the observable XBΩ∆Ω describing subsystem lsquoBrsquo Here the term

lsquorealityrsquo has the meaning as defined by EPR [Einstein et al (1935)] Simi-larly the detection of one photon of a two photon entangled number statenot only produces the reality of this photon but also that of a second oneA discussion of Einstein-Podolsky-Rosen entanglement can also be found in[Schnabel (2015)] Note that the EPR paradox can also be described aslsquoquantum steeringrsquo [Schrodinger (1935) Cavalcanti et al (2009) Handchenet al (2012)] It should also be mentioned that two-mode squeezing beingdetected with BHDs and not with photon counters cannot be used to violatea Bell inequality The latter topic is outside the scope of this Review

Bi-partite squeezed states were first characterized with balanced homo-dyne detectors by the group of J Kimble in 1992 [Ou et al (1992)] Gener-ally the EPR paradox becomes more pronounced the stronger the bi-partitesqueezing is A measure of the strength of EPR entanglement was introducedby M Reid [Reid and Walls (1985)] According to this measure the resultin Fig 5 can be quantified to ε2 = 00309 where the critical value is one Itcorresponds to the strongest Gaussian EPR entangled state generated so far

For a long time it looked like that two-mode squeezed states are notuseful for laser interferometers The reason for that belief was that a laserinterferometer as any other measurement device too is built to measureone observable It seems to be ideal already if the quantum noise in thissingle observable is squeezed The increased quantum noise in the orthog-onal observable is not harmful in this case and squeezing in two differentobservables useless Only recently realistic scenarios were discussed in whichtwo-mode squeezing in fact does improve the performance of a laser inter-

18

-12

-10

-8

-6

-4

-2

0

0 1 2 3 4 5 6 7 8 9 10

100

dB

109

dB

Δ (XAvac + X B

vac ) = Δ (Y

Δ (XA + X B)

Avacminus Y B

vac )2 2

2

Δ (YAminus Y B)2

Nor

mal

ized

noi

se v

aria

nce

[dB

]

Times [s]

Figure 5 Two-mode squeezing measurement ndash For this measurement the outputs oftwo balanced homodyne detectors are added or subtracted and the variances (noise pow-ers) of the results recorded The upper trace was measured with modes lsquoArsquo and lsquoBrsquo beingin their ground states This measurement served as a reference level Strong two-modesqueezing was observed as shown by the lower two traces The sideband frequency wasΩ(2π) = 8 MHz and the resolution bandwidth was ∆Ω(2π) = 200 kHz The measure-ment results were first published in Ref [Eberle et al (2013)]

ferometer [Steinlechner et al (2013)] The proof-of-principle experiment isreviewed in Sec 7

25 Observations using photon counters

Alternatively to field quadratures an optical mode in a squeezed statecan also be characterized at least partly by detecting its photon numberdistribution For a pure squeezed vacuum state such a measurement wouldreveal the existence of solely even photon numbers including a large prob-ability for zero photons The average photon numbers of squeezed vacuumstates with feasible squeeze factors are very small of the order of one persecond and bandwidth in hertz see Fig 13 (a) ndash (c) A distribution with closeto zero probability of odd photon numbers however has not been measuredso far The reason is the lack of ideal photon counters First of all theefficiency of these detectors ie their probability of converting one photoninto one click and no photon into no click must be almost perfect lsquoLostrsquophotons as well as dark counts wash out the oddeven oscillations Further-more most detectors available can only distinguish between zero and one

19

photon This problem can be solved by distributing the squeezed mode ontoa large number of single photon detectors using an array of beam splitterssuch that all paths have a low probability of carrying more than one pho-ton Photon number measurements on squeezed vacuum states neverthelessplay an extremely important role in quantum optics When the squeezingstrength is very low the probability of detecting more than 2 photons can beneglected and the detection of a photon heralds the existence of a second one

0

300

600

900

1200

-40 -20 0 20 400

5

10

15

20

25

Two-

fold

coi

ncid

ence

s [1

(4s

)](a) (b)

Non-degeneratetwo-mode squeezing

5050

APDA1

APDA2

APDHerald

Photon coincidencedetection

Delay [ns]

Thr

ee-f

old

coin

cide

nces

[1

(4s)

]1500

Figure 6 Coincidence clicks from non-degenerate photon pairs ndash The first suchexperiment was reported in Ref [Hong and Mandel (1986)] (a) shows a setup with threeavalanche photo-diodes (APDs) for proving the successful heralding of a single photonnumber state (b) Histograms of the two-fold coincidence detections at APDHerald andAPDA1 (red) and at APDHerald and APDA2 (yellow) with theoretical models (solid lines)If the two-mode squeezing just carried one photon in each spatial subsystem the three-foldcoincidence detection should be zero Indeed the according histogram (grey points righty axis) shows only a few events These are produced by false (dark) counts of the APDsThe delay for the three-fold coincidences is defined as the time between counts at lsquoA1rsquo andlsquoA2rsquo given that the trigger APDHerald detected a photon (within a 100 ns time window)The data was taken on photons that were up-converted from 1550 nm to 532 nm and itwas first published in Ref [Baune et al (2014)]

If a mode of light is always excited by either zero or two photons lsquocondi-tionalrsquo or lsquoheraldedrsquo one-photon Fock states can be realized (Measurementson an ensemble of the n-photon Fock state would always produce the mea-surement result n ie Fock states have a zero photon number uncertaintyThey are also called lsquonumber statesrsquo) The above concept of producing aone-photon Fock state obviously requires the deterministic and balanced dis-tribution of the down-converted signal and idler fields into two different paths

20

In order to achieve this the signal and idler fields need to be non-degenerateUsually a mode in a squeezed state is composed of degenerate signal andidler fields and this degeneracy thus needs to be removed Possible waysare producing the down-converted fields at well separated wavelengths [Vil-lar et al (2005) Su et al (2006) Li et al (2010) Samblowski et al (2011)]separating the upper and lower sidebands belonging to an ordinary squeezedmode by frequency filters [Schori et al (2002) Hage et al (2010)] and us-ing spatial filters [Hong et al (1987)] A frequently used approach is usingtype II parametric down-conversion where the photons within a pair are al-ways orthogonally polarized [Ou et al (1992) Kiess et al (1993) Kwiat et al(1995)]The list of experiments with conditional or heralded photon number states islong They showed for instance nonclassical g(2)-functions [Hong et al (1987)]and violations of Bell inequalities [Weihs et al (1998)] Fig (6) shows a re-sult from a more recent experiment in which a bipartite-squeezed state withsubsystems at 1550 nm and 810 nm was produced the subsystem at 1550 nmsubsequently up-converted to 532 nm and the lsquoquantum non-Gaussianityrsquo ofheralded up-converted single photons demonstrated [Baune et al (2014)]Squeezed states are also the resource for the conditional generation of super-positions of coherent states [Ourjoumtsev et al (2006) Neergaard-Nielsenet al (2006)] and so-called N00N-states [Afek et al (2010)]

The generation of nonclassical states mentioned in the paragraph aboveis not stationary but relies on a probabilistic trigger event The produc-tion of squeezed states themselves usually happens in a stationary fashionThis distinction has an important consequence for applications of nonclassi-cal states in measurement devices Only (stationary) squeezed states allowfor a continuous improvement of a measurement Avoiding any loss of mea-suring time is generally of high relevance for the detection of short-livedsignals with unknown arrival time as well as for the detection of long-livedquasi-monochromatic signals since the signal-to-noise-ratio (SN) improveswith measuring time

26 Conclusions

The detection of squeezed light produces measurement results that canbe considered as remarkable Let us focus on experiments where a mode in abright coherent state is overlapped with a mode in a squeezed vacuum state

21

as shown in Figs (1) and (3) In both setups the squeezed vacuum field caneasily be blocked which allows us to compare the measurement results ona bright coherent state with and without the interference with the squeezedvacuum state Without squeezing the photo-electric detectors measure alarge number of photon events with a large quantization noise (shot noise)The large noise reflects the fact that all photon events were independent fromeach other as shown in Fig 2 (bi) With squeezing the photo-electric de-tectors again measure a large number of photon events with an expectationvalue that is even slightly higher but nevertheless the quantization noise ofall detected photons is significantly reduced Fig 2 (bii)Based on the discussion of EPR entanglement in Subsec 24 the photo-electric detection of the output light of a squeezing-enhanced laser inter-ferometer (with αlowastα1) produces the reality of photons This way we cankeep the lsquowave picturersquo in which no photons exist when light travels alongthe interferometer arms and when it interferes at the beam splitter Whenthe energy of the beam is elevating electrons to the conductance band ofthe photo-diodersquos semi-conductor n photon events simultaneously appearwithin the measuring interval with probability P (n) What conclusion hasto be drawn if the probabilities resemble a sub-poissonian statistic ndash Theoccurrence of photon events is still truly random but in this case not forindividual photons The occurrence of photons is correlated in such a waythat the probability of detecting an additional photon in the same time in-terval reduces the larger the number of already detected photons is Whatfollows from the discussion of EPR entanglement for a photon counting ex-periment with pure squeezed vacuum and ideal photon counters Here theprobabilistic detection of one photon entails the detection of a second onewith certainty With some smaller probability a third photon is detectedwhich entails the detection of a fourth photon with certainty and so on

If a photon of a mode that was not interrogated by the environment beforeis absorbed its reality is created in this very moment If the photon belongsto a squeezed state this process instantaneously influences the probability ofother photons becoming reality

Of course a more general statement can be made based on the insightthat interaction with the environment creates the reality of any kind ofquanta including electrons atoms and molecules

22

3 Theoretical description of squeezed states

31 The quadrature amplitude operators

Consider a single mode of light at optical frequency ω Its Hamiltonoperator reads

Hω = ~ω(n+

1

2

)= ~ω

(adaggerωaω +

1

2

)= ~ω

(X2ω + Y 2

ω

) (7)

where n is the photon number operator and aω and adaggerω are the annihilationand creation operators which obey the commutation rule

[aω a

daggerω

]= 1 The

operator aω has a complex-valued dimensionless eigenvalue spectrum andcorresponds to the complex amplitude αω in classical optics Xω and Yω arethe hermitian amplitude and phase quadrature operators The eigenvaluesof the quadrature operators are also dimensionless and proportional to theelectric fields at the oscillationrsquos antinode and at the oscillationrsquos node Inthe above equation they are defined such that their variances are ∆2Xω =∆2Yω = 14 if the oscillator is in its ground state ie if 〈n〉 = 0

Although Eq (7) simply describes the energy of an harmonic oscillator itis the essence of quantum theory since it mathematically describes the wave-particle dualism Whereas the eigenvalues of n have a discrete spectrum theeigenvalues of Xω and Yω have a continuous spectrum In classical opticsthe phase quadrature is zero In quantum optics its expectation value is alsozero but its uncertainty contributes to the overall energy

Eq (7) describes a cavity mode as well as a section that is cut from apropagating quasi-monochromatic light beam The latter example is of highrelevance in actual experiments By setting the sectionrsquos time window ie themeasuring time interval the time-frequency (lsquomodulationrsquo) mode is defined

The quadrature operators introduced in Eq (7) and displayed in Fig 7 donot correspond to lsquoXrsquo and lsquoY rsquo that are of relevance in laser interferometry andin optical communication and which were already discussed in Subsec 22and 23 The optical frequency of visible and near-infrared light is far too highto be transferred to an oscillation of photoelectric voltage Quite general alaser interferometer targets signals at audio or radio band frequencies Ωi ωSuch a measurement is achieved as stated before by decomposing the photo-electric voltage from the photo diode at the interferometer output into asingle-sided spectrum (positive frequencies only) of intervals of Ωplusmn∆Ω2

23

(a)

(b)

(c)

(d)

2π 4π

2π 4π

2π 4π

2π 4π

Figure 7 Phase spaces and electric field oscillations of monochromatic lightndash Top Left Monochromatic light in a coherent state is represented by a phasor (whitearrow) including its quantum uncertainty (white dashed circle and fuzzy area) located inthe phase-space spanned by the quadratures Xω and Yω When the phase space rotateswith optical frequency ω2π the projection of the quantum phasor onto a fixed (vertical)axis corresponds to the electric field E(t) as shown on the right side (a) Weakly displacedcoherent state (b) Corresponding amplitude squeezed state The electric field uncertaintyaround the zero average field region is anti-squeezed (c) Vacuum state at the same opticalfrequency (d) Corresponding squeezed vacuum state The meaning of the uncertaintycould be carved out by supplementing them with monochromatic waves all having theoptical frequency ω2π Changing amplitudes then display amplitude quadrature noiseChanging shifts along the time axis model the electric field uncertainty at the expectedzero crossing They are not implemented in the graphics here however since any of thosewaves does not exist due to Heisenbergrsquos uncertainty relation

24

The signals as well as the quantum uncertainties carried by a beam oflight are thus described by a spectrum of pairs of non-commuting quadratureoperators Mathematically every such operator is defined by an integral overthe Fourier components within the bandwidth The spectral weighting of theFourier components is called the lsquowindow functionrsquo By going to sidebandintervals a spectrum of a new type of optical mode is defined which describesthe modulation of the electric field in the respective frequency interval Ω plusmn∆Ω2 In this Review we call it a lsquomodulation modersquo

The quadrature operators that are defined around a modulation frequencyΩ with a bandwidth of ∆Ω are the quadrature amplitude operators thatare relevant in laser interferometry Whenever they are not related to aspecific band we use the short form XΩ∆Ω(t) equiv X and YΩ∆Ω(t) equiv Y cfEqs (1) and (2) These operators can slowly vary with time where thetime dependence is limited by ∆Ω (The time dependence is not due toquantum uncertainty which usually is time independent but for instancedue to the time dependence of the signal eg a passing gravitational wave)Let us consider now a pair of quadrature operators for a particular sidebandΩ plusmn ∆Ω2 The Hamilton operator of the corresponding modulation modeis found by switching to the frame rotating at optical frequency ω Thetransition is done by applying the unitary transformation U = exp(iωadaggerat)generating a new Hamiltonian H = U daggerHωU minus i~UpartU daggerpartt The Hamiltonianof the modulation mode reads

H = ~Ω

(nΩ +

1

2

)= ~Ω

(adaggera+

1

2

)= ~Ω

(X2 + Y 2

) (8)

where nΩ is the (occupation) number operator for the modulation modeand a and adagger its annihilation and creation operators The commutation rule[a adagger

]= 1 is unchanged X and Y are the amplitude and phase quadra-

ture amplitude operators respectively They correspond to the depth of theamplitude modulation and for weak excitations to the depth of the phasemodulation respectively They are the conventional hermitian field operatorsin experimental quantum optics Note that modulation modes at angularfrequency Ω can be described by a superposition of three optical frequenciesa carrier at ω an upper sideband at ω + Ω and a lower sideband at ω minus ΩThe quantum mechanical description of modulation states in connection tooptical carrier and upper and lower sidebands is known as the lsquoTwo-PhotonFormalismrsquo [Caves and Schumaker (1985) Schumaker and Caves (1985)]

The quadrature amplitude operators in Eq (8) are again defined such

25

that the variances of the uncertainty of a modulation field in its ground stateor in a coherent state are

∆2Xvac = ∆2Yvac = 14 (9)

Generally quadrature operators X and Y as defined in Eqs (7) and (8) arethe real and imaginary parts of the annihilation operator

a = X + iY hArr adagger = X minus iY (10)

hArr X =1

2

(a+ adagger

) Y =

1

2i

(aminus adagger

) (11)

They satisfy the commutation relation[X Y

]=i

2 (12)

and their variances are limited by a Heisenberg uncertainty relation of thefollowing form

∆2X∆2Y ge 1

16 (13)

A quantum state is called a ldquosqueezed staterdquo [Bachor and Ralph (2004)]if ∆2Xϑ lt 14 for an arbitrary field quadrature Xϑ = X cosϑ + Y sinϑ seeEq (3) The angle of the lowest variance below 14 is called the squeeze angleθ The largest factor by which the variance is below 14 is called the squeezefactor often given on a decibel (dB) scale using the following transformation

minus 10 middot log10

(∆2Xθ

∆2Xvac

) (14)

The squeeze factor can also be described by the squeeze parameter r

eminus2r =∆2Xθ

∆2Xvac

(15)

When a squeezed state experiences optical loss it remains squeezed butthe squeeze factor is reduced Also the statersquos purity is reduced ie theproduct of the quadrature uncertainties increases above the minimum valueOptical loss corresponds to mixing the state with the vacuum state Let∆2Xϑ be the variance of a quadrature amplitude ∆2Xvac the variance of the

26

(quadrature angle independent) ground state uncertainty and (1 minus η2) therelative energy loss Then the resulting quadrature variance reads

∆2Xprime

ϑ = η2∆2Xϑ + (1minus η2)∆2Xvac (16)

To maximize the benefit from squeezed states in applications stronglysqueezed states need to be generated and optical loss minimized Opticalloss occurs due to absorption and scattering in the optical components inthe path of the squeezed beam including the squeezing resonator itself anddue to non-perfect matching to the interferometer mode non-perfect inter-ference contrast of the interferometer and non-perfect quantum efficiency ofthe photo diodes The sum of all losses including those outside the inter-ferometer need to be less then 10 to allow a nonclassical quantum noisesuppression of a factor of 10 in power ie 10 dB

32 Phase space representations of squeezed states

The Wigner function ndash The properties of squeezed states are nicely dis-played by the Wigner function W (X Y ) [Wigner (1932)] An example interms of a squeezed vacuum state is shown in Fig 8 It is a quasi-probabilitydistribution which contains the statersquos full information including its quan-tum statistic There are two ways how a Wigner function provides a sufficientcriterion for nonclassicality First by containing negative values second byfeatures that have a smaller (squeezed) width compared with the Wignerfunction of the ground state Integrating the Wigner function over Y pro-vides the probability density of measurement results ie of the eigenvaluesof the observable X and vice versa

infinintminusinfin

W (X Y )dY = p(X)

infinintminusinfin

W (X Y )dX = p(Y ) (17)

where p(X) and p(Y ) are the observed probability distributions also exem-plarily shown in Fig 8

