early history and fundamentals of optomechanics...maxwell’s theory of electromagnetism: in terms...

Early history and fundamentals

of optomechanics

Antoine Heidmann

Optomechanics and Quantum measurement team

Laboratoire Kastler Brossel Ecole Normale Supérieure, Université P. et M. Curie,

Collège de France, CNRS, France

New and emerging research field:

interaction between a laser beam and …

… a mechanical resonator via radiation pressure

Suspended mirror

Micro-resonator

Introduction to quantum optomechanics

Addresses some fundamental aspects of Quantum Mechanics

2

Introduction to quantum optomechanics 3

Radiation pressure: energy carried by an electromagnetic field Radiation pressure: energy carried by an electromagnetic field

Sun's light at Earth : 1 kW m 3 Pa 10 pN = 1 mm

Solar wind and radiation pressure

Sun s light at Earth : 1 kW m 3g1 W laser force of few nN

First attempts to observe radiation pressure

Crookes radiometer (1873)

Improved versions in the 1900s with an accuracy ~ 1% (Nichols radiometer)

Considered as a demonstration of Maxwell's equations

of of

Laser Resonator

Couple a laser beam to a mechanical resonator via radiation pressure:

Radiation pressure drives the mechanical object (mirror…) into motion

The motion changes the path followed by light

Momentum exchange at every photon reflection

Orders of magnitude for 1W: ; quantum fluctuations: Orders of magnitude for 1W: ; quantum fluctuations:

Control the resonator motion: Laser cooling down to the fundamental quantum state of the resonator

Control the light: Radiation-pressure back-action in optical measurements, Quantum optics


Use a mechanical resonator sensitive enough to radiation pressure

High quality factor, low mass

Use a very sensitive displacement sensor

High finesse optical cavity

Couple a laser beam to a mechanical resonator via radiation pressure:

Radiation pressure drives the mechanical object (mirror…) into motion

The motion changes the path followed by light


2 – Use light to control mechanical motion: cold damping and laser cooling

3 – Quantum optics with optomechanics

: 2 Use light to

1 – Use optomechanical sensors to detect tiny mechanical motion

Introduction to quantum optomechanics

3

Applications of quantum optomechanics:

4 – Many recent applications!

6

Early history and fundamentals of optomechanics…

History: from the 70's to the 21th century… Emergence of a new research field at the frontier of quantum optics

Fundamentals of optomechanics Main theoretical concepts: classical, semi-classical and quantum levels

Optomechanical sensors: going to zeptometer measurements High-sensitivity measurements from gravitational-wave interferometers to tabletop experiments

Control and cooling of a mechanical resonator Down to the quantum ground state

Resonators, cavities and other devices A short story of systems, devices and technologies

Towards recent evolutions

Outline of the course 7

“The direct rays of the Sun strike upon the comet, penetrate its substance, draw away with them a portion of this matter, and issue thence to form the track of light we call the tail.”

Two tails: dust (radiation pressure) and ions (solar wind)

Johannes Kepler (1619) postulates the existence of a force which pushes away from the sun the tail of comets

Discovery of radiation pressure

Two tails: dust (ra ons (solar wind) adiation pressure) and ioust (ra ons (so

Electromagnetic waves carry energy and momentum described by the Poynting vector =

Momentum transfer = for an absorbing media

Maxwell’s theory of electromagnetism:

8

Photons ( )

“The direct rays of the Sun strike upon the comet, penetrate its substance, draw away with them a portion of this matter, and issue thence to form the track of light we call the tail.”

Two tails: dust (radiation pressure) and ions (solar wind)

Johannes Kepler (1619) postulates the existence of a force which pushes away from the sun the tail of comets

Discovery of radiation pressure

Two tails: dust (ra

PhPhototoons

ons (solar wind)adiation pressure) and ioust (ra ons (so

Electromagnetic waves carry energy and momentum described by the Poynting vector =

Momentum transfer = for an absorbing media

Maxwell’s theory of electromagnetism:

In terms of photons: = where = number photons/second

9 First observations of radiation pressure

Crookes radiometer (1873): mill with black/white vanes eter (1873): mill with black/white vaneseter (1873): mill with black/white vaneseteRadiation pressure pushes counter-clockwise but rotation is dominated by thermal effects in the residual gas around the vanes

Lebedev (1901), Nichols and Hull (1903): torsion balance nceLebedevL (1901) Nichols and Hull (1903): torsion balanLebedevL (1901), Nichols and Hull (1903): torsion balan- Vanes with silver coating to improve reflectivity - Proper choice of gas pressure to reduce thermal effects - Transient exposure (quarter of balance period)