The ground state coherent states as well as (quadrature) squeezed stateshave quadrature eigenvalue probability densities that are Gaussian TheirWigner functions are also Gaussian and thus entirely positive Wigner func-tions of other nonclassical states for instance Fock states exhibit negativevalues For this reason the Wigner function is called a quasi -probabilityfunction

27

X Y XY

Figure 8 Wigner function and its projections ndash Displayed is the full information ofa squeezed vacuum state The quasi probability density of the Wigner function (centre)is plotted along a third dimension and is color-coded Also shown are the actual mea-surement results from which the Wigner function is reconstructed They are representedby the squeezed and anti-squeezed Gaussian projections onto the X and Y axes Theirprobability densities p(X) and p(Y ) are given on the vertical axes The Gaussian measure-ment statistic of the first clearly shows squeezing compared to the ground state statistic(dashed) The squeeze factor is 116 dB and the squeeze parameter r = 1335 [Mehmetet al (2010)]

Fig 9 shows the Wigner functions for (a) a coherent state (b) the ground(vacuum) state (c) a displaced squeezed state and (d) a squeezed vacuumstate All Wigner functions describe a modulation of the carrier light atsideband frequency Ω integrated over the frequency interval ∆Ω The car-rier light is not part of these Wigner functions The displacement in (a)represents a classical amplitude modulation (b) corresponds to the absenceof any photons with a frequency offset of plusmnΩ from the local oscillator field(c) and (d) represent states whose amplitude modulation depth is more pre-cisely defined than that of the ground state Fig 10 shows Wigner functionspectrum for a broadband squeezed vacuum field Every Wigner functiondescribes the modulation field at some modulation frequency Ωi integrated

28

X

Y

X

(a) (b)

X X

(c) (d)

Y

Y Y

Figure 9 Simplified representation of Wigner functions ndash The darker the shadedareas the larger is the phase-space quasi-probability Shown are four different (time-independent) states of a modulation mode at frequency Ω for a specific resolution bandwidth ∆Ω Panel (a) represents a coherent state the displacement (α) corresponds to aclassical amplitude modulation Panel (b) represents the ground (vacuum) state (c) adisplaced squeezed state and (d) a squeezed vacuum state both with squeeze angle θ = 0The latter is in analogy to Fig 8 Again the light field that carries the modulation is notpart of the pictures

over the resolution bandwidth (RBW) of ∆Ω

The Glauber-Sudarshan P -function ndash The P -function [Glauber (1963)Sudarshan (1963)] is calculated by de-convoluting the Wigner function fromthe ground state uncertainty [Gerry and Knight (2005)] For displaced vac-uum states (coherent states) the P -function corresponds to a displaced δ-function The mathematical expression of the P -function of a squeezed statecontains infinitely high orders of derivatives of the δ-function [Vogel andWelsch (2006)] Such a function contains negativities but cannot be dis-played It is possible however to define a phase-space quasi probabilityfunction for squeezed states that can be displayed and that does show neg-

29

0

Ω1

Ω2

Ω

Single-sided spectrum

Y

X

Figure 10 Hint of a Wigner function spectrum ndash A single-sided spectrum (positivefrequencies only) with respect to the carrier field can be used to visualize a broadbandsqueezed field Shown are two examples displaying a squeezed vacuum state at Ω1 anda displaced amplitude squeezed state at Ω2 The individual Wigner functions cover theresolution bandwidth ∆Ω gt 0 (not shown) In general the squeezing strength as well asthe squeeze angle and the displacement are a function of sideband frequency

ativities as a sufficient and necessary condition for certifying the squeezingeffect This lsquononclassicality functionrsquo is calculated by de-convoluting theWigner function from an uncertainty distribution that is steeper than theGaussian distribution A pronounced negativity of a squeezed vacuum stateof up to 69 standard deviations was found [Kiesel et al (2011)]

The double-sided phasor picture ndash This phasor picture links quantumstates of modulations with the quantum states of the contributing opticalfields [Bachor and Ralph (2004)] and is mathematically described by thetwo-photon-formalism [Caves and Schumaker (1985) Schumaker and Caves(1985)] Generally a weak amplitude or phase modulation at frequency Ω ofa carrier field at optical frequency ω can be understood as the carrierrsquos beatwith two optical frequencies at ωplusmnΩ The double-sided phasor picture is ableto display a spectrum of different and independent modulation frequenciesin the rotating frame of the carrier field The carrier light field is time-independent but the upper and lower sidebands are not They rotate withplusmnΩi(2π) respectively around the frequency axis

30

ω0

Yω

Xω

Upper sideband

Lower sideband

ω0+Ω1

ω0+Ω2

ω0-Ω2

ω0-Ω1

ω

Upper sideband

Lower sideband

Yω

Xω

ω0+Ω1

ω0+Ω2

ω0-Ω2

ω0-Ω1

ω

ω0

Figure 11 Double-sided phasor pictures ndash Phase spaces at optical frequency ω0 plusmnΩirotate around the frequency axis with sideband frequency plusmnΩi Its sign determines thedirection of rotation with respect to the local oscillator in the rotating frame at ω0 Apair of phase spaces need to be superposed to provide a description of a modulation fieldat |Ωi| Top Amplitude quadrature squeezed field with a coherent displacement at |Ω2|The displacement corresponds to a classical amplitude modulation The uncertaintiesof all optical frequencies are circular and larger than that of the ground state (dashed)Quantum correlations are indicated by additional symbols More details are given in themain text Bottom Corresponding spectrum of (displaced) vacuum states which do nothave any quantum correlations

31

Fig 11 shows such a double-sided phase space picture where the carrierrsquosmodulation at Ω1 is in a squeezed vacuum state and where the modulationat Ω2 is in a displaced squeezed state The picture shows how a classicalamplitude modulation as well as the quantum statistic of a modulation fieldis decomposed into contributions from upper and lower sidebands For asqueezed modulation field the upper and lower sidebands show no squeezedbut circular thermally excited quantum uncertainties The uncertainties of apair of sidebands however show correlations as well as anti-correlations InFig 11 these (anti-) correlations are marked with times and + for the modulationfrequency Ω1 and with N and bull for the modulation frequency Ω2

33 Covariance matrix representation of (single-party) squeezed states

Since squeezed states have a Gaussian quantum statistic four numbersare sufficient for their full description These numbers are the second mo-ment of the quadrature amplitude showing the strongest squeezing and thesecond moment of its orthogonal quadrature amplitude as well as their firstmoments describing the displacement These four numbers are sufficientto calculate the Wigner function shown in Fig 8 In general the quadra-ture of strongest squeezing is not perfectly aligned with one of the axesof the measurementrsquos coordinate system The so-called covariance matrix(VXXVXY VY XVY Y ) [Simon et al (1994)] accounts for phase space rotationsand enables the calculation of how these states evolve within an interfero-metric arrangement Their components are normalized to the vacuum noisevariance ∆2Xvac = ∆2Yvac and read

VXY =

langXY + Y X

rangminus 2

langXranglang

Yrang

2∆2Xvac

(18)

The following examples represent the ground state a pure 10 dB am-plitude quadrature squeezed state and a pure 10 dB squeezed state with asqueeze angle of 45

Vvac =

(1 00 1

) V0

01 =

(01 00 10

) V45

01 =

(505 495495 505

) (19)

with V4501 =RT

45V001R45 where Rα = (cosα minussinα sinα cosα) is the rota-

tion matrix

32

34 Phase space representation of two-mode (bi-partite) squeezed states

XA

YA

XB

YB

Figure 12 Bi-partite squeezed vacuum state ndash Shown is a Wigner-function-basedphase space representation in close analogy to the single party version in Fig 9 (d) Thepicture describes a single modulation at frequency Ω with bandwidth ∆Ω Measurementsat party A and B reveal local Wigner functions that correspond to thermal states sincethe uncertainties (indicated by the color and by the large circles) are symmetric and largerthan that of the ground state (indicated by the small dashed circles) The uncertaintieshowever show correlations and anti-correlations here indicated by + and times respectivelyThe strength of these (anti-)correlations are indicated by ellipses Bi-partite squeezingie entanglement is present if the short axes of the ellipses are shorter than the diameterof the ground state uncertainty The picture in fact represents Einstein-Podolsky-Rosenentanglement [Einstein et al (1935)] From a measurement of XA or YA the correspondingmeasurement result at party B can always be inferred with an uncertainty that is smallerthan the ground state uncertainty

A bi-partite state enables a measurement on subsystem A and simulta-neous a measurement on subsystem B For a large number of simultaneousensemble measurements of the same quadrature amplitude Xϑ the followingtwo joint quadrature variance can be calculated

∆2(XAϑ plusmn XB

ϑ ) (20)

A state that is symmetrically shared between two parties (A and B) is calleda two-mode squeezed state if the variances of joint quadrature measurementsfulfill the following inequality [Duan et al (2000)] ie

∆2(XA minus XB)

∆2(XAvac minus XB

vac)+

∆2(Y A + Y B)

∆2(Y Avac + Y B

vac)lt 2 (21)

33

with ∆2(XAvac plusmn XB

vac) = ∆2(Y Avac plusmn Y B

vac) = 2∆2Xvac A lsquotwo-mode squeezedstatersquo reveals entanglement in the second moments of the measurement statis-tics It is thus a lsquobi-partite Gaussian entangled statersquo

Fig 12 displays a (pure) bi-partite squeezed vacuum state distributedbetween A and B The state shows full symmetry regarding its subsystemsat the two sites The large circles and the colored area represent Wignerfunctions of the subsystems Measurements of the quadrature amplitudesXA YA XB and YB show identical variances and the correlations and anti-correlations have identical strength since ∆2(XA minus XB) = ∆2(YA + YB) lt12 for our normalization of quadrature amplitudes having a ground statevariance of 14

Generally a symmetric bi-partite squeezed state fulfills another quantita-tive (Gaussian) entanglement criterion if less than 50 of the vacuum state issymmetrically mixed into the initially pure state Bi-partite squeezed statesare always entangled but in this case they are even Einstein-Podolsky-Rosen(EPR) entangled [Reid (1989)] allowing the demonstration of the quantumsteering effect [Einstein et al (1935) Schrodinger (1935) Reid (1989) Cav-alcanti et al (2009)] The first such experiment was performed by Ou etal [Ou et al (1992)] using type II parametric down-conversion (PDC) Laterexperiments produced bi-partite squeezed vacuum states by overlapping twosqueezed vacuum states each produced with type I PDC on a balanced beamsplitter and used the entangled output for the demonstration of quantumteleportation [Furusawa et al (1998) Bowen et al (2003ca)] The criterionin Eq (21) and the EPR criterion from [Reid (1989)] was experimentallycompared in Ref [Bowen et al (2003b)] The steering effect in asymmet-ric bi-partite squeezed states were recently experimentally characterized inRef [Handchen et al (2012)]

Fig 12 shows features similar to those in the top part of Fig 11 Thisis not a coincidence and shows that a bi-partite squeezed state can also begenerated by spatially splitting the upper and lower sideband of a (single-party) squeezed state This was first experimentally demonstrated by thegroup of E Polzik [Schori et al (2002)] and later used for EPR multiplex-ing of a single longitudinal mode of a squeezing resonator [Hage et al (2010)]

35 Covariance matrix representation of bi-partite squeezed states

Also the full information of bi-partite states including the entanglementcan be cast by the covariance matrix [Simon et al (1994)] which can be used

34

to calculate the propagation of these states in laser interferometers Againall variances are normalized to the vacuum noise variance in full analogy toEq (18) The generic bi-partite covariance matrix has dimension 4times4 andreads

Vbp =

VXAXA

VXAYA VXAXBVXAYB

VYAXAVYAYA VYAXB

VYAYBVXBXA

VXBYA VXBXBVXBYB

VYBXAVYBYA VYBXB

VYBYB

with

VXAYB =

langXAYB + YBXA

rangminus 2

langXA

ranglangYB

rang2∆2Xvac

ϑ

(22)

Due to the symmetry in Eq (22) the 4times4 covariance matrix is fully spec-ified by just ten independent coefficients If the phase spaces at A and Bare aligned along the strongest correlations and anti-correlations the ma-trix components referring to different quadrature amplitudes eg VXAYA arezero Such entangled states can be produced by overlapping two squeezedfields with a squeeze angle difference of 90 on a balanced beam splitter

A symmetric bi-partite squeezed vacuum state which is also called anlsquoS-classrsquo [DiGuglielmo et al (2007)] bi-partite squeezed vacuum state shows(anti-)correlations in two joint quadratures as defined in Eq (21) For a puresuch state of 10 dB squeezing the covariance matrix reads

Vbp10|10 =

505 0 495 0

0 505 0 minus495495 0 505 0

0 minus495 0 505

The following covariance matrix describes a so-called lsquoV-classrsquo 10 dB bi-partite squeezed vacuum state Here only one joint quadrature shows 10 dBsqueezing whereas the orthogonal joint quadrature shows vacuum noise Thestate is obtained by overlapping one 10 dB squeezed state with a vacuumstate on a balanced beam splitter

35

Vbp0|10 =

055 0 045 0

0 55 0 minus45045 0 055 0

0 minus45 0 55

The first measurement of all elements of such a covariance matrix was achievedin [DiGuglielmo et al (2007)]

36 Photon numbers of squeezed states

In contrast to the ground state squeezed vacuum states do have photonexcitations As said earlier quantum theory links the wave and the particlepictures Indeed the squeeze factor of a modulation mode is directly con-nected to a certain photon number excitation Squeezed states of light areproduced via spontaneous photon pair generation eg by parametric down-conversion The following operator S is called the lsquosqueeze operatorrsquo [Gerryand Knight (2005)] It creates and annihilates photon pairs

|r θ〉 = S(r θ) |0〉 (23)

where |r θ〉 is a squeezed vacuum state with squeeze parameter r and squeezeangle θ and |0〉 is the vacuum state The definition of the squeeze operatoris

S(r θ) = exp

[1

2

(reminusiθa2 minus reiθadagger 2

)] (24)

The following shows that this definition indeed results in a state with squeezedquadrature amplitude variances Lets set θ = 0

〈0| Sdagger(r 0) X S(r 0) |0〉 =1

2〈0| Sdagger(r 0)

(a+ adagger

)S(r 0) |0〉 (25)

〈0| Sdagger(r 0) Y S(r 0) |0〉 =i

2〈0| Sdagger(r 0)

(aminus adagger

)S(r 0) |0〉 (26)

Using the Baker-Hausdorff formula we get

Sdagger(r θ) a S(r θ) = a cosh r minus adaggereiθsinh r (27)

Sdagger(r θ) adagger S(r θ) = adaggercosh r minus aeminusiθsinh r (28)

36

Since 〈0| X |0〉 = 〈0| Y |0〉 = 0 also Eqs (25) and (26) are zero To finallycalculate the variances we need

〈0| Sdagger(r 0) X2 S(r 0) |0〉 =1

4〈0| Sdagger(r 0)

(a2 + adaggera+ aadagger + adagger2

)S(r 0) |0〉

〈0| Sdagger(r 0) Y 2 S(r 0) |0〉 = minus1

4〈0| Sdagger(r 0)

(a2 minus adaggeraminus aadagger + adagger2

)S(r 0) |0〉

Given that SSdagger is the identity and using again Eqs (27) and (28) we get theexpected variances

∆2X =1

4

[cosh2 r minus 2cosh r sinh r + sinh2 r

]=

1

4eminus2r

∆2Y =1

4

[cosh2 r + 2cosh r sinh r + sinh2 r

]=

1

4e2r

Since the squeeze operator can only create and annihilate photon pairsa squeezed vacuum state without photon loss must correspond to an evennumber of photons But not only photon loss also a coherent displacementleads to flattening out the odd-even oscillations The probability of detectingN photons in a pure displaced squeezed state are derived for instance in[Gerry and Knight (2005)] and read

P (N) = |〈N |α r θ〉|2 =(05 tanh r)N

N cosh rexp

[minus|α|2 minus 1

2(αlowast2eiθ + α2eminusiθ)tanh r

]times∣∣∣HN

[(αcosh r + αlowasteiθsinh r)

radiceiθsinh(2r)

]∣∣∣2 (29)

where HN is the N th Hermite polynomialFig 13 shows the photon number distributions for 5 different pure squeezed

states according to Eq (29) Panels (a) to (c) show squeezed vacuum stateswith 43 dB 86 dB and 172 dB of squeezing Panel (d) shows the moregeneral case of a squeezed state with a coherent displacement α 6= 0 Dueto θ = 0 the state is amplitude quadrature squeezed Panel (e) refers to thecorresponding phase quadrature squeezed state For comparison panel (f)shows the photon number distribution of the coherent state with the samedisplacement

The panels in Fig 13 represent the diagonal elements of the statersquos den-sity matrix in number basis Only the latter also contains the coherences

37

002

004

006

000 000

002

004

006

008

403020100 Photon number n

403020100Photon number n

Pro

babi

lity

403020100000

005

010

015

100 2 4 6 8100 2 4 6 800

04

06

02

100 2 4 6 800

03

02

01Pro

babi

lity

Pro

babi

lity

08

00

04

06

02

08

(a) (b)

(c) (d)

(e) (f)

Figure 13 Photon number distributions ndash All panels represent pure states (a) 43 dBsqueezed vacuum state (r = 05 α = 0) (b) 86 dB squeezed vacuum state (r = 1 α = 0)(c) 172 dB squeezed vacuum state (r = 2 α = 0) (d) displaced 86 dB squeezed state(r = 1 θ = 0 α = 4) (e) displaced 86 dB squeezed state (r = 1 θ = π2 α = 4) (f)Coherent state (r = 0 α = 4) The average photon numbers are about 027 14 131174 174 and exactly 16 see Eq (30)

between photon numbers [Gerry and Knight (2005)] Figures as shown heregenerally do not give full descriptions of the states

38

A squeezed vacuum state (r 6= 0) always has a non-zero photon numberand can not be the ground state The average photon number of a puresqueezed vacuum state can be calculated using Eq (8) With the maximallysqueezed quadrature variance ∆2Xsqz the average photon number is given by

n = 〈n〉|α=0θr〉 = ∆2Xsqz +(∆2Xsqz)

minus1

16minus 1

2=eminus2r

4+e2r

4minus 1

2 (30)

with the vacuum noise variance normalized to one quarter A coherent dis-placement further adds |α|2 photons on average

4 Squeezed-light generation

41 Overview

Squeezed light was first produced in 1985 by Slusher et al using four-wave-mixing in sodium atoms in an optical cavity [Slusher et al (1985)]Shortly after squeezed light also was generated by four-wave-mixing in an op-tical fibre [Shelby et al (1986)] and by degenerate parametric down-conversion(PDC) in a 2nd-order nonlinear crystal placed in an optical cavity [Wu et al(1986)] The pumped cavity was operated below its oscillation thresholdie the parametric gain did not fully compensate the round trip losses whichis also called lsquocavity-enhanced optical-parametric amplification (OPA)rsquo

The early day experiments achieved squeeze factors of a few percent up toabout 3 dB Today squeeze factors of more than 10 dB are directly observedin several experiments [Vahlbruch et al (2008) Eberle et al (2010) Stefszkyet al (2012) Vahlbruch et al (2016)] All of them are based on cavity-enhanced OPA (below threshold) The parametrically amplified mode isdegenerate ie signal and idler modes are identical In particular the down-conversion process is of lsquotype Irsquo which means that the amplified mode has awell-defined polarization Squeezed states can also be generated above oscil-lation threshold In Refs [Villar et al (2006) Jing et al (2006)] bi-partitesqueezing was generated with above-threshold PDC Both experiments usedtype II PDC which provides orthogonally polarized signal and idler fieldsType II PDC below threshold was also used to generate squeezed and bi-partite squeezed fields [Grangier et al (1987) Ou et al (1992)] All these

39

experiments were performed in the continuous-wave regime which is also thefocus of this Review Squeezed states of modulations of trains of laser pulseshowever have been also generated since the 1980s using either PDC or theoptical Kerr effect [Slusher et al (1987) Bergman and Haus (1991) Our-joumtsev et al (2006) Dong et al (2008)] For an overview of the develop-ments in squeezed-light generation in the continuous-wave as well as pulsedregime see Ref [Bachor and Ralph (2004)] Squeezed-light generation inopto-mechanical setups [Aspelmeyer et al (2014)] which use the intensitydependent phase shift from radiation pressure was discussed in Refs [Paceet al (1993) Rehbein et al (2005) Corbitt et al (2006)] and recently ex-perimentally achieved by several groups [Brooks et al (2012) Safavi-Naeiniet al (2013) Purdy et al (2013)]

42 Degenerate type I optical-parametric amplification (OPA)

This section provides a graphical description of how degenerate type IOPAPDC turns a vacuum state into a squeezed vacuum state and a coher-ent state into a displaced squeezed state The process requires a bright pumpfield and a 2nd-order nonlinear crystal For simplicity we set all nonlinearitiesabove 2nd-order to zero

Let us consider a short segment of the second-order nonlinear crystalpumped with light of optical frequency 2ν All other modes that enter thecrystal shall not contain any photons ie are in their vacuum states Ofthese the only mode of interest is that at optical frequency ν which spatiallyoverlaps with the pump mode Fig 14 shows the total electric field of theoptical input E in and the 2nd-order nonlinear dielectric polarisation of thecrystal P The latter is proportional to the total electric field of the outputEout The pump field at 2ν periodically drives the vacuum field at ν betweenregions of low and high polarisation This process transforms the vacuumstate into a squeezed vacuum state in the output [Bauchrowitz et al (2013)]The output further contains the hardly depleted pump field and frequencydoubled parts of the pump field at 4ν It is again emphasized that Fig 14displays OPA in a small segment of the crystal In reality the nonlinear effectaccumulates over the crystal length or even over several passages since thecrystal is usually put into an optical resonator A noticeable effect is achievedif all infinitesimal contributions constructively interfere This is achieved incase of phase matching ie if the wave fronts of the modes at 2ν and νpropagate with the same speed and thus do not run out of phase Note that

40

t

t

Squeezing

Figure 14 Optical parametric amplification of a vacuum state ndash The upper leftcorner shows the crystal polarization P(E) = ε0

(χ(1)E + χ(2)E2

) ie the separation of

charge carriers by the electric component of an optical field E The graph illustrates howan input quantum field (from below) is projected into an output quantum field (towardsthe right) In the example shown here the input field is composed of a classical pumpfield E in

2ν at frequency 2ν and zero-point fluctuations E inν of a field at frequency ν cf

Fig 7(c) The superposition E in of these two fields is transferred into a time-dependentdielectric polarization that is the source of (and thus directly proportional to) the elec-tric component of the output field Eout The quantum uncertainty of the output fieldshows a phase-dependent (parametric) amplification at frequency 2ν Spectral decompo-sition of the output field Eout reveals coherent amplitudes at frequencies 2ν and 4ν and asqueezed vacuum state Eout

sqzν cf to Fig 7(d) The concept of this figure was published inRef [Bauchrowitz et al (2013)]

41

t

t Amplitudequadraturesqueezing

Figure 15 Optical parametric amplification of a coherent state ndash The pictureshows how a displaced vacuum state is transformed into a displaced squeezed state Thepumprsquos electric field is maximal when the input field at fundamental frequency ν shows azero crossing The phase relation described results in an output state that is amplitudequadrature squeezed If the pump field at the input in phase was shifted by half of itswavelength the squeezed output field were a phase quadrature squeezed The squeezinggeneration displayed here corresponds to the transition from Fig 7(a) to Fig 7(b) but withan additional reduction of the displacement The concept of this figure was published inRef [Bauchrowitz et al (2013)]

in actual squeezing experiments the 4ν component is usually suppressed byphase miss-matching

Fig 15 shows the same process but now for an input field at frequencyν in a coherent state In this case the relative phase between the two inputstates ∆ϕ = ϕ2ν minus 2ϕν is relevant In Fig 15 the relative phase is set suchthat the expectation value of the field at frequency ν is zero when the pump

42

(a) (b)

(c) (d)

X

Y

X

YΔϕ = ndash50deg

X

Y

Δϕ = 0deg

X

Y

Δϕ = 90deg

Figure 16 Phase-space illustration of degenerate OPA ndash The (displaced) dashedcircle in each diagram represents the uncertainty of the initial state at optical frequencyν The (displaced) shaded area represents the state after degenerate optical parametricamplification All quantum uncertainties shown correspond to pure states The boldgreen arrow describes the bright second-harmonic pump field whose uncertainty can beneglected The phase between the 2nd harmonic pump and the initial state (∆ϕ = ϕ2ν minus2ϕν) determines the result of the parametric amplification

field reaches its maximum (∆ϕ = 90) The output at the fundamentalfrequency is then an amplitude squeezed state with a deamplified coherentamplitude

Fig 16 summarizes the squeezing operation on the vacuum state as wellas on displaced vacuum states for different phase relations ∆ϕ between thetwo input fields

43 Cavity-enhanced OPA

Placing the nonlinear crystal inside a cavity can greatly enhance thedown-conversion efficiency but not only that A cavity introduces a thresholdfor the pump power above which the parametric gain is infinite just limited

43

by the finite pump power In this case the vacuum uncertainty of the inputfield at frequency ν is amplified to a bright laser field at frequency ν Thedevice is then called an optical-parametric oscillator (OPO) For the gener-ation of squeezed states however the pump power is usually kept (slightly)below threshold Due to nonzero optical loss there exists a pump powersmaller than the threshold above which the tiny improvement of squeezingis not noticeable anymore Getting the pump power closer to the thresholdcould even reduce the observed squeeze factor if a fluctuating squeeze an-gle projects anti-squeezing into the observed quadrature amplitude [Franzenet al (2006) Suzuki et al (2006) Dwyer et al (2013)] The cavity has an-other important purpose It confines the transverse spatial mode usuallyto TEM00 This mode confinement is crucial for any efficient application ofthe squeezed state in laser interferometry since it allows the suppression ofanti-squeezing from other transversal modes The squeezing process requiresa nonlinear material that should show negligible absorption at both opticalfrequencies involved in particular at the wavelength of the squeezed modeIn Refs [Vahlbruch et al (2008) Mehmet et al (2009)] 10 dB and 116 dBof squeezing were achieved using MgOLiNbO3 The highest squeeze factorstoday are produced in (quasi phase matched) periodically poled KTP [Eberleet al (2010) Mehmet et al (2011) Stefszky et al (2012) Vahlbruch et al(2016)]

The optical cavity that is built around the nonlinear crystal is vital forsqueezed-light generation and it deserves a detailed consideration Gener-ally the mode propagating away from a cavity is the result of interferenceat the cavity coupling mirror One contribution is given by the intra-cavityfield attenuated by the amplitude transmission coefficient t of the outcouplingmirror The second contribution is given by the outside field that is reflectedby the same mirror with amplitude reflectivity r =

radic1minus t2 and spatially

overlapped with the first Also the mode from a squeezing resonator is suchan interference product

The impedance matched resonator

Let us consider first an empty optically stable and loss-less Fabry-Perotresonator built from two identical mirrors each with amplitude reflectivityr = r1 = r2 lt 1 A propagating field be perfectly mode-matched to one ofthe cavity resonances In this setup the resonator shows zero reflection andthe resonator is said to be impedance matched (for all such input fields)

44

|α〉

r1α

r12

= +

|α〉 r22 = r1

2

minusr1α equiv t1 αcav = 1minus r12 minusα sdot r1

1minus r12

0

XY

Figure 17 Empty impedance-matched resonator ndash Mode-matched and resonant lightthat enters the cavity from the left here displayed by a quantum phasor for a coherentstate is fully transmitted including its quantum uncertainty The back-reflected statedestructively interferes with itself for all frequencies well within the cavity linewidth The180 phase shift of the transmitted cavity field amplitude minusr1α is a necessary conditionin order to fulfill energy conservation on cavity resonance Impedance matching is forinstance achieved for a resonant cavity with matched mirror reflectivities (r2

2 = r21) and

zero optical loss The complex amplitude of the field traveling towards left inside the cavityis readily deduced from the figure and reads αcav = minusαr1(1minus r2

1)minus12 (highlighted by thedashed circle) In the displayed setup zero field uncertainties are reflected however alsothe vacuum state that enters the cavity from the right is fully transmitted (not shown)

r1α

XY

r12 r2

2 =1

= + |α〉

minus 1+ r1( )α equiv t1 αcav = 1minus r12 minus 1+ r1( )α

1minus r12

|α〉

Figure 18 Empty maximally overcoupled resonator ndash Maximal overcoupling isachieved for a resonant cavity with a perfect end-mirror reflectivity (r2

2 = 1) and zerooptical loss For a given input-mirror reflectivity r2

1 the intracavity light power is maximalMode-matched and resonating light entering the resonator from the left is fully reflectedThe complex amplitude of the field traveling towards left inside the cavity is readilydeduced from energy conservation to αcav = minusα(1 + r1)(1 minus r2

1)minus12 (dashed circle) Inthis setup no uncertainty from the right couples to the left side of the cavity

45

Obviously the interference described in the previous paragraph is fullydestructive The same resonator also shows zero reflection of the input fieldrsquosquantum uncertainty since the interference happens between parts of thesame quantum state The mode propagating away from such a resonatorhowever is not in a nonclassical but in a vacuum state because the vacuumstate that enters the cavity through the opposite site is also fully transmittedThe interference at the coupling mirror of an impedance matched resonatoris displayed in Fig 17

r1α

r12

+

=

OPA PDC

Xcavg r1(1+r1) Xcav

Δ2Xg 0 Δ2Yg

XY

r22 =1

|α〉

Figure 19 Squeezing resonator ndash Shown is the interference at the zero-loss squeezingresonator operated at threshold The lower line represents the perfectly squeezed modepropagating away from the cavity towards the left The parametric gain medium inside thecavity deamplifies the X quadrature of the cavity mode (Xcav) by the factor r1(1 + r1)which is the ratio of the intra-cavity field amplitudes of the two previous figures The Xquadrature of the field that is back-reflected towards the left destructively interferes withitself similar to the situation of the impedance matched cavity in Fig 17 The parametricpower gain per resonator round-trip (G) needs to mimic the effect of an end mirror withreflectivity r2

2 = r21 For this reason the deamplification of Xcav corresponds to a round-

trip deamplification factor of r1 equivradic

1G The round-trip amplification factor for Ycav

then is 1r1 equivradicG which exactly compensates for the outcoupling and thus determines

the parametric oscillation threshold (threshold for bright lasing) The variances of thequantum uncertainties ∆2Ycav and ∆2Y are thus infinite In this setup no field uncertaintyfrom the right couples to the left of the cavity and a perfectly X-quadrature-squeezed fieldoutside the squeezing resonator is produced

46

The perfectly over-coupled single-ended resonator

We now increase the reflectivity of the far mirror lsquo2rsquo to being perfect (r2 = 1)This way the counter-propagating vacuum state can not enter the cavityAgain a propagating field be perfectly mode-matched through mirror lsquo1rsquo toone of the cavity resonances For frequencies well inside the cavity linewidththe situation is displayed in Fig 18 The setup protects the left side of thecavity against vacuum fluctuations entering through mirror lsquo2rsquo but of coursedoes not squeeze quantum noise The intra-cavity built-up factor is too highfor achieving destructive interference below the vacuum uncertainty on theleft side of the resonator

The impedanced-matched single-ended squeezing resonator

Building on the two previous concepts the straight forward approach now isto start from the perfectly over-coupled single-ended resonator and insert anattenuator into the cavity that does not couple the cavity mode to any bathbut still results in a roundtrip efficiency of precisely r1(lt 1) in amplitudeOptical loss is not appropriate since it increases the coupling of the cavitymode to a thermal bath neither would any phase-insensitive atenuator beappropriate It is easy to show that a phase-insensitive attenuator adds ad-ditional uncertainty since otherwise the commutation relation [a adagger] = 1 isviolated The amplification process that matches our requirement is OPA Toachieve infinite squeezing in X on cavity resonance a second-order nonlinearcrystal needs to be put into the cavity and pumped such that the intra-cavityamplitude quadrature is attenuated by the factor (1 + r1)r1 (on cavity reso-nance) with respect to the empty cavity This factor is readily deduced fromFigs 17 and 18 Due to the symmetry in parametric amplification the intra-cavity phase quadrature is then amplified by (1 + r1)r1 and the round-tripgain has a value of 1r1 in amplitude In this situation not only infinitesqueezing but also the (laser) threshold of the resonator is achieved sincethe round-trip gain of the intra-cavity phase quadrature equals its roundtriploss here fully given by the incoupling mirror

The physical descriptions in Figs 17 to 19 are fully consistent with ob-servations in squeezing experiments The consideration above in particularshows that the intra-cavity field shows a finite squeezing strength while theexternal field shows infinite squeezing The strongest intra-cavity squeezefactor possible is (1 + r1)2r2

1 In the high reflectivity limit this factor corre-

47

sponds to 6 dB Averaged over the full cavity mode the squeeze factor of thecavity mode is in this limit even limited to 3 dB [Walls and Milburn (2008)]Higher intra-cavity squeeze factors are possible for lower mirror reflectivities

44 The generation of squeezed light for laser interferometry

With the insights gained in the previous subsection we now turn to ac-tual experiments The application of squeezed states in laser interferometrycertainly requires large squeeze factors (idealy accompanied with the highestpossible purity) to maximize the impact in terms of sensitivity improvementIn cavity-enhanced OPA the highest parametric gain is achieved on cavityresonance ie at zero sideband frequency But this is not the main reasonwhy this Subsection focusses on the generation of squeezed states at lowsideband frequencies The application of squeezed states in a laser inter-ferometer requires that their sideband frequencies cover the devicersquos signalband Ground-based gravitational wave (GW) detectors have a detectionband from about 10 Hz to 10 kHz frequencies which can be considered aslsquolowrsquo compared to typical frequencies in quantum optics experiments

Squeezing at MHz sideband frequencies is easier to observe than at acous-tic frequencies because the latter are often polluted with excess noise fromlight beams that serve as control beams [Bowen et al (2002) McKenzie et al(2004)] and parasitic interferences from back-scattered light [Vahlbruch et al(2007)] Furthermore the observation of squeezing at low sideband frequen-cies requires a more stable setup since larger measuring times are necessaryThe observation of strong squeezing at MHz frequencies however alreadysets an upper limit to the optical loss of the setup At least the same squeezefactor can be observed at lower frequencies

There are two different main topologies for squeezing resonators TheFabry-Perot-type standing-wave resonator consists of a minimum number ofmirror surfaces and has the advantage of being compact and thus robustagainst mechanical vibrations Usually one or even two mirror coatings aredirectly placed on the spherical and polished surfaces of the nonlinear crys-tal itself [Wu et al (1986) Grangier et al (1987) Breitenbach et al (1998)Vahlbruch et al (2008) Eberle et al (2010) Vahlbruch et al (2016)] TheBowtie traveling-wave resonator has the advantage of providing a separa-tely accessible counter propagating mode for cavity length control [Ou et al(1992) Takeno et al (2007)] It shows no direct back-reflection of incoupledlight which helps reducing parasitic interferences [Stefszky et al (2012)]