Demonstration of radiation pressure with 1% accuracy

10

First observations of radiation pressure

Crookes radiometer (1873): mill with black/white vanes Radiation pressure pushes counter-clockwise but rotation is dominated by thermal effects in the residual gas around the vanes

Lebedev (1901), Nichols and Hull (1903): torsion balance - Vanes with silver coating to improve reflection - Proper choice of gas pressure to reduce thermal effects - Transient exposure (quarter of period)

LebedevL (1901) Nichols and Hull (1903): torsion balanLebedevL (1901) Nichols and Hull (1903): torsion balanLebedevL (1901), Nichols and Hull (1903): torsion balan- Vanes with silver coating to improve reflection- Proper choice of gas pressure to reduce thermal effects-- Transient exposure (quarter of period)Tra

Crookes radiometer (1873): mill with black/white vanesCrookes radiometer (1873): mill with black/white vaneseter (1873): mill with black/white vaneseteRadiation pressure pushes counterR -rr clockwise but rotation is dominated by thermal effects in the residual gas around the vanesd

Demonstration of radiation pressure with 1% accuracy

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Since the Gordon Research Conference (Mechanical systems in the quantum regime) in March 2010, proposed by Florian Marquardt and Khaled Karrai: - Come up with a low-cost (1000€) direct demonstration of radiation pressure, with instructions to build the apparatus, evidence of radiation pressure effect! - The experiment will have to be assembled by Florian (or an equivalent theoretician)

No winner yet!

11

- Orbital perturbations (dust, asteroids, spacecrafts…) - Possible spacecraft propulsion with solar sail!

Solar radiation pressure:

Consequences of radiation pressure

rafts…)

IKAROS mission: 14 × 14 m sail with embedded LCD panels of variable reflectance to control the trajectory.

6-months journey to Venus

Laser cooling of atoms

Magneto-optic trap (MOT) Doppler cooling

1997 Nobel prize to S. Chu, C. Cohen-Tannoudji, W. Phillips

12

- Mechanical effects of radiation pressure on mechanical resonators - Many consequences now envisioned from fundamental concepts to applications

Optomechanics!

Consequences of radiation pressure

Studied in the classical domain since the 70's, in the context of gravitational-wave detection:

- 70's: modification by light of the stiffness and damping of a mechanical resonator - 80's: light-induced optical bistability - 90's: very-sensitive displacement sensors

Walther et al, 1983 Ligo, Virgo, Geo, Tama… Braginsky, 1970

13 Classical description of optomechanical coupling

I. Favero

Simplest system: A moving mirror (single-mode resonator) embedded in a Fabry-Perot cavity

Mechanical system:

Optical single mode:

Position

Complex amplitude (classical counterpart of the annihilation operator )

Intensity (as number of photons/second): =

Radiation pressure: = 2

14

= with the mechanical susceptibility: = 1

Free mass submitted to an external force :

Harmonic oscillator (pendulum):

Mechanical response of the resonator

=

+ + =

Frequency

Susc

eptib

ility

Linear response theory: = = 1

with : resonance frequency : mechanical damping Q: mechanical quality factor

15

Laser telemetry: measures the propagation time towards a target

Robotics Measurements in building trade

Applications:

How to detect displacements? Use light! 16

Accuracy about 1 mm (limited by atmospheric perturbations)

Laser reflection on artificial satellites or on retroreflectors installed on the Moon during Apollo missions

Accurate study of the Earth-Moon dynamics Definition of an absolute reference frame Terrestrial geodesy Fundamental tests of gravitation

eriiccc pepepe ttrtrtrturururbbbababatititititiononon )s)s)s)

ectors installedt

Example: Lunar Laser Ranging 17

Interferometer: measurement of relative lengths between the 2 arms

Output intensity variation:

Moving mirror

Sensitive optical measurement of displacements

O

Michelson – Morley interferometer

18

D. Rugar (IBM San Jose), Nature 430, 329 (2004)

Interaction between a micro-magnet and a single electron spin

Sensitivity about 1 better than the atto-newton on the detected force

Interferometric measurement of the cantilever oscillations

Example: spin flip of a single electron 19

Moving mirror

Use of a high-finesse cavity:

(F : cavity finesse) Resonance width:

Phase of the reflected light:

Ultra-sensitive displacement sensor

Fabry – Perot cavity

20

Moving mirror

Use of a high-finesse cavity:

(F : cavity finesse) Resonance width:

Phase of the reflected light:

Sensitivity at 10 20 m:

Ultra-sensitive displacement sensor 21

Lossless cavity (perfect mirrors) with a totally reflecting moving mirror

Relevant parameters:

Basic equations of a cavity with a movable mirror

Phase measurement

Cavity

Moving mirror

High-finesse cavity : mirror reflection = 1 and transmission = with 1 Resonances: = frequency resonances ( ) = Free spectral range = We assume a near-resonant cavity: = + with

,

22


Input-output relations:


Phase measurement

Cavity

Moving mirror

,

Cavity round trip time: = We neglect retardation effect due to the mirror motion

Input mirror Propagation

23


Intracavity field evolution:


Phase measurement

Cavity

Moving mirror

,

Usual equation for a Fabry-Perot cavity with = + : depends on the mirror motion!