48

(a)

(b)

(c)

(d)Squeezingresonator

To inter-ferometer

OPA

SHG

BHD

Mode cleaner

LO

Laser

DBS

DBS

Figure 20 Generation of squeezed light ndash (a) Example of a 2nd-order nonlinearcrystal for the squeezed-light generation at 1064 nm Shown is a bi-convex 65 mm long7MgOLiNbO3 crystal whose polished surfaces also carry the mirror coatings of the res-onator The crystal thus realizes a monolithic squeezing resonator as it was used forthe first demonstration of 10 dB squeezing [Vahlbruch et al (2008)] (b) Optical con-figuration of a half-monolithic (hemilithic) standing-wave squeezing resonator Here thecavity length can be adjusted by displacing the coupling mirror The crystal surface insidethe cavity is anti-reflection coated The photograph shows a 10 mm long PPKTP crystalsqueezing resonator as used for the GEO 600 squeezed-light source [Abadie (2011)] (c)Mechanically stable housing of a standing-wave squeezing resonator The crystalrsquos temper-ature is stabilized at its phase matching condition using Peltier elements (d) Schematicfor the squeezed-light generation After spatial filtering of continuous-wave laser lighttwo hemilithic standing-wave resonators are employed The first generates second har-monic pump light (SHG) The second (OPA) generates a squeezed vacuum field at theinitial wavelength The squeezed states are observed by a balanced homodyne detector(BHD) or alternatively sent and mode-matched to the optical mode of an interferometerbeforehand LO local oscillator DBS dichroic beam splitter

49

Fig 20 (a) and (b) show photographs of typical nonlinear crystals used forsqueezed-light generation at near infra-red wavelengths The crystals shownhere form a monolithic standing-wave squeezing resonator (a) or are partof a half-monolithic standing-wave squeezing cavity (c) shows a tempera-ture stabilized and mechanically stable housing of the squeezing resonator(d) shows a schematic of a full setup for the generation of squeezed vacuumstates of light for an application in a laser interferometer The only brightinput required for the squeezing resonator (OPA) is the second-harmonicpump field The resonator mode at fundamental frequency is thus initiallynot excited by photons ie it is in its ground state characterized by vacuumfluctuations due to the zero point energy see Fig 7 (c) [Gerry and Knight(2005)] The pump field spontaneously decays in the degenerate pair of sig-nal and idler fields The combined down-converted field leaving the resonatorexhibits quantum correlations which give rise to a squeezed photon countingnoise when overlapped with a bright coherent local oscillator beam Thedetection is done either in a balanced homodyne detector (BHD) or with asingle photo diode The squeeze factor increases the closer the pump powerof the squeezing resonator gets to the oscillation threshold and the lower theoptical loss on down-converted photon pairs is

441 High squeeze factors ndash minimizing decoherence

Squeezed states of light have significant impact on the sensitivity of laserinterferometers if large squeeze factors can be produced Squeezing of 3 dBimproves the signal-normalized quantum-noise spectral density by a factorof 2 This factor corresponds to doubling the (coherent state) light powercirculating inside the interferometer Squeezing of 10 dB corresponds to aten-fold power increase The experimentally demonstrated squeeze factorswere considerably improved in recent years [Takeno et al (2007) Vahlbruchet al (2008) Polzik (2008) Eberle et al (2010) Stefszky et al (2012)]culminating in a value of as large as 150 dB [Vahlbruch et al (2016)] Thisvalue corresponds to the same reduction of signal-normalized quantum noisethat is achieved by increasing the light power by a factor of 32 (At this pointit is already noted that squeezing the quantum noise can simultaneouslyreduce quantum measurement noise (shot noise) as well as quantum backaction noise (radiation pressure noise) This is not possible with scaling thelight power of coherent states see Subsec 55)

50

Ideally a parametric squeezed-light source can produce an infinite squeez-ing level see Fig 19 fundamentally just limited by the energy provided bythe pump field In practice the limit is set by decoherence mechanismsThe by far most important one is optical loss Optical loss occurs duringsqueezed-light generation its propagation through the interferometric setupincluding imperfect mode matchings and finally the photo-electric detec-tion Also detector dark noise [Schneider et al (1998)] phase noise [Takenoet al (2007)] and excess noise [Bowen et al (2002)] impair the observablesqueezing strength

Optical loss is usually understood as coupling the squeezed mode to a zerotemperature bath ie overlapping it with a vacuum mode For any amountof loss the resulting state is still squeezed But to be able to directly observesay 10 dB of squeezing the total loss on the state needs to be less than 10in this example cf Eq (16) To minimize optical loss the nonlinear crys-tal as well as lenses and beam splitters in the interferometric path need toshow very low absorption and scattering at the wavelength of the squeezedlight PPKTP shows absorption of about 10minus4cm and below at near-infraredwavelengths Low OH content fused silica is a suitable material for all otheroptics Absorptions of less than 10minus6cm were measured [Hild (2007)] Coat-ings on crystal surfaces and on all other optical components should also showlowest optical loss Total loss of the 10minus6 level are available today Superpol-ished surfaces which show roughnesses with less than 1 A root mean square(integrated over spatial scales from approximately 1 micron to 100 microns)and thus very low scattering are necessary to achieve these low numbersMinimizing the total number of optical components is essential From thisperspective a monolithic squeezing resonator as shown in Fig 20 (a) is theoptimum choice The squeezed mode needs to be matched to the mode ofthe laser interferometer or to the mode of the balanced homodyne detectorVisibilities of up to 998 have been achieved [Eberle et al (2010)] whichcorresponds to a loss of about 04 Of great importance also is the quantumefficiency of the photo-diodes used for detecting the squeezed field (togetherwith the interferometric signal) Recently a quantum efficiency of photo-diodes in a squeezing experiment of (995plusmn 05) was measured [Vahlbruchet al (2016)] To minimize photon loss the photo-diodes had no protectionwindow an anti-reflection coating on the semi-conductor material and theremaining reflection was re-focussed with an external mirror

Also the dark-noise spectral density of the detection electronics reducesthe observable squeezing and needs to be as low as possible Similar to optical

51

noise it also provides a contribution to the observed variance The dark noiseof the detection electronics needs to be much lower than the detected photoncounting noise In [Vahlbruch et al (2016)] it was 28 dB below shot noisebut still reduced the observable squeeze factor from 153 dB to 150 dB

Excess noise emerges if the squeezed mode couples to a nonzero tem-perature bath or to a mode whose excitation is strongly fluctuating (Thecoupling process can always be understood as a beam splitter coupling andis physically described by overlapping electric fields Coupling to a zero tem-perature bath leads to Eq (16)) The captured excess noise variance thenneeds to be added to the initial squeezing variance which deteriorates theobserved squeezing stronger than just mixing in the vacuum mode Excessnoise is less likely to occur at MHz frequencies but can be significant ataudio-band sideband frequencies and below and is thus a serious issue ingravitational-wave detectors [Chua et al (2014)] The reason for that isthat acoustically or thermally excited motions of surfaces produce frequencyshifts of back-scattered light mainly at these low frequencies [Vahlbruch et al(2007)]

Phase noise corresponds to stochastic phase fluctuations between thesqueezed field and the local oscillator within the measuring time It cor-responds to mixing the squeezed mode with itself with a fluctuating squeezeangle [Suzuki et al (2006) Franzen et al (2006)] Phase noise in squeez-ing experiments typically is less of an issue than optical loss [Dwyer et al(2013) Oelker et al (2016) Vahlbruch et al (2016)] The setuprsquos phasenoise can be reduced by making the squeezing resonator more compact andthus mechanically more stable against acoustic and thermal fluctuations ofthe environment and by improving the quality of the servo loops for cavitylength and propagation length controls Operating a squeezed-light resonatorin vacuum might also be beneficial The ability to run a high performancesqueezed-light generator in vacuum was demonstrated in [Wade et al (2015)]

442 Squeezing in the gravitational-wave (GW) detection band

High squeeze factors have been first demonstrated at sideband frequen-cies of a few MHz and above where excess noise is generally negligible whenworking with visible or near-infra-red light Today we know that extendingthe squeezing spectrum towards the audio-band and even below is technicallynot always easy but straight forward once a high squeeze factor is achieved

52

at MHz frequencies In most squeezing experiments the main laser light

Figure 21 Photograph of the GEO 600 squeezed-light source ndash The breadboarddimensions are 135 cm times 113 cm The squeezing resonator is high-lighted by the white ar-row and is set up as a standing-wave hemilithic cavity containing a plano-convex PPKTPcrystal of about 10 mm length (see also Fig 20 b) It is pumped with continuous-wave532 nm light that is produced by frequency doubling of light (at angular frequency ω)from a commercial NdYAG laser Two more laser fields at about 1064 nm having fre-quency offsets of more than 10 MHz with respect to ω(2π) serve as optical control fieldsBoth fields are mode-matched and injected into the squeezing resonator together with thesecond-harmonic pump field

at the squeezing wavelength is accompanied by significant noise up to thelaser relaxation oscillation For this reason laser control fields at the opticalcarrier-frequency in the optical path of the squeezed mode need to be avoided[Bowen et al (2002) Schnabel et al (2004) McKenzie et al (2004)] and thesqueezing resonator length and the optical path stabilized by other means[McKenzie et al (2005) Vahlbruch et al (2006)] Furthermore and mostimportantly excess noise due to back-scattering is an issue Back-scattering(also called lsquoparasitic interferencesrsquo) is produced if DC light scatters out ofthe optical path hits a vibrating surface and re-scatters back into the opticalpath [Vahlbruch et al (2007)] Significant back-scattering can be produced

53

in interferometers for the detection of gravitational waves since light powersof several hundreds of kilowatts are used Even back-scattering from the mil-liwatt local oscillator of balanced homodyne detectors is an issue at acousticsideband frequencies and below The recipe for avoiding parasitic interfer-ences turns out to be threefold (i) avoiding scattering by using ultra-cleansuperpolished optics with close to perfect anti-reflex coatings (ii) avoidingback-scattering by carefully blocking all residual (faint) light fields and (iii)reduce the vibrationally and thermally excited motion of all mechanical andoptical parts that could potentially act as a re-scattering surface with re-spect to the optical path [Vahlbruch et al (2007) McKenzie et al (2007)]The insights described above led to the first demonstration of audio-bandsqueezing at frequencies down to 200 Hz [McKenzie et al (2004)] and laterto the first demonstration of squeezing over the full gravitational-wave de-tection band even from as low as 1 Hz [Vahlbruch et al (2007)] Whilea standing-wave squeezing resonator [Ou et al (1992)] can be built in avery compact way that is rather insensitive against mechanical vibrations[Chelkowski et al (2007)] a traveling-wave bow-tie squeezing resonator [Wuet al (1986)] is more tolerant against back-scattered light [Chua et al (2011)]The strongest squeezing in the audio-band of up to 116 dB was reported inRef [Stefszky et al (2012)]

443 The first squeezed-light source for GW detection

The first squeezed-light source for the continuous operation in GW de-tectors had been designed and completed between 2008 and 2010 [Vahlbruch(2008) Vahlbruch et al (2010)] Since then this source has been producingsqueezed vacuum states in a fully phase controlled way using co-propagatingfrequency-shifted bright control beams [Vahlbruch et al (2006)] as an inte-gral part of the GW detector GEO 600 The source is a turn-key device witha fully automated re-lock system [Vahlbruch et al (2010) Khalaidovski et al(2012)] Re-locking is required if the temperature of the environment changessignificantly which drives the actuators outside their dynamic ranges

Up to 9 dB of squeezing over the entire GW detection band was observedusing a balanced homodyne detector (BHD) located close to the squeezingresonator The squeeze factor has been limited by optical loss due to ab-sorption in the PPKTP crystal transmission of the back-surface and thenon-perfect AR-coating of the crystalsrsquos intra-cavity surface The adjustableair gap has been necessary to allow for an easy way to apply length control

54

-12

-8

-4

0

4

8

12

16

20

10 100 1k 10k

Rel

ativ

e no

ise

pow

er [

dB]

Frequency [Hz]

shot noise (a)

squeezed noise (b)

anti-squeezed noise (c)

~ 9dB

Figure 22 Broadband squeezing spectrum ndash Noise power spectra measured on theoutput of the GEO 600 squeezed-light source shown in Fig 21 with a balanced homodynedetector The traces correspond to the spectra of quadrature amplitude variances normal-ized to vacuum noise The resolution bandwidth used increases towards higher frequenciesto reduce the measurement time (a) Shot noise normalized to unity which serves as thereference level (0 dB) (b) Squeezed noise covering the complete detection band of ground-based GW detectors (c) Anti-squeezed noise Peaks at 50 Hz and 100 Hz are the electricmains frequency and its first harmonic The data was published in Ref [Vahlbruch et al(2010)]

Additional optical loss in the path to the balanced homodyne detector mainlyarose due to a Faraday isolator that eliminated parasitic interferences Fi-nally the mode missmatch to the BHD as well as its non-perfect quantumefficiency provided additional loss Inferring the squeeze factor without theBHD detection loss more than 10 dB of squeezing are provided by the sourceSince 2010 it has been used in basically all observational runs of the GEO 600GW detector see Section 6

444 Generation of two-mode (bi-partite) squeezing

lsquoTwo-mode squeezed lightrsquo or lsquobi-partite squeezed lightrsquo is light that allowsfor joint measurements at two locations A and B These joint quadraturemeasurements reveal correlations and anti-correlations with a remaining un-certainty smaller than the ground-state uncertainty which certifies the pres-

55

ence of entanglement cf Subsec 34 Bi-partite squeezed light has been gen-erated by type I and by type II parametric down-conversion In case of type Ithe squeezed fields from two squeezing resonators as described in Subsec 44are overlapped on a balanced beam splitter with a 90 phase shift The twooutput fields together represent the entangled mode [Furusawa et al (1998)Bowen et al (2003c) Eberle et al (2013)] In case of type II signal and idlerfields are non-degenerate regarding polarisation and a single cavity contain-ing an appropriate crystal and a polarising beam splitter are sufficient for theproduction of bi-partite squeezing Also in this case the measurements ofthe quadrature amplitudes of signal and idler fields show large uncertaintiestogether with bi-partite correlations and anti-correlations that are strongerthan the ground state uncertainty of individual subsystems [Ou et al (1992)Villar et al (2006) Jing et al (2006)]

To date the strongest entanglement of bi-partite squeezed light has beenproduced based on type I parametric down-conversion [Eberle et al (2013)]The requirements of producing strong entanglement are identical to those ofproducing strong squeezing outlined above The strength of bi-partite en-tanglement can be given in decibels in full analogy to the squeeze factorPractically the strength of bi-partite squeezing is always somewhat smallerthan that of single party squeezing since it requires an additional mode-matching that results in additional optical loss

45 Conclusions

The first observation of squeezed light was achieved in 1985 [Slusher et al(1985)] Shortly after cavity-enhanced optical parametric amplification forsqueezed-light generation was demonstrated [Wu et al (1986)] which todayenables the observation of up to 15 dB of squeezing [Vahlbruch et al (2016)]Quite generally the maximum squeezing level that is observed does not de-pend on the strength of the optical nonlinearity Squeezing cavities can easilybe operated at their oscillation threshold where they should provide infinitesqueezing if decoherence is zero The main limiting factor is optical lossincluding that of the photo-electric detection

Dedicated experimental research and development towards a squeezed-light source for applications in gravitational-wave detectors can be tracedback to 2002 [McKenzie et al (2002) Bowen et al (2002)] Since then a sur-prising amount of progress has been made culminating in the first squeezed-light source specifically built for the integration into a gravitational-wave

56

detector For the future squeeze factors above 15 dB will certainly be possi-ble by further reducing optical loss This statement is supported by the highdegree of matching between experimental data and a theoretical loss modelas presented in Fig 3 of [Vahlbruch et al (2016)]

5 Quantum noise in laser interferometers

51 Interferometric measurements

The purpose of a laser interferometer is the precise measurement of smallchanges of an optical path length with respect to a reference path Forthis the interferometer transfers the change of the phase difference betweentwo light fields into an amplitude quadrature change of the interferometerrsquosoutput light The latter can easily be detected by a single photo diode Ofgeneral interest are differential changes of the optical path length that aremuch smaller than the laser wavelength ie differential phase changes thatare much smaller than 2π In this case the differential phase change can bedescribed in very good approximation as a differential change of the phasequadrature instead

In order to transfer the phase quadrature signal with minimum loss ahigh interference contrast at the interferometerrsquos beam splitter is essentialAdditionally instrumental noise in terms of unwanted excitations of the out-putrsquos amplitude quadrature needs to be reduced to a minimum Noise arisesdue to power fluctuations of the input laser light back-scattered laser lightinside the interferometer thermally driven displacements of mirror surfacesand in many more ways The important measure of the sensitivity of aninterferometer obviously is its signal-to-noise-ratio The most useful measureis given in terms of the noise spectral density S(f=Ω2π) that is normalizedto the physical unit of the signal S(f) is in fact a lsquonoise-to-signal-ratiorsquoand can be seen as the signal-normalized variance of the photo diode out-put decomposed into spectral components versus sideband frequency f withthe resolution bandwidth of 1 Hz As an example S(100 Hz) = 10minus39 m2Hzmeans that the instrumental noise in the one hertz band around 100 Hz equalsa signal that would be produced if the mirror of one interferometer arm os-cillates with an amplitude of just

radic10 middot 10minus20 m in the very same band Such

small spectral densities are achieved by gravitational-wave detectors [Abbott(2016)]

57

52 Quantum measurement noise and shot noise

The most fundamental noise source in laser interferometers is due to thequantum noise of light which is in fact two-fold [Caves et al (1980)] Firstof all there is lsquoquantum measurement noisersquo which arises in the processof photo-electric detection For coherent states the quantum measurementnoise is the lsquophoton counting noisersquo of mutually independent photons andusually simply called lsquoshot-noisersquo Fig 2 (b i) shows a time series of suchnoise hiding the actual signal The frequency components of the shot noiseare well described by the quantum uncertainty of the output fieldrsquos amplitudequadratures XΩ∆Ω see Subsec 22 (Recall this quantity corresponds to the

differential phase quadrature YΩ∆Ω of the light beams in the interferometerarms) The photon counting noise has a white Fourier spectrum howeverthe lsquoshot noisersquo of an interferometer is usually normalized to the signal whosetransfer function is usually not white for instance due to the presence of armcavities or a signal-recycling cavity

All current and planned gravitational-wave detectors are Michelson-typelaser interferometers with operating points very close to a dark fringe Thelight power in the output port is just a couple of tens of mW which canbe handled by a single photo diode In this configuration the signal-to-shot-noise-ratio is actually maximized which can be shown in three steps [Bachorand Ralph (2004)] For the first step we use plane waves to describe theelectric field in the output port of a Michelson interferometer For perfectinterference contrast at the balanced beam splitter ie for perfect modematching and for defining φ = 0 as the dark port condition we get

Eout(t φ) =1

2E0 sin(ωt+ φ)minus 1

2E0 sin(ωt) (31)

where E0 is the amplitude of the total internal field whose two parts hasaccumulated a differential phase It directly follows for the squared fields

E2out(t φ) =

(sin

φ

2

)2

(E0 cos(ωt+ φ2))2 (32)

We now turn to a light beam with a localized transversal mode that can befocussed onto a photo-electric detector The photo diode has perfect quantumefficiency ie the rate of photo electrons is not only proportional to the rateof output field photons but also has a unity slope efficiency Since the optical

58

frequency is too high to be resolved we consider the averaged light power

P out(φ) =

(sin

φ

2

)2

P (33)

The next step is a formulation of the signal being the derivative of detectedphoton number versus phase Let n be the average value of the photonnumber per measuring time interval Eq (33) can then be rewritten as

nout(φ) =

(sin

φ

2

)2

n (34)

rArr dnout(φ) = n sinφ

2cos

φ

2dφ (35)

The final step is the calculation of the signal-to-shot-noise-ratio Shot noiserefers to coherent states which have a standard deviation of the photonnumber of σ(n) =

radicn

dnout(φ)

σ(nout)=n sinφ

2cosφ

2dφ

radicn sinφ

2

(36)

and find for a signal-to-noise ratio of unity for coherent states and for anon-zero but still small phase difference ∆φCoh 2π

1 =radicn cos

φ

2∆φCoh with φ 6= 0 (37)

In this equation the smallest measurable phase difference is given for φrarr 0

∆φCohmin =

1radicn (38)

This is the well-known shot-noise limit of high-precision phase sensing ∆φCohmin

is the smallest phase shift that can be measured with a signal-to-noise ra-tio of one when using n mutually independent photons per measuring time(those of a coherent state) when the loss of photons is assumed to be zeroThe typical purpose of a laser interferometer is the continuous sensing (moni-toring) of a continuously changing phase An illustrative example is the phasesignal produced by the black hole merger measured by Advanced LIGO onSept 14 in 2015 (Fig 1 in [Abbott (2016)]) The measuring interval should

59

be short to be able to resolve the time-evolution of the signal Generally themeasurement of an arbitrary signal that lasts for a finite time thus needs tobe understood as l subsequent measurement intervals using n photons eachIt can be shown that Eq (38) is also valid for interferometers operated athalf fringe ie when each output port contains the same light power In thiscase photo diodes need to be placed in both output ports and the actualsignal is provided by their difference voltage

Due to its importance the shot-noise limit deserves some remarksThe phase φ in Eq (31) is the phase difference of two mode-matched fieldsand might be accumulated by a single pass along the length L such as ina Mach-Zehnder interferometer or in a double pass such as in a (simple)Michelson interferometer or in four passes as realized in a Michelson inter-ferometer with folded arms [Grote (2005)] The shot-noise limit in Eq (38)and its scaling therefore holds independent of the number of passes Theclaim in Ref [Higgins et al (2007)] that the scaling according to Eq (38) canbe surpassed by multiple passes is not justifiedThe fact that Eq (38) is derived by approaching φ rarr 0 correctly describesthe actual operation point of gravitational-wave detectors which is close tobut not exactly at a dark port In practice a tiny offset from dark port ischosen at which the shot noise is well above the photo diodersquos electronic darknoiseEq (38) solely depends on the number of quanta but not on the lightrsquos wave-length λ Of course the shot-noise limit for the change of an optical pathlength ∆L does depend on the wavelength and ∆φCoh

min needs to be replacedby ∆φCoh

min = 2π∆LCohminλ

Finally an essential result of the shot-noise limit is that the ideal precisemeasurement should use lsquoas much quanta as possible per measuring inter-valrsquo which translates to lsquoas much light power in the interferometer armsas possiblersquo Eq (38) is indeed the one and only reason why gravitational-wave detectors use high power lasers power-recycling and arm resonatorsExtending the measuring time for a given light power can also improve thesensitivity but only if the signal repeats ie is periodic Let us assume thatone period of the signal is resolved by l intervals using n photons each Inthis case repeating the overall measurement k times improves Eq (38) by1radick The fundamental statement of Eq (38) however does not change

since the actual photon number n may then simply incorporate the factor k

60

For a given average photon number the shot-noise limit in Eq (38) canonly be surpassed by using photons that are quantum correlated ie by usingnonclassical states of light How is the shot-noise limit surpassed with thehelp of squeezed states A Michelson interferometer that is operated closeto a dark fringe acts like an almost perfect mirror for both input ports Allthe input light is back-reflected towards the laser source This also accountsfor the quantum uncertainty of the input light The quantum uncertaintythat impinges onto the photo diode thus (mainly) enters the interferometerthrough its (almost) dark port An interferometer that uses displaced co-herent states entering from one port can thus be improved by replacing theordinary vacuum entering the signal output port by a squeezed vacuum stateThis was the proposal by CM Caves in 1981 [Caves (1981)] which is labeledhere with lsquoCSVrsquo If the differential phase quadrature of the interferometer issqueezed Eq (38) then within the limit of large coherent state displacementα sinh2r improves to

∆φCSVmin asymp

eminusrradicn (39)

(The above expression is an approximation since the squeezing operationproduces a small number of photons that are not accounted for here) Ofcourse the mode of the squeezed vacuum needs to be precisely matchedto the mode of the interferometer The first experimental demonstrations ofsqueezed phase measurements used a Mach-Zehnder [Xiao et al (1987)] and apolarization interferometer [Grangier et al (1987)] Fig 2 shows how spatialdegeneracy between an externally generated squeezed mode and the signalmode in a Michelson interferometer is achieved using a polarizing beam split-ter and a Faraday rotator Again the limit in Eq (39) can only be achievedif optical loss is zero Optical loss not only reduces the signal but here alsoreduces the squeeze parameter see Eqs (15) and (16)

Let us consider an example The sensitivity of a laser interferometer thatuses coherent states with an excitation of 1023 photons per second can beimproved by a factor of

radic10 by either adding 09 middot 1024 photons per second

or by adding about just 2 photons per second and bandwidth in hertz thatbelong to the 10 dB squeezed vacuum confer Eq (30) Since the full sig-nal band of ground-based GW detectors covers sideband frequencies up to10 kHz just 2 middot 104 photons per second are necessary At a wavelength ofλ = 1064 nm these values correspond to a power increase by 168 kW and

61

37 fW respectively

The question arises whether a scaling of the sensitivity better than propradic1n is possible It was theoretically shown that in principle the scaling

can indeed considerably be improved yielding the so-called Heisenberg limitor Heisenberg scaling [Bondurant and Shapiro (1984) Yurke et al (1986)Braunstein (1992) Holland and Burnett (1993)]

∆φHLmin prop

1

n (40)

The Heisenberg scaling requires nonclassical states that have a certain num-ber of quanta similar to Fock states ie n = n The theoretically optimalstates describe a superposition of n (N) indistinguishable photons in oneinterferometer arm while having zero (0) photons in the second arm and viceversa and were named lsquoN00Nrsquo-states [Dowling (2008)] A specific propertyof these states is lsquosuper-resolutionrsquo The output ports of the interferometershow an n-times faster oscillation of the interference fringes when changingthe phase between the two interferometer arms Super-resolution correspondsto an n-times improved signal transfer function and was demonstrated forinstance in Refs [Rarity et al (1990) Kuzmich and Mandel (1998) Mitchellet al (2004) Afek et al (2010)] The presence of this nonclassical phe-nomenon however does not prove a sensitivity better than the semi-classicalbound according to Eq (38) Sensitivity is rather related to the signal-to-noise-ratio and needs to take into account all imperfections as well as theprobability of a successful detection of the sensing state [Thomas-Peter et al(2011)] All experiments so far used post-selection on particular measure-ment outcomes and neglected the typically large probability that nothingwas detected

Super-resolution was demonstrated with up to n = 5 [Afek et al (2010)]In addition to the fact that super-resolution does not prove a sensitivity bet-ter than the semi-classical bound photon numbers in state-of-the-art super-resolution experiments are extremely small compared to the photon numberof about 1023 (within a measuring interval of one second) of coherent statesused in Ref [Abbott (2016)] and of about 1022 using coherent states plussqueezed vacuum states used in Ref [Abadie (2011)]( which did prove a sen-sitivity better than the semi-classical bound)

Another interesting and related question is what the smallest phase is

62

that can be estimated in a single measurement again using a given numberof quanta Taking into account that no prior information about the phaseshift exists still a scaling proportional to 1n is possible In the limit of largen however an additional factor of π is required in the nominator of Eq (40)[Sanders and Milburn (1995) Berry and Wiseman (2000)] yielding

∆φHLmin asymp πn (41)

The state that can actually achieve this bound is different from the N00Nstate and was found in [Summy and Pegg (1990) Luis and Perina (1996)Berry and Wiseman (2000)] A N00N state is not the optimum state forphase estimation (via a single measurement) since it only provides one bit ofinformation A recent review on generell aspects on phase measurements isgiven by Ref [Demkowicz-Dobrzanski et al (2015)]

It is important to note that Eqs (38-40) do not consider photon lossExperiments that demonstrated super-resolution and aimed for achievingthe scaling in Eq (40) were conditioned on zero photon loss Let η gt 0 bethe average efficiency of detecting (all) photons Eq (38) then reads

∆φηCohmin =

radic1

η n (42)

Eq (39) turns into

∆φηCSVmin asymp

radicηeminus2r + 1minus η

η n(43)

and Eq (40) turns into [Demkowicz-Dobrzanski et al (2012 2013)]

∆φηmin =

radic1minus ηη n

for 0 lt η lt 1 (44)

For non-zero photon loss most interestingly the ultimate sensitivity of aphase measurement for a given photon number also shows a 1

radicn -scaling

The difference between the CSV strategy of using bright coherent states incombination with squeezed vacuum states which is bounded by Eq (43) andthe strategy of using the optimal nonclassical state which is bounded byEq (44) is marginal in practice [Demkowicz-Dobrzanski et al (2013)] Forgravitational-wave detectors and for any other laser interferometer using in-tense light there is no need for an alternative to the CSV strategy

63

We now turn back to the shot noise according to Eq (38) Generally noisecan be decomposed into its spectral contributions For a simple Michelsoninterferometer without arm resonators and without a signal-recycling cavitythe square-root of the single-sided shot-noise spectral density normalized tothe differential arm length change x in units of m

radicHz is given by [Saulson

(1994)] radicSMI

SNx =

radic~c2

2ωPprop 1radic

P (45)

where ω is the optical angular frequency of the quasi-monochromatic carrierlight and P the total light power in both arms including the built-ups fromcavities In combination with a squeezed vacuum whose relative phase gen-erates squeezing of the output lightrsquos amplitude quadrature the right handside reduces according to the factor eminusr Note that the single-sided spectraldensity is only defined for positive sideband frequencies and thus twice aslarge as the double-sided spectral density

The spectral density of the measurement of a GW induced strain is givenby the same expression but normalized to h = xL (If the gravitational waveis oriented in an optimal way with respect to the Michelson interferometerone arm is squeezed while the other is expanded by the same amount of ∆L =x2 and h then corresponds to the actual gravitational-wave amplitude)The square-root of the single-sided shot-noise spectral density normalized tostrain in units 1

radicHz is given by

radicSMI

SNh =

radic~c2

2L2ωP (46)

Equations (45) and (46) show that the smallest measurable signal (corre-sponding to unity signal-to-shot-noisendashratio) is inversely proportional to thesquare root of the laser power and has a white spectrum for sideband frequen-cies much smaller than the carrier frequency see horizontal line in Fig 23

All first- and second-generation GW detectors use power-recycling andadditional cavities to improve their sensitivities Fabry-Perot arm resonatorsdo not only increase the light power but additionally also increase the signalfor signal frequencies inside the resonator linewidth For lossless Fabry-Perot

64

arm resonators the spectral densities in Equations (45) and (46) need to bemultiplied by the following factor [Kimble et al (2001)]

HFP =

radicL2(γ2

FP + Ω2)

c2 (47)

where γFP = cTFP(4L) is the Fabry-Perot arm resonatorrsquos half bandwidthand TFP is the light power transmission of the input mirror The end mirrorsare assumed to have perfect reflectivity A similar expression can be derivedfor describing the improvement due to signal-recycling [Buonanno and Chen(2001)]

In summary shorter laser wavelengths higher light powers and squeezingof the amplitude quadrature of the interferometer output reduce shot noise ina broadband way ie for all signal frequencies Fabry-Perot arm resonatorsas well as signal-recycling provide improvements mainly for frequencies insidethe resonator linewidths

53 Quantum back-action and quantum radiation pressure noise

In laser interferometers quantum back-action noise results from the un-certainty of the lightrsquos radiation pressure force on the interferometer mirrorsand is also called lsquo(quantum) radiation pressure noisersquo (RPN) Its origin isthe quantum uncertainty of the differential amplitude quadrature XΩ∆Ω ofthe fields in the interferometer arms It results in an uncertain momentumtransfer to the mirrors and thus in an position uncertainty of the mirrors atfuture times with respect to their differential mode of motion [Caves et al(1980)] The physical mechanism of radiation pressure corresponds to anintensity dependent phase shift [Pace et al (1993)]The higher the light power in the arms of a laser interferometer the loweris its shot-noise spectral density see Eq (45) Unfortunately the spectraldensity of quantum back-action noise increases with light power The single-sided force noise spectral density reads

radicSRPNF =

radic8~ωPc2

(48)

Whereas the force noise of the quantum radiation pressure has a white spec-trum the RPN does not since the mirrorrsquos reaction to external periodicforces depends on frequency The link between the Fourier component of an

65

external force F (Ω) and the Fourier component of the displacement x(Ω) isgiven be the mechanical susceptibility HM It reads for an harmonic oscillatorwith mass M

HM(Ω) =1

M | minus Ω2 + Ω2M + iΩΩMQ|

(49)

where ΩM is the oscillatorrsquos resonance frequency and Q its quality factorThe square root of the single-sided spectral density of the RPN normalized

to the displacement of an harmonic oscillator with mass M is then given by

radicSRPNx = HM(Ω)

radic8~ωPc2

(50)

In GW detectors the test mass mirrors are suspended as pendula with highmechanical Q-factors and their centre of mass motion corresponds to that ofa harmonic oscillator The resonance frequencies of the pendula are lowerthan the detection band of interest The mechanical susceptibility is thereforeoften approximated for the so-called free-mass regime as H fm

M (Ω) = (mΩ2)minus1The square root of the single-sided spectral density of the RPN normalizedto differential displacement of two mirrors with each of mass M in a simpleMichelson interferometer is given by [Saulson (1994)]radic

SfmMIRPNx =

radic2~ωPc2m2Ω4

propradicP (Ω ΩM) (51)

where m = M2 is the mirrorsrsquo reduced mass In case of a simple Michel-son interferometer that is enhanced with arm cavities the spectral density inEq (51) needs to be multiplied with the expression given in Eq (47) In com-bination with a squeezed vacuum whose relative phase generates squeezing ofthe output lightrsquos phase quadrature the right hand side reduces according tothe factor eminusr Note if the radiation pressure noise is squeezed the shot noisemust be anti-squeezed or vice versa The radiation pressure noise calibratedto strain of space time is given by the right side of Eq (51) divided by theinterferometer arm length LIn summary heavier masses longer laser wavelengths lower light powersand squeezing of the amplitude quadrature in the interferometer arms re-duce radiation pressure noise

66

1 10 100 1000Frequency [Hz]

Radiation pressure noise

Shot noise

SQL

10-21

10-18

10-15

10-12

radicSx

[mradic

Hz]

ndashndash

ndashndash

Figure 23 Displacement-normalized quantum noise spectral densities ndash Consid-ered is a simple Michelson interferometer with neither arm cavities nor signal recyclingThe two end mirrors (m = 100 g) of the interferometer arms are suspended as pendulahaving a resonance frequency of ΩM2π = 1 Hz and a Q-factor of 107 The interferometeruses quasi-monochromatic light (in coherent states) with a total power of 4 kW Opticalloss and the offset from a dark output fringe is assumed to be negligible Wavelengthλ = 1550 nm The standard quantum limit (SQL) corresponds to the lowest noise achiev-able at a given sideband frequency when varying the light power without using quantumcorrelations