Evolution / 1 round trip Source term Phase shift Damping

Input-output relation: Interference between incident and intracavity fields

Radiation pressure:

24


Usual optomechanical equations

Phase measurement

Cavity

Moving mirror

,

Relevant parameters:

Cavity damping rate: =

Intracavity field as a number of photon rather than a flux: =

Optical frequency shift per displacement: = with = for a Fabry-Perot cavity

= 2 + + ( ) + = = =

Optomechanical equations:

25

= +

Lossless cavity: no intensity change =

The phase of the reflected field depends on the mirror position

Airy peak as a function of the detuning , of width Finesse : = =

Amplification at resonance:

= =

Intracavity mean field:

with = +

Reflected mean field:

Mean fields and cavity finesse

= 0 = 2

Intracavity intensity =

Detuning

= /

26

Detuning

Bistabilityturning points

Unstable

Detuning

Due to radiation pressure, the mirror position depends on the intensity:

= = 0 = = 0

+ + 0 =

Up to 3 solutions for

Intracavity mean field: with = +

Effect on light: (1) bistable behavior

= 0 = 2

Intracavity intensity =

Incident intensity

Intra

cavi

ty intens

ity 27

Microwave cavity with a movable wall (A. Gozzini et al, 1985)

Fabry-Perot cavity with a movable mirror (H. Walther et al, 1983)

Pin

Pout

Firsts observations of radiation-pressure bistability

Pin Pout

- Suspended 60-mg quartz plate - Few Hz resonance frequency - Cavity finesse = 15 - 1 to 2W incident power

28


of optomechanics

~ 2nd course ~

Antoine Heidmann





History: from the 70's to the 21th century… Discovery and first observations of radiation pressure

Fundamentals of optomechanics Main theoretical concepts: classical description

Optomechanical sensors: going to zeptometer measurements Bistability, thermal motion at the zeptometer scale




Summary of the 1st course

Early hi anics…

OOp saleBistability, yy thermal motion at the zeptometer scaale

~ 2nd course ~ Going to quantum:

Main theoretical concepts Consequences for optomechanical sensors and in quantum optics

30

Optomechanical effect on light: displacement sensor

Virgo, Ligo, Geo600, TAMA, …

Sensitivity about 10 m

Motivated since the 80's by gravitational-wave detection

31

Measurement of mirror displacements with a very high finesse cavity ( )

Homodyne detection working at the quantum level

High-finesse cavity (20 ppm losses+transmission): collab.

Ultra stable laser source

Phys. Rev. Lett. 83, 3174 (1999) Phys. Rev. Lett. 99, 110801 (2007)

Tabletop optomechanical sensor 32

Measurement of mirror displacements with a very high finesse cavity ( )

Sensitivity: Phys. Rev. Lett. 83, 3174 (1999)

Phys. Rev. Lett. 99, 110801 (2007)

Tabletop optomechanical sensor 33

Displacement noise spectrum

Many peaks corresponding to the Brownian motion of the various vibration modes of the mirrors

Background mainly due to the quantum noise of light (shot noise)

34

+ + =

A mechanical mode is equivalent to a harmonic oscillator: Mass , frequency , damping Evolution equation for the mechanical motion at temperature :

Friction

Quality factor:

Friction is responsible for the widening of the resonance:

Coupling to a thermal bath (Langevin force)

The Langevin force is responsible for position fluctuations:

Thermal noise of a vibration mode 35

Cavity

Laser

Phase detection

Brownian motion of vibration modes of the micro-resonator

Observation of sharp mechanical resonances

Thermal noise of a micro-mirror

Phys. Rev. Lett. 97, 133601 (2006)

36

Cavity

Laser

Phase detection

Brownian motion of vibration modes of the micro-resonator

Observation of sharp mechanical resonances

Thermal noise of a micro-mirror

Phys. Rev. Lett. 97, 133601 (2006)

Lorentzian fit

37

Optical excitation by an intensity-modulated beam

Spatial scan over the mirror surface

Determination of resonance frequencies, quality factors, masses, and spatial profiles

Spatial characterization of vibration modes 38

Optical excitation by an intensity-modulated beam

Spatial scan over the mirror surface

Spatial characterization of vibration modes

Modes of a cylindrical mirror:

39

Temporal evolution of a high-Q single mode:

Oscillation at resonance frequency with slowly varying amplitude and phase

Time Trajectory in phase space:

Temporal evolution: phase-space monitoring

Eur. Phys. J. D 22, 131 (2003)

Mirror

Generator

= cos + = cos + sin

40



Time Trajectory in phase space:


Eur. Phys. J. D 22, 131 (2003)

= cos + = cos + sin

2D Brownian motion

41



Time


Eur. Phys. J. D 22, 131 (2003)

Thermal gaussian distribution:

42

Requires a very sensitive measurement of the recoil, and a very movable system

Bohr – Einstein discussions on double-slit experiment

Do interferences disappear?