54 Interferometer total quantum noise and the standard quantum limit

Both shot noise and radiation pressure noise contribute to the total quan-tum noise of a given interferometer If they are not quantum correlatedwhich is the case for a conventional Michelson interferometer when detectingthe output lightrsquos amplitude quadrature their variances add up (The resultis not shown in Fig 23) It can easily be deduced from the previous sectionsthat changing the laser power will shift the two quantum noise contributionsHowever the total quantum noise never goes below the standard quantumlimit (SQL) [Braginsky and Manukin (1967)]

Let us consider Fig 23 for sideband frequencies much greater than thependulum resonance Here the test mass mirrors react as free masses whenexerted to external forces The SQL in this free-mass regime is calculated

67

by minimizing the sum of the squares of Eqs (45) and (51) [Saulson (1994)]

Sfmtotx =

~c2

2ω

[1

P+

4ω2

c4m2Ω4P

] (52)

Its derivative reads

dSfmtotx

dP=minus1

P 2+

4ω2

c4m2Ω4 (53)

Setting the above equation to zero provides the optimum laser power versussideband frequency in order to achieve the lowest total quantum noise

P fmopt =

c2mΩ2

2ω (54)

Inserting the optimal light power into Eq (57) provides the square root ofthe single-sided noise spectral density of the free-mass SQL in m

radicHzradic

SfmSQLx =

radic2~mΩ2

(55)

Again m is the reduced mass and dividing by the interferometer arm lengthL yields normalization to the GW-induced strain h Eq (55) shows that theSQL falls off with sideband frequency The corresponding equation for aMichelson interferometer that uses arm cavities readsradic

SfmFPSQLx =

radic~

mΩ2

(1

HFP

+HFP

) (56)

with HFP according to Eq (47)Using the expression for the SQL the square root of the total quantum

noise spectral density of a Michelson interferometer in the free-mass approx-imation can be written asradic

SfmFPtotx =

radicSfmFP

SQLx

2

[1

k+ k

] (57)

with the radiation pressure coupling parameter

k(Ω) =2ωP

mc2Ω2 (58)

68

For a fixed light power and fixed reduced mass of the mirrors the quantumnoise limited interferometer reaches the SQL when k = 1 which is realizedat the angular sideband frequency ΩSQL =

radic2ωP(mc2)

Note that neither squeezing the phase quadrature nor squeezing the am-plitude quadrature of the interferometer light leads to sub-SQL performance[Caves (1981)] also confer Ref [Schnabel (2005)] Fig 3 (left) As we willsee in the next sections the standard quantum limit can be surpassed if shotand radiation pressure noise are correlated Then the total quantum noise isnot given by the sum of the variances ie the sum of the squares in Eqs (45)and (51)

55 Squeezed light for surpassing the standard quantum limit

A measurement with sensitivity better than the standard quantum limit(SQL) is also called a lsquoquantum non-demolition (QND)rsquo measurement [Bra-ginsky and Khalili (1995 1996) Kimble et al (2001)] Several QND tech-niques for laser interferometers were proposed in recent decades [Jaekel andReynaud (1990) Kimble et al (2001) Purdue and Chen (2002) Chen (2003)McClelland et al (2011) Danilishin and Khalili (2012) Graf et al (2014)]What they have all in common is they exploit quantum correlations betweenobservable uncertainties

Arguably the most extensive way of introducing quantum correlationsand surpassing the SQL is the injection of squeezed states of light [Jaekeland Reynaud (1990)] If the squeezed quadrature angle of the injected statesis neither 0 nor 90 the quantum uncertainties of the amplitude and phasequadrature amplitudes that describe the differential field in the two interfe-rometer arms become correlatedLet us consider a very simplified setup that just consists of a quasi-mono-chromatic light field that is back-reflected from a quasi-free mirror Thelight power and the mass be such that reflected light in a coherent stateresults in a measurement of the mirror position with a noise spectral den-sity at the SQL at sideband angular frequency ΩSQL At this frequencyquantum measurement noise and back-action noise are of the same sizeie the uncertainty in X produces an equally large additional uncertaintyin Y Upon reflection the quadrature amplitude variances change from∆2X = ∆2Y = 14 to 2∆2X = ∆2Y = 12 This result corresponds tothe situation in Fig 23 at the crossing frequency of shot noise and radiation

69

pressure noise The coupling of the uncertainty variances can be describedby the matrix K = (1 minusk 0 1) where k = 1 at the SQL If the modulationstate at ΩSQL is the ground state its variances are transferred according to

KT

(1 00 1

)K =

(1 0minus1 1

)(1 00 1

)(1 minus10 1

)=

(1 minus1minus1 2

) (59)

In accordance with Fig 23 the variance of YΩSQLis twice as large as the vac-

uum noise varianceNow let the quantum noise of the light field be 10 dB squeezed at 45

(Eq (19)) The projection of the quantum uncertainty onto the X-observableproduces the radiation pressure noise by being transferred with the couplingfactor k = 1 at the SQL into the Y -observable in fact with negative signsince a larger value of X produces a larger optical path length and thus aretardation of the phase Due to the squeezing at 45 the initial uncertaintyin Y cancels with the additional uncertainty that originates from the onein X The following calculation shows that the strength of the cancellationcorresponds to the initial squeezing strength Upon reflection the quantumuncertainties transform in the following way(

1 0minus1 1

)(505 495495 505

)(1 minus10 1

)=

(505 minus01minus01 02

) (60)

The state of light after reflection has a squeezed phase quadrature amplitudeThe improvement in comparison to Eq (59) is exactly 10 dB The quantumnoise improvement corresponds to the input squeeze factor and is also a mea-sure by what factor the SQL is surpassed Squeezed vacuum injection thusallows surpassing the SQL upon measuring the conventional Y -quadrature(which is realized by a single photo diode in the interferometerrsquos output port)as first realized by [Unruh (1983) Yuen (1983) Jaekel and Reynaud (1990)]

In the example above the input squeeze angle is optimized for a sin-gle sideband frequency Injecting a broadband squeezed vacuum field withfrequency-independent squeeze angle of 45 would result in a rather bad inter-ferometer quantum-noise performance at frequencies far smaller or largerthan ΩSQL Fig 24 shows the quantum-noise performance if the input fieldhas squeeze angles that are optimized for every k(Ω) as given in Eq (58)Injected squeezing can thus lead to a broadband sub-SQL performance ifthe quantum measurement noise (shot noise) and the quantum back-actionnoise (radiation pressure noise) are correlated in an optimal way Due to the

70

10 50 100 500 1000 5000Frequency [Hz]

Dis

plac

emen

t noi

se sp

ectra

l den

sity

[mradic

Hz]

10-20

10-19

10-18

Total quantum noise 0dB

Total quantum noise -10dB

SQL

Y

X

Y

X

Y

X

Figure 24 Surpassing the SQL with squeezed-light injection ndash At shot-noise lim-ited sideband frequencies squeezing of the Y -quadrature amplitude improves the noisespectral density of the interferometer At radiation-pressure-noise limited sideband fre-quencies squeezing of the X-quadrature amplitude improves the noise spectral density ofthe interferometer If both kinds of quantum noise contribute equally (at the SQL markedwith a dot) a squeeze angle of 45 results in surpassing the SQL by the full squeeze factorsee Eq (60) In the graph here the squeeze angle is optimized for all frequencies result-ing in a broadband quantum noise reduction [Jaekel and Reynaud (1990)] Measurementsensitivities beyond the SQL (shaded area) are in the so-called quantum non-demolition(QND) regime [Kimble et al (2001)] Dashed horizontal lines represent the (squeezed)shot noise Dashed straight lines with negative slope represent the (squeezed) radiationpressure noise The calculations use 10 dB of squeezing a conventional Michelson inter-ferometer with neither arm resonators nor signal recycling a light power at the beamsplitter of 1 MW at a wavelength of λ = 1550 nm and mirror masses of 1 kg

correlation shot noise and radiation pressure noise can be squeezed simulta-neously

71

Light with a frequency-dependent squeeze angle

The discovery that shot noise and radiation pressure noise can be squeezedsimultaneously and thus a broadband reduction of quantum noise beyond theSQL be achieved required the insight that the spectral analysis of light definesa spectrum of many lsquosideband modulation modesrsquo that all can be in differ-ent quantum states An ordinary squeezing resonator which is on resonancefor light at twice the pump wavelength produces a spectrum of modulationmodes that all have the same squeeze angle A frequency-dependent squeezeangle can be introduced by reflecting such a field from a detuned single-ended filter cavity which was suggested by Kimble and coworkers [Kimbleet al (2001)] They showed that the optimal frequency dependence thatleads to the broadband improvement shown in Fig 24 can be realized byusing altogether two filter cavities as shown in Fig 25 Motivated by this re-sult research and development on filter cavities for optimizing the frequencydependence of broadband squeezed fields has been very active in recent years[Corbitt et al (2004) Chelkowski et al (2005) Dwyer et al (2013) Kweeet al (2014) Straniero et al (2015) Oelker et al (2016)]

Photo diode Squeezed vacuum

Faraday Rotator

Coherent light

YΩΔΩ

Figure 25 Frequency dependent squeezing injection ndash A broadband squeezed fieldwith a frequency-dependent squeeze angle that is optimal for gravitational-wave detectorsis produced by reflecting off an ordinary broadband squeezed field from two detuned opticalfilters [Kimble et al (2001)]

A light field with a frequency-dependent squeeze angle was first demon-strated in Ref [Chelkowski et al (2005)] see Figs 26 and 27 The experi-

72

Detuned filter cavityLockingphoto diode

Isolator

EOM R asymp 1

SHG

EOM

DBSOPA

LaserIsolator

Mode cleaner

LO

Homodyne detector

Figure 26 Generation of a frequency-dependent squeezing ndash A frequency-dependent orientation of the squeeze ellipse was first demonstrated in Ref [Chelkowskiet al (2005)] Initially a conventional spectrum of squeezed vacuum states of light was gen-erated in a squeezing resonator (lsquoOPArsquo) The squeezed vacuum was transmitted throughan optical isolator to a detuned filter cavity After reflection the squeezed vacuum stateswere absorbed in a balanced homodyne detector (BHD) The phase of the BHDrsquos localoscillator (LO) was changed for quantum state tomography of the squeezed states in diffe-rent regions of the spectrum The result showed a frequency-dependent orientation of thesqueeze ellipse see Fig 27 SHG second harmonic generation EOM electro-optical mod-ulator for applying phase modulation sidebands for cavity length control DBS dichroicbeam splitter R mirror reflectivity λ4 quarter wave plate for turning linear polarizedlight into circular polarised light and vice versa

ment consisted of a standing-wave squeezing resonator which produced ans-polarized broadband amplitude quadrature squeezed field accompanied bya dim continuous-wave DC control field with a wavelength of λ = 2πcω =1064 nm The squeeze bandwidth covered sideband frequencies up to aboutΩ(2π) = 30 MHz which corresponded to the linewidth of the squeezing res-onator The optical cavity for producing the frequency dependence of thesqueeze angle was a standing-wave cavity composed of a plane incouplingmirror of reflectivity r1 =

radic097 and a concave end mirror of reflectivity

r2 =radic

09995 The cavity length was L = 50 cm resulting in a linewidthof 147 MHz The squeezed field first passed a Faraday isolator to preventinterference effects between the filter cavity and the squeezing resonator A

73

Phase quadrature

min

max

-2 -1 0 1 2

141 MHz

-2

-1

0

1

2

Am

plit

ud

e q

uad

ratu

re

Figure 27 Frequency-dependent squeezing ndash Picture top right Reconstructed con-tour plot of the Wigner function of the sideband modulation at Ω(2π) = 141 MHz afterreflection from a 1515 MHz detuned filter cavity The state shows quantum correlationsbetween phase and amplitude quadratures ie squeezing at an angle of here about 40The white circle visualizes the standard deviation of the vacuum state uncertainty Thewhite ellipse represents the standard deviation of the squeezed uncertainty Small picturesMeasurement results on the same continuous-wave laser beam at various sideband frequen-cies around 15 MHz For each tomographic picture noise histograms of 100 equidistantquadrature angles were measured In each case the laser beam was phase locked to a ref-erence beam and the quadrature angle stably controlled and stepwise rotated The phasereference was given by a phase modulation at 198 MHz [Chelkowski et al (2005)] Thepicture was first published in Ref [Schnabel (2005)] (copyright ccopy2007 by Imperial CollegePress)

λ4-waveplate turned the s-polarized field into a circularly polarized beamwhich was then mode matched into the detuned cavity The retro-reflectedfield was analyzed by a balanced homodyne detector (BHD) for quantumstate tomography The filter cavity was electro-optically controlled to be de-tuned by 1515 MHz with respect to the DC control field The cavity length

74

control was achieved by the Pound-Drever-Hall (PDH) locking techniqueutilizing a circularly polarized laser beam that carried 15 MHz phase mod-ulation sidebands and was coupled into the filter cavity from the back Theoutput voltage of the BHD was characterized by a spectrum analyser as wellas used to perform quantum state tomography In the latter case the BHDoutput voltage was mixed down with an electronic local oscillator at differentradio-frequencies around 15 MHz and low-pass filtered to set the resolutionbandwidth to ∆Ω(2π) = 100 kHz The final electric signal corresponds to atime series of quadrature amplitude measurements XθΩi∆Ω Quantum statetomography is a method to reconstruct the phase space quasi-probability dis-tribution (Wigner function) of quadrature amplitudes from sets of measuredXθΩ∆Ω distributions when varying the angle θ [U Leonhardt (1997)] Forevery sideband frequency Ωi 100000 quadrature values were measured di-vided up on 100 equidistant quadrature angles Each quadrature angle wasstably controlled with a precision of plusmn1 Fig 27 shows the reconstructedWigner functions which were all measured on the same laser beam but atdifferent sideband frequencies For these measurements the detuned filtercavity was locked to the lower sideband at minus1515 MHz The result clearlyshows the frequency-dependent orientation of the squeeze ellipse In a morerecent experiment a frequency-dependent squeeze angle was also realized inthe kHz regime [Oelker et al (2016)]

56 Optomechanically induced (ponderomotive) squeezing

The radiation pressure of light when acting on a movable mirror re-sults in an intensity dependent phase shift [Pace et al (1993)] The cou-pling produces a so-called lsquoponderomotive effectrsquo [Braginsky and Manukin(1967)] which is of third order optical nonlinearity and which transforms abright coherent state inside an interferometer into a squeezed state of light[Vyatchanin and Matsko (1993)] This type of squeezed-light generationis usually called lsquoponderomotive squeezingrsquo or lsquooptomechanical squeezingrsquoConsequently even if no squeezed field is injected into the interferometercorrelations between the quadrature amplitudes are generated that allow forsurpassing the SQLPonderomotive squeezing as produced by the interferometer itself can onlybe exploited for evading back-action (radiation pressure noise) It can notbe used to squeeze the interferometer shot noise This is why ponderomotivesqueezing is fundamentally less extensive than injecting externally producedsqueezed states of light [Corbitt et al (2006)] suggested an external pon-

75

deromotive squeezing source for gravitational-wave detectors In this casedue to its external generation also the interferometerrsquos shot noise can besqueezed Recently ponderomotive squeezing was observed for the first time[Brooks et al (2012) Purdy et al (2013)] The achieved squeeze factors aremuch smaller than those produced by optical-parametric down-conversion[Vahlbruch et al (2016)]

Let us have a look again at Eq (59) Rotating the covariance matrix onthe right by arctan(minus

radic54minus 12) asymp minus58 indeed reveals squeezing

(cos 58 minussin 58

sin 58 cos 58

)(1 minus1minus1 2

)(cos 58 sin 58

minussin 58 cos 58

)asymp(

262 00 038

) (61)

The vacuum-noise normalized variance of 038 corresponds to about 42 dBof ponderomotive squeezing This is the general value that is produced atthe angular sideband frequency ΩSQL At higher frequencies the squeezefactor gets smaller at lower frequencies higher The squeezing strength of42 dB can be observed if the photo diode in the interferometer output portis replaced by a balanced homodyne detector using a local oscillator phase ofabout minus58 It can be shown however that the optimal signal-to-quantum-noise-ratio at the SQL is achieved for a local oscillator phase of exactly 45At this angle back-action is fully evaded

Full evasion of radiation pressure noise at all frequencies requires an op-timized frequency dependence of the relative local oscillator phase This canbe achieved by reflecting off the interferometer output field from two detunedfilter cavities [Kimble et al (2001)] The scheme was called lsquovariational out-putrsquo In the case of zero optical loss this scheme can fully evade radiationpressure noise just leaving the shot noise as the only quantum noise contri-bution

The variational-output scheme can be used to enhance the frequency-dependent squeezed input scheme The right site of Eq (60) shows thatthe output statersquos squeezing is not optimally detected in the Y -quadratureRather than with a single photo diode the detection should be done witha balanced homodyne detector with optimized phase of its local oscillatorIn this case the output lightrsquos quantum noise is solely given by squeezedshot noise The total quantum noise in Fig 24 would then be given by thelowest (dashed) horizontal line This combined scheme was called lsquosqueezed

76

variationalrsquo [Kimble et al (2001)] It can be realized by reflecting off theinterferometer output light from in total two optical filter cavities placed infront of the balanced homodyne detector

57 Conclusions

The highest quantum-noise-limited sensitivities of high-precision laserinterferometers are achieved by employing a large number of quanta tomaximize the signal strength in combination with strongly squeezed statesto minimize the quantum noise From this perspective it is clear that thequantum-noise-limited sensitivity of future gravitational-wave detectors willbe further improved ndash by increasing the light power and the squeeze factorTo be able to do so the optical loss in these devices needs to be reduced

In principle the optical loss in laser interferometers can be made smallbut never zero Recent theoretical research has shown that for any non-zero loss the sensitivity scales proportional to 1

radicn at best where n is the

average photon number per measurement This scaling is efficiently achievedby combining strongly displaced coherent states with squeezed vacuum statesof light

If a repeated measurement is not only limited by quantum measurementnoise but also by quantum back-action noise squeezed states of light can beused to simultaneously reduce both ie in the case of an interferometer shotnoise and radiation pressure noise

6 The first application of squeezed light in an operating gravita-tional-wave detector

Squeezed states of light have been successfully used to improve the sensi-tivity of the gravitational-wave detector GEO 600 from 2010 up to the pointwhen this Review was written [Abadie (2011) Grote et al (2013)] Afterdecades of proof-of-principle experiments [Xiao et al (1987) Grangier et al(1987) McKenzie et al (2002 2004) Vahlbruch et al (2005 2006 20072008) Goda et al (2008)] the implementation of a squeezed-light source inGEO 600 has resulted in the first sensitivity improvement beyond shot noiseof a measurement device that targets new observations in nature The im-plementation of squeezed states in GEO 600 was not done to provide anotherproof-of-principle demonstration but was realized because it offered a rela-tively cheap way of further improving the measurement sensitivity Of course

77

the sensitivity of GEO 600 can also be further increased by purely classicalmeans however the implementation of arm resonators to enable higher lightpowers without increasing the thermal load on the beam splitter or eventhe realization of longer interferometer arms are much more expensive Inthis respect the sensitivity improvement of GEO 600 with squeezed light canarguably be regarded as the first lsquotruersquo application that developed out of thefield of lsquononclassical (quantum) metrologyrsquo (Note that the term rsquoquantummetrologyrsquo is currently defined in different ways [Giovannetti et al (2006)Gobel and Siegner (2015)] and the term rsquononclassicalrsquo referring to a non-classical P-function gives a distinct description)

61 Gravitational waves

Einsteinrsquos General Theory of Relativity [Einstein (1916)] or simply lsquoGen-eral Relativityrsquo (GR) predicts that accelerating mass distributions producegravitational radiation analogous to electromagnetic radiation from accel-erating charges Experimental evidence of their existence is given by theobservation of the slow spiraling together of two neutron stars caused bythe loss of orbital energy to gravitational waves The inspiral rate exactlymatches the predictions of Einsteinrsquos theory [Weisberg and Taylor (2005)]Recently Advanced LIGO observed gravitational waves for the first time[Abbott (2016)] thereby giving the go-ahead for gravitational-wave astron-omy The gravitational-wave source was the final inspiraling and the mergerof two black holes 13 billion light years away from earth

Gravitational-waves evolve in the far field of the source propagate withthe speed of light and are measurable on earth with laser interferometersFig 28 displays a gravitational wave propagating along a certain directionGravitational waves are dynamical changes of space-time They are transver-sal and quadrupolar in nature and have two polarization states

A variety of known astrophysical and cosmological sources are predictedto emit gravitational radiation that should reach the Earth with a measur-able strength [Sathyaprakash and Schutz (2009)] The first gravitational waveevent detected was produced by two black holes of 36 and 29 solar massesDuring the final 02 seconds of their inspiraling they produced a peak gravi-tational strain in our solar system of 10minus21 covering frequencies up to 250 Hz[Abbott (2016)] Other predicted sources are mergers of neutron stars supernovae and background signals from the Big Bang According to GR GWsfrom complex astrophysical sources carry a plethora of information that willhave a major impact on gravitational physics astrophysics and cosmology

78

L +Δ L

Binary system

fBS

Figure 28 Space-time oscillation ndash Gravitational waves are dynamical deformations ofspace-time that form in the plane perpendicular to the direction of wave propagation Asa result distances between free-falling test masses in a transverse plane will change witha strain h = ∆LL For black hole or neutron star binary systems with orbital frequencyfBS distances will oscillate at frequency fGW = 2fBS The wavelength of this oscillation isgiven by λGW = cfGW where c is the speed of light The wave of orthogonal polarizationwith respect to the one shown is rotated by 45 around the propagation axis

62 Interferometric detection of gravitational waves

Current gravitational wave detectors are kilometre-scale laser interfero-meters [Dooley et al (2016) Aasi (2015) Acernese (2015) Aso et al (2013)]Continuous-wave laser light is split into two beams traveling in orthogonaldirections Both beams are reflected back towards the central beam splitterwhere they interfere Gravitational waves change the optical path lengthdifference and thus the light power directed towards the photo-diode thatis positioned in the signal output port of the beam splitter A gravitationalwave at frequency fGW = ΩGW(2π) reveals itself as a light-power modula-tion at the same frequency The spectral decomposition of the output signalis described by a spectrum of the quadrature amplitude YΩ∆Ω introduced inSec 3 It corresponds to the amplitude quadrature amplitude of the outputlight and relates to the differential phase quadrature of the interferometerarms

79

Photo diode

Squeezed vacuum

Faraday Rotator

Signal

Quantum noise

Coherent light XΩΔΩ

YΩΔΩ

(a) (b)

YΩΔΩFigure 29 Squeezed-light-enhanced interferometric measurement ndash (a) Michelsoninterferometer with arm cavities power recycling and signal recycling (see main text forexplanation) The interferometer is operated close to a dark fringe such that the quantumnoise entering from the dark port is back-reflected The squeezed field is mode-matched tothe signal output field (b) Phase space diagram of the gravitational-wave signal outputat sideband frequency Ω(2π) The quantum noise is squeezed below the ground stateuncertainty and thus the signal to quantum noise ratio improved

The first key ingredient of an interferometric gravitational-wave detectorare suspended heavy mirrors that can be regarded as quasi-free in the di-rection of laser light propagation thereby acting as test masses that probespacetime Being on ground current detectors are located in rather noisy en-vironments that allow the realization of undisturbed quasi-free mirrors onlyabove a sideband frequency of the order of 10 Hz Since sufficiently strongGW signals are expected up to a frequency of 10 kHz todayrsquos gravitationalwave detectors target at signals in the acoustic band from 10 Hz to 10 kHzThe quasi-free motion of the test mass mirrors in this frequency regime isachieved by suspending the mirrors as sophisticated multiple-stage pendulain vacuum chambers [Aasi (2015)] Far above the pendularsquos resonant frequen-cies which are typically around 1 Hz the centre of masses of the mirrors areisolated from vibrations of the ground and they react on frequency compo-nents of small external forces approximately as free masses The mirrors andtheir suspensions are built from materials having exquisitely high mechani-cal quality factors This helps to concentrate the thermal energy that causes

80

displacements of the mirror surface into well-defined vibrational frequencymodes At these particular very sharp frequencies no gravitational wavescan be detected

The second key ingredient of an interferometric gravitational-wave de-tector is laser light with a power of up to hundreds of kilowatts or evenmegawatts The light is quasi-monochromatic and needs to show very lowamplitude (quadrature) noise and phase (quadrature) noise at sideband fre-quencies within the detection band Low amplitude noise is necessary toprovide a shot noise limited output field It is also necessary to avoid classi-cal radiation pressure noise which becomes an issue if the light power or themirror masses in the two arms are not identical Low phase noise is requiredif the storage time of the light in the two arms is not identical This mightaccidentally occur due to different linewidths of the arm cavities or mightbe part of the interferometer design to allow for the length control schemeproposed by Schnupp [Heinzel et al (1998)] To maximize the light power in-side the interferometerrsquos cavities it should be produced in an almost perfecttransversal spatial distribution of a Gaussian TEM00 mode

Light sources of gravitational-wave detectors are ultra-stable NdYAGmaster-slave systems that provide up to 200 W of light at 1064 nm [Winkel-mann et al (2011) Kwee et al (2012)] The high power in the interferometerarms is achieved by cavity built-ups in the so-called power-recycling cavityand in the arm cavities Power recycling uses a partially reflective mirror thatis located between the light source and the interferometer beam splitter Itssurface is matched to the lightrsquos wave front and forms an optical cavity to-gether with the rest of the interferometer Since gravitational-wave detectorsare operated close to a dark fringe large power built-ups can be achievedThe highest power built-up is achieved for a mirror transmission equal to the(given) interferometer round trip loss In this case an impedance-matchedcavity is achieved The power-recycling cavity as well as the arm cavitiesare stabilized on resonance for the input light The difference between theirfunctionality is that the power-recycling cavity does not limit the detectionbandwidth of the interferometer GEO 600 as well as Advanced LIGO em-ploy a third type of cavity the so-called signal-recycling cavity Similarlyto power recycling a partially reflecting mirror that is placed between theoutput port of the beam splitter and the photodiode is used to resonantly en-hance the GW signal [Meers (1988)] The signal-recycling cavity resonantlyenhances the signal modulation fields within its linewidth without furtherenhancing the carrier light power In combination with low linewidth Fabry-

81

Perot arm resonators it can also be used to extract the signal by reducing theeffective finesse of the arm resonators for the signal sidebands This schemeis called resonant sideband extraction [Heinzel et al (1996)] The signal-recycling cavity has also been tested in a detuned setting in which just theupper or lower sideband is extracted or resonantly enhanced respectively[Heinzel et al (2002)] Current gravitational-wave detectors however usecarrier-tuned signal recycling

All these techniques are lsquoclassicalrsquo approaches for maximizing the signal-to-shot-noise ratio At frequencies above a few hundred Hertz howevershot-noise is still the limiting noise source in gravitational-wave detectorsFuture gravitational-wave detectors will therefore use even higher light pow-ers but further increasing the light power becomes more and more challeng-ing Optical materials with less light absorption need to be found to coun-teract an increasing thermal load inside the mirrors Mirror masses need tobe further increased to counteract the increasing radiation pressure noiseNonclassical approaches are superior and become more and more attractivethe farther classical approaches are pushed to the extremes Nonclassicalapproaches allow for simultaneously increasing the signal-to-shot-noise ratioand the signal-to-radiation-pressure-noise ratio without changing light poweror mirror masses see Fig 24 They also allow for a complete evasion of radia-tion pressure noise [Braginsky and Khalili (1995 1996) Kimble et al (2001)]see Subsec 56

63 Squeezed-light enhancement of the gravitational-wave detector GEO 600

In 2010 GEO 600 was equipped with the squeezed-light source shownin Fig 18 The location of the squeezed-light source close to the outputport is shown in Fig 30 It was known that GEO 600 was shot-noise limitedat sideband frequencies above about 700 Hz In this frequency regime thereplacement of the ordinary vacuum states that entered the interferometerfrom the output port by a spectrum of squeezed vacuum states was expectedto reduce the noise spectral density into the nonclassical regime It was notprecisely clear what squeezing factors could be expected since the opticalloss upon mode-matching an external field into the output port propagationalong the arms and the final photo-electric detection was not determined

Critical components were the quantum efficiency of the photo-diode aswell as the optical loss of Faraday rotator for overlapping the squeezed fieldwith the output mode Also the transversal output mode of GEO 600 was an

82

issue since it contained significant contributions from higher order modesto which a good mode matching of the squeezed mode was not possible Theimplementation of the squeezed-light source thus was accompanied with anew custom made InGaAs photo-diode with 3 mm diameter The goal was aquantum efficiency of greater 99 [Vahlbruch et al (2016)] Also the Fara-day rotator was custom-made and optimized for lowest optical loss which

Figure 30 GEO 600 ndash View into the central building of the British-German GW detectorlocated close to Hannover Germany The vacuum chambers contain the suspended beamsplitter power- and signal recycling mirrors additional input and output optics as well asmirrors to realize a double pass of the laser light through the 600 m long interferometerarms By courtesy of the AEI

83

involved a rather precise rotation of the polarisation of 45 plusmn 05 over anaperture of more than 15 mm Finally a ring cavity (output mode cleaner)was placed in front of the photo-diode which acted as a passive filter forhigher transversal modes Since GEO 600 was not limited by radiation pres-sure noise and since it used a carrier-tuned signal-recycling cavity a frequencyindependent orientation of the squeezing angle was optimum After several

100 200 300 500 700 1000 2000 3000 500010-22

10-21

10-20

Obs

erva

tory

noi

se c

alib

rate

d to

str

ain

[1radic

Hz]

10-19

10-18

10-17

Noise calibrated to test m

ass displacement [m

radicHz]

Sideband frequency f

Figure 31 Nonclassical reduction of the GEO 600 instrumental noise ndash Shownare the square roots of the noise spectral densities without squeezed vacuum states (top)and with squeezed vacuum states (bottom) [Abadie (2011)] Their calibrations [Affeldtet al (2014)] to gravitational-wave strain and differential arm length change are shownon the left and right y-axes respectively Note that both traces increase towards higherfrequencies due to the frequency-dependent signal enhancement of the signal-recyclingcavity The injection of squeezed vacuum states leads to a broadband noise reductionof up to 35 dB at shot-noise limited frequencies The spectral features are for instancecaused by excited violin modes of the mirror suspensions (600ndash700 Hz and harmonics)Data by courtesy of the AEI

months the combination of the squeezed-light source and the gravitational-wave detector succeeded The main laser of the squeezed-light source wasphase locked to the main laser of GEO 600 and a stable mode matching be-

84

tween the squeezed field and the interferometer output field was achieved[Abadie (2011)] The effect on GEO 600rsquos (strain normalized) noise spectraldensity is shown in Fig 31 At frequencies above about 700 Hz the squareroot of noise spectral density was reduced by up to one third This value cor-responds to a quantum noise variance of 045 (minus35 dB) with the shot noisevariance normalized to unity For isotropically distributed gravitational-wavesources this factor produces a detection rate increase by a factor of 153 asymp 34After its integration into GEO 600 the squeezed-light source was used in allscientific runs seeking for gravitational waves for instance in the observa-tional run S6eVSR4 that was undertaken from June 3rd to September 5th

in 2011 [Grote et al (2013)]Towards the end of 2011 right before the start of the detector upgrade

to Advanced LIGO a nonclassical sensitivity improvement was also demon-strated in one of the LIGO detectors [Aasi (2013)] In this experiment anonclassical sensitivity improvement corresponding to up to 215 dB abovefrequencies of about 150 Hz was achieved The successful test is a strongmotivation for a squeezed-light upgrade of Advanced LIGO Note that thedesign of Advanced LIGO was completed in 1999 and squeezed-light sourceswere not mature at those times

In the past years the squeezing enhanced GEO 600 detector was not onlyused for observations but also was the control of the injected squeezed modefurther improved Stabilizing the overlap between squeezed mode and brightmode of the interferometer to close to perfect is necessary to reduce theeffective optical loss and to maximize the measurable squeezing factor Re-cently superior methods for stabilizing the longitudinal phase of squeezedvacuum mode were found [Dooley et al (2015)] and the first automatic align-ment system for stabilizing and optimizing the transversal mode overlap wasdemonstrated [Schreiber et al (2016)]

64 Are squeezed states the optimal nonclassical resource ingravitational-wave detectors

For a given number of photons Eq (41) quotes the ultimately smallestphase change that can be measured with a signal-to-noise-ratio of one Thescaling with number of photons per measuring time of this Heisenberg limitseems appealing compared to the scaling achievable with coherent states orsqueezed states according to Eqs (38) and (39) respectively The Heisenberglimit however is only valid for precisely zero photon loss Since the non-classical states required to achieve Eqs (40) and (41) show an exponentially

85

increasing sensitiveness to loss when increasing the photon number the ac-tual scaling can not be deduced from Eq (40) Proposals to use Fock statesand the so-called N00N states for optimizing interferometer sensitivities [Hol-land and Burnett (1993) Dowling (1998) Mitchell et al (2004) Afek et al(2010)] are thus only applicable when the experiment is conditioned on zerophoton loss As discussed in recent publications the correct expression forthe fundamental sensitivity limit needs to consider not only the total photonnumber inside the interferometer but also the total photon loss [Dorner et al(2009) Ko lodynski and Demkowicz-Dobrzanski (2010) Knysh et al (2011)Escher et al (2011) Demkowicz-Dobrzanski et al (2012)]

Based on these earlier works Ref [Demkowicz-Dobrzanski et al (2013)]proved that the nonclassical sensitivity enhancement of GEO 600 reportedin Ref [Abadie (2011)] has been exceedingly close to fundamental quantuminterferometry bound under given energy constraints and photon loss lev-els More than that it was generally proven that the approach of com-bining displaced coherent states and squeezed vacuum states is optimal forgravitational-wave detectors

In Ref [Abadie (2011)] the gravitational-wave detector GEO 600 used aneffective number of photons per second of approximately n = 2 middot 1022 whichcorresponded to a total optical power inside the interferometer arms of P asymp37 kW at a wavelength of 1064 nm The total optical loss was 1minus η asymp 038The injected squeezing factor was eminus2r asymp 01 For these numbers the ratioof Eqs (43) and (44) is calculated to

∆φCSVmin

∆φgenmin

asymp

radicηeminus2r + 1minus η

1minus ηasymp 108 (62)

which is a good approximation within the limit of large coherent state dis-placements α sinh2r The quantum noise of GEO 600 including thesqueezed-light source was just 8 above the fundamental quantum inter-ferometry bound An increased squeezing strength of 16 dB (eminus2r asymp 0025)which is in reach would bring the approach based on coherent states andsqueezed vacuum states to within just 2 above the fundamental bound

Future GW detectors will have significantly reduced optical loss values(1 minus η) lsquoLossrsquo includes scattering and absorption at mirrors non-perfectfringe contrasts and the non-perfect quantum efficiency of the photo detec-tor Optical loss reduction is important for at least four reasons First itleads to an increased signal second it leads to a reduced quantum noise when

86

employing squeezed states third less absorption reduces the thermal load onthe test mass mirrors and fourth less scattering reduces the probability ofback-scattered light which produces disturbance signals [Billing et al (1979)Vahlbruch et al (2007) Punturo et al (2014)] The higher the finesse valuesof the arm and signal-recycling cavities are the more significant is opticalloss at mirror test masses the beam splitter and the signal-recycling mirrorThe finesse value of the power-recycling cavity and the loss of mirrors andlenses that guide the output field to the photo-diode are less critical Suitablephoto detectors of 995 quantum efficiency are available today [Vahlbruchet al (2016)] but achieving a total optical loss of 10 is still challengingThe reason for that is that first of all a measurement device aiming for bestabsolute sensitivity should use as much quanta (photons) as possible Highfinesse values for the enhancement cavities are thus essential but results inan unavoidable scaling-up of the effect of mirror losses A realistic exampleof future gravitational wave detectors thus considers η = 09 with a squeezingfactor of 20 dB (eminus2r = 001) In this case the quantum noise will be about4 above the ultimate fundamental bound for a given photon number

From Eq (62) it can be concluded that there is no need for any moresophisticated nonclassical states than squeezed states In particular non-classical states with a defined photon number such as N00N states are notrequired Within the approximation quoted this result is independent ofthe photon number This result is also independent of the physical systemused for interferometric phase estimation and can also be made for quantum-enhanced atomic clock calibration in the presence of dephasing Here the-oretical results indicate that the precision of Ramsey interferometry withspin-squeezed states is close to the optimal one in the asymptotic regime ofa large number of atoms [Huelga et al (1997) Ulam-Orgikh and Kitagawa(2001) Escher et al (2011)] as already stated in Ref [Demkowicz-Dobrzanskiet al (2013)] More sophisticated nonclassical states with fixed number ofn quanta might still be useful for the exceptional case when the absorptionof one quantum already results in zero measurement sensitivity anyway Anexample is an ensemble measurement where the absorption of a single photondemolishes the source of the phase change to be characterized A typicallyused approach of conditioning the measurement result on n clicks of n singlephoton counters conditions on precisely zero loss and is thus able to use theadvantage of Eq (40) over Eq (39)

87

65 Conclusions

Squeezed states of light will contribute to realizing gravitational-wave ob-servatories with much higher sensitivities than existing or planned ones Tobenefit from squeezed states in a most efficient way optical loss in terms ofabsorption and scattering must be minimized In particular the optical lossof mirror coatings and mirror substrates need to be minimized The rele-vant mirrors include the test masses the balanced beam splitter the signalrecyclingextraction mirror and all optical components between the latterand the photo diode Excellent spatial mode matching between the brightinterferometer field and the squeezed vacuum field is also of great impor-tance Achieving this requires further improvement of the surface figures ofall reflective optical components of the interferometer as well as improvedhomogeneity of all optical components that the light passes through

The quantum noise reduction achieved in a gravitational-wave detectoris of course always smaller than the highest squeeze factor provided by thesqueezed-light source As an example let us consider the observation of 15 dBof nonclassical noise suppression directly at the source If the squeezed fieldsenses an additional loss of 5 when propagating through the interferometerwhich is a very challenging number from todayrsquos point of view the remainingsqueezing level is about 11 dB see Eq (16)

7 The application of 2-mode-squeezed light in laser interferome-ters

71 Quantum Dense Metrology

At first glance the application of bi-partite (two-mode) squeezed statesto a device whose goal is measuring a single observable seems meaninglessSqueezing the uncertainty of that observable should be the optimum one cando This is indeed true when concerning just quantum noise but recentlyit was discovered that in the presence of classical disturbances bi-partitesqueezing can improve such measuring devices [Steinlechner et al (2013)]The concept was named quantum dense metrology (QDM) The potentialimprovement of a gravitational-wave detector with bi-partite squeezed statesis shown in Fig 32 (a) A description is given in the caption The pre-condition for a potential improvement can be best understood within a phasespace diagram Fig 32 (b) contains two different kinds of lsquosignalsrsquo The firstis the actual signal which always shows up as a phase space displacement

88

along the Y axis The second is a disturbance signal that can produce adisplacement in arbitrary direction in phase space A prominent example

Faraday Rotator

Coherent light

Photo detectors Balanced homodyne detectors

Squeezed vacuum

Squeezed vacuum

Quantum noise in

Y (A) and X (B)

XΩΔΩ(B)

YΩΔΩ(A)

(a) (b)

Disturbance projected

onto Y

and X

XΩΔΩ(B)

YΩΔΩ(A)

Example of (unknown) disturbance

Signal

Figure 32 Bi-partite-squeezed-light-enhanced measurement ndash (a) Setup for theapplication of bi-partite (two-mode) squeezed light in a laser interferometer on the basisof QDM Two squeezed vacuum fields are overlapped on a balanced beam splitter withprogrammable squeeze angles for instance with a relative angle of 90 which produces abi-partite state as shown in Fig 12 The beam splitter outputs are entangled for any rela-tive angle greater than zero One part is matched to the interferometer mode The secondpart is kept outside as a reference beam The interference of the interferometer outputand the reference beam is arranged with such a phase difference that it reproduces the twosqueezed inputs on the photo detectors The two squeezed beams are photo-electricallydetected measuring the respective squeezed quadrature (using balanced homodyne detec-tors) Both beams carry half of all interferometer induced modulations which includesignals as well as disturbances A single readout as shown in Fig 21 cannot distinguishbetween the two kinds The double readout shown here provides additional informationand allows for recognition of the disturbance [Steinlechner et al (2013)] as well as in prin-ciple a modeling of the disturbance and with a correct model an improvement of thenoise spectral density of the interferometer [Ast et al (2016)] (b) Phase space diagramdescribing phase quadrature readout A as well as amplitude quadrature readout B Bothshow squeezed quantum noise The amplitude quadrature readout does not contain anygravitational-wave signal ie any feature in this channel must be due to disturbancesThis information can be used to improve the interferometer

89

for such a disturbance is parasitic interference due to back-scattered laserlight [Vahlbruch et al (2007)] Back-scattering is a limiting noise at lowsignal frequencies of gravitational-wave detectors [Billing et al (1979) Vinetet al (1997) Hild (2007) Ottaway et al (2012) Punturo et al (2014)] Notethat all noise that couples in via unwanted motions of the test mass mirrorsso-called lsquodisplacement noisersquo always produces a phase space displacementalong the Y axis and cannot be tackled with QDM

Fig 33 shows measurement results obtained in Ref [Steinlechner et al(2013)] In a table-top experiment one part of a bi-partite squeezed state ofa continuous-wave mode at 1064 nm was mode-matched into the output portof a Michelson laser interferometer operated at its dark fringe in full analogyto Fig 32(a) A lsquosignalrsquo was produced by driving the piezo behind one of theend mirrors at a frequency of 555 MHz The lsquodisturbancersquo was introducedby re-injecting a small amount of light that leaked through the second endmirror with an additional piezo-mounted mirror The piezo was driven ata frequency of 517 MHz to produce a phase modulation An additionalDC voltage defined an arbitrary and unknown optical path length of thelight before being re-injected and as such the phase space orientation of thedisturbance signal This mechanism of a parasitic interference is realizednaturally in any interferometric device due to back-scattering of quanta frommoving surfaces in the environment

The interferometer output consisted of the signal as well as the distur-bance with a quantum uncertainty given by one subsystem of the bi-partiteentanglement It was overlapped with the second subsystem of the entan-gled state on a balanced beam splitter and the two outputs were analysedwith balanced homodyne detectors The phases of the bi-partite entangle-ment and the BHD local oscillators were controlled to resemble Fig 32(b)ie both BHDs measured a squeezed uncertainty regardless of the phase ofthe (generally unknown) disturbance

The beam splitter that combines interferometer output and the entangledreference beam unavoidably splits the signal as well as the disturbance intotwo paths For a balanced beam splitter this generally reduces the signaland disturbance power by 3 dB for both quadrature measurements Fig 33shows however that both BHDs performed about 6 dB below shot noisewhich demonstrates the usefulness of the scheme The squeeze factor can inprinciple be infinite which thus qualifies the lsquo3 dB penaltyrsquo In the abovefigure the additional information from the second BHD output was used torecognize the parasitic interference in the first BHD output providing a lsquovetorsquo

90

50 51 52 53 54 55 56 57

50 51 52 53 54 55 56 57-70

-75

-80

-85

-90

-70

-75

-80

-85

-90

Y (A) (Ω2π)Ω∆Ω

Frequency ( f )

Noi

se p

ower

(dB

m)

Noi

se p

ower

(dB

m)

X (B) (Ω2π)Ω∆Ω

~ ndash6 dB

~ ndash6 dB

Shot noise reference

Shot noise reference

Signal Projected disturbance

Projected disturbance

Vet

o

No signal

Figure 33 Bi-partite-squeezed-light-enhanced measurement ndash The result wasachieved in a table-top setup [Steinlechner et al (2013)] In the two panels the lower

(blue) traces show the squeezed quadrature noise-power spectra ∆2Y(A)Ω∆Ω(Ω2π) (top) and

∆2X(B)Ω∆Ω(Ω2π) (bottom) as simultaneously measured with balanced homodyne detectors

lsquoArsquo and lsquoBrsquo respectively The conventional Y -measurement (top) cannot distinguish be-tween signal and disturbances The additional X-measurement (bottom) does not detectany phase quadrature signal thus any feature in this measurement is a parasitic signal dueto a disturbance The respective projection onto the Y -measurement can thus be lsquovetoedrsquoIn a more sophisticated approach the X-data might be used to model and then to elim-inate the disturbance as well as its projections on both quadrature measurements Theresult is a reduced spectral density of the actual phase quadrature measurement [Ast et al(2016)] Traces shown here are slightly sloped due to the decreasing transfer functions ofthe balanced homodyne detectors The resolution bandwidth was ∆Ω(2π) = 10 kHz thevideo bandwidth was 100 Hz All traces were averaged three times

signal to trigger its removal from the data streamThe question arose whether the additional information can be used to

reduce the actual noise spectral density of the first measurement ie to re-cover signals that were buried by parasitic interferences Very recently it wasshown that this is indeed possible Ref [Ast et al (2016)] reports a table-top proof-of-principle experiment in which the additional information in the

91

QDM approach could be used for improving the sensitivity of an interfero-meter The measurement sensitivity was improved from above-shot-noise tosub-shot-noise (sub-Poissonian) performance This result was possible notbecause the way the parasitic interference arose was known but because theadditional information provided by QDM allowed for fitting a model of theexcess noise to the readout data

Quantum dense metrology (QDM) as shown in Fig 32 improves a mea-surement by simultaneously reading out two conjugate observables Bothreadout observables show a squeezed quantum noise and act as estimatorsof independent physical quantities This situation was recently described aslsquoquantum-mechanics freersquo [Tsang and Caves (2012)] QDM is based on anEinstein-Podolsky-Rosen (EPR) entangled [Einstein et al (1935)] bi-partitesystem as described in Subsec 34 EPR entanglement was previously con-sidered for the quantum-informational task of dense coding which doublesthe capacity of quantum communication channels [Bennett et al (1992)Braunstein and Kimble (2000)] The application of EPR entanglement inmetrology was first proposed by DrsquoAriano et al [DrsquoAriano et al (2001)]

72 Conclusions

A single beam that carries an optimized spectrum of squeezed vacuumstates and that is injected into the interferometerrsquos dark port provides themost efficient and practically optimal approach to reduce the quantum noisein laser interferometers by means of nonclassical states (see Section 5) Theconclusion of the section here is that two entangled beams provide a superiorapproach if the interferometerrsquos sensitivity is limited by classical noise thatis not exclusively restricted to the actual observable which is the phasequadrature amplitude Y Parasitic interferences due to laser light that isbackscattered from vibrating surfaces are an example Current gravitational-wave detectors use light fluxes of about 1024 photons per second [Abbott(2016)] Just a single photon per second and hertz which leaves the mainlight beam and is backscattered from a vibrating surface and in this way getsfrequency shifted into the detection band produces a significant disturbancesignal The lsquoquantum-dense metrologyrsquo approach might provide a powerfultechnique to tackle this problem

Very recently it turned out that QDM is not the only technique that mayexploit EPR entanglement to improve phase measurements Ref [Ma et al(2017)] proposes to use EPR entanglement to simultaneously suppress shotnoise and radiation pressure noise in a gravitational-wave detector without

92

the need for an additional filter cavity (confer subsection 55) In this caseEPR entanglement is exploited that is carried by one broadband squeezedbeam and that is present between quadrature amplitudes defined with re-spect to different optical frequencies ω and ωprime as investigated in Ref [Hageet al (2010)] Such lsquofrequency multiplexedrsquo EPR entanglement might resultin considerably lower costs of building a gravitational-wave detector with abroadband simultaneous squeezing of shot noise and radiation pressure noiseAlso this proposal does not lead to a fundamentally lower quantum noise butrather improves on classical aspects of an interferometer

8 Summary and Outlook

In many cases experiments that involve interference of quantum statescan be described in a semi-classical way This description uses the classicalwave picture for the interference part of the experiment and subsequently theclassical particle picture when the states transfer their energy to a detectoror more generally to a thermal bath This semi-classical description is notpossible when using the specific class of lsquononclassicalrsquo states Squeezed statesof light are a prominent example of these Squeezed states and other nonclas-sical states allow for observations that made Einstein Podolsky and Rosenformulate their critical and seminal paper on quantum theory [Einstein et al(1935)]

In the review here it is argued that after many successful proof-of-principle experiments with nonclassical states in the past decades the routineuse of squeezed-light in observational runs of the gravitational-wave detectorGEO 600 goes beyond proof-of-principle and is a true application of nonclassi-cal light Since 2010 the squeezed-light source has improved the measurementsensitivity of GEO 600 in basically every observational run [Abadie (2011)Grote et al (2013) Dooley et al (2016)] At quantum noise limited frequen-cies ie above a few hundreds of hertz the sensitivity has been improvedcorresponding to a squeezing strength in the noise spectral density of up to37 dB which corresponds to an increase of the average gravitational-wavedetection rate by a factor of 043minus32 = 36 This success is a strong motiva-tion to also equip the Advanced LIGO Virgo and Kagra gravitational-wavedetectors with squeezed light Similar improvement factors even down tolower signal frequencies are expected [LSC (2013)] The achievable improve-ment factors are mainly limited by the optical loss on the squeezed states

93

and much higher factors are achievable in principleUp to now squeezed states have not been used to reduce the radiation-

pressure noise in gravitational-wave detectors The reason is that so far othernoise sources are larger than radiation pressure noise and such an effect can-not be observed It is expected however that future gravitational-wavedetectors will eventually be partly limited by radiation pressure noise Fromthis point on squeezed light will be used to simultaneously reduce shot noiseand radiation pressure noiseSqueezed states are the optimum nonclassical states for gravitational-wavedetectors or more generally for all laser interferometers operating with largeaverage photon numbers per measuring interval [Demkowicz-Dobrzanski et al(2013)] In addition to using higher light powers and heavier test mass mir-rors higher squeeze factors will thus contribute to mitigate the lightrsquos quan-tum noise in laser interferometers

Two-mode (bi-partite) squeezed light has not been used in gravitational-wave detectors so far They are not capable of further reducing the quantumnoise in laser interferometers but they can be used to mitigate classicalnoise that originates from fluctuating phase space displacements A well-known such noise source is back-scattered light Proof-of-principle experi-ments were performed recently [Steinlechner et al (2013) Ast et al (2016)]This new technique could turn out to be valuable in next generations ofgravitational-wave detectors in particular in those targeting high sensitivi-ties at low sub-audio signal frequencies and using high light powers Suchan implementation in gravitational-wave detectors does not require any newtechnology Compared to a squeezed-light enhanced interferometer just asecond squeezed-light source is required

It is certainly remarkable that those quantum states that made EinsteinPodolsky and Rosen falsely think quantum theory incomplete are now ex-ploited as new technologies in measurement devices targeting new observa-tions in nature

Acknowledgements

RS thanks M Ast J Bauchrowitz C Baune S Chelkowski J DiGugliel-mo A Franzen B Hage J Harms A Khalaidovski L Kleybolte NLastzka M Mehmet S Steinlechner and H Vahlbruch for their contri-butions many fruitful discussions and their support with the figures and JFiurasek for many valuable comments on the manuscript Thanks are also

94

due to Y Chen F Khalili and H Miao for fruitful discussions within thequantum noise working group of the LIGO Scientific Collaboration (LSC)Special thanks are due to H Vahlbruch and H Grote together with theGEO 600 team for their pioneering work on the squeezed-light implementa-tion in GEO 600 RS is supported by the Deutsche Forschungsgemeinschaft(Grant No SCHN 757-6) and by the European Research Council (ERC)project lsquoMassQrsquo (Grant No 339897)

References

References

Aasi J et al Jul 2013 Enhanced sensitivity of the LIGO gravitationalwave detector by using squeezed states of light Nature Photonics 7 (8)613ndash619URL httpwwwnaturecomdoifinder101038nphoton2013177

Aasi J et al Apr 2015 Advanced LIGO Classical and Quantum Gravity32 (7) 074001URL httparxivorgabs14114547httpstacks

ioporg0264-938132i=7a=074001key=crossref

20895763c84bce3f8929251031b2475c

Abadie J et al Sep 2011 A gravitational wave observatory operatingbeyond the quantum shot-noise limit Nature Physics 7 (12) 962ndash965URL httparxivorgabs11092295httpwwwnaturecom

doifinder101038nphys2083

Abbott B P et al Feb 2016 Observation of gravitational waves from abinary black hole merger Phys Rev Lett 116 061102URL httplinkapsorgdoi101103PhysRevLett116061102

Acernese F et al 2015 Advanced virgo a second-generation interferome-tric gravitational wave detector Classical and Quantum Gravity 32 (2)024001URL httpstacksioporg0264-938132i=2a=024001

95