Experiment not done yet! Measures which path photon takes via recoil (due to radiation pressure)

First quantum optomechanical system in 1920 ?

Interest of optomechanics at the quantum level 43

Laser noise

Mirror motion

Two conjugate quantum noises:

Laser noise (shot noise) Mirror motion due to radiation pressure

First theoretical studies of the standard quantum limit in 1980 First theoretical studies of the standard quantum limit in 1980

Quantum noise in interferometric measurements 44

Observe quantum fluctuations of a macroscopic mechanical system

Can a macroscopic object behaves like a microscopic one: Observation of quantum fluctuations, state entanglement, decoherence...

2 distinct worlds: classical and quantum domains…

Displacement sensor at the quantum level 45

Fundamental quantum state of a harmonic oscillator:

Harmonic potential:

Heisenberg uncertainty relation:

Residual quantum fluctuations of position

Energy quantification:

with

Fundamental energy

Resonator as a quantum harmonic oscillator

for ~ 1 g and ~ 5 MHz

Temperature: = < 1 < 200 200

~10 m

46

Quantum behavior of optomechanical systems

Quantum treatment of light noises and resonator motion

I. Favero

Insight into the usual quantum approach based on a Hamiltonian description of the system and on the input-output formalism

Simple model: corpuscular approach, valid when only photons play a role (no phase or interference effects)

An equivalent and intuitive approach, the semi-classical description of quantum noises, valid in presence of strong mean fields

47

Uncoupled Hamiltonian:

Optomechanical coupling:

The optomechanical Hamiltonian

with = phonon number

(zero-point fluctuation amplitude)

Full Hamiltonian: Also contains terms describing dissipation (photon decay and mechanical friction), fluctuations (thermal phonons) and driving (external laser)

with =

= + where =

= +

Phase-shift: = ; Radiation pressure: =

Interaction Hamiltonian = +

where = is the vacuum optomechanical coupling strength

48

Full Hamiltonian: in the frame rotating at

Input-output formalism: quantum Langevin equations

The optomechanical Hamiltonian

= 2 + + +

= 2 + +

= + + +

Similar to classical equations, but work with operators and quantum input noises

Input terms characterized by noise correlations functions: ( ) = ( ) , ( ) = + 1

with = is the mean number of phonons

49

Photons treated as discrete events localized in time Intensity measurement (with a photodiode or a photomultiplier) corresponds to a detection of the photon flux

A light beam is considered as a stream of photons (no phase)

Instantaneous intensity: = ( )

The set of photon events completely characterizes the light beam

Simple corpuscular approach

Simple model with ‘photons’ as corpuscular objects: Cannot describe interferences (cavity…) but useful to understand some basic quantum features of radiation pressure

JOSA B 10, 745 & 1637 (1993)

50

Photons emitted by a laser are randomly distributed in time (Poisson distribution)

Photon noise

Low intensity beam (channel < )

Intense beam (channel > )

Distribution of photon number per channel: = ! =

Noise spectrum : [ ] = (white) shot noise

Time

Time

Frequency /

[ ] JOSA B 10, 745 & 1637 (1993)

51

Phase-shift not relevant. Take into account delay on photon reflection, usually neglected:

Reflection on a single movable mirror

Resonator

Incident photons

Reflected photons Intensity measurement

Phys. Rev. A 50, 4237 (1994)

Monte-Carlo simulation of the incident photon flux: = ln (1 )

Elastic scattering at every reflection, free motion between two photons

Computation of reflected photon flux

52


Resonator

Incident photons


Reflected photon flux is more regular than the incident one!

Incident flux

Mirror position

Reflected flux

Time

Phys. Rev. A 50, 4237 (1994)

53

Intensity squeezed state

Monte-Carlo simulation for an intense laser beam (width channel > and 1)

Temporal redistribution of photons induces a photon noise reduction below the shot noise

Intensity noise spectrum:

Temporal redistribution of photons

Incident flux

Mirror position

Reflected flux

Incident beam

Reflected beam

Time

Frequency

Phys. Rev. A 50, 4237 (1994)

Order of magnitude: = 2 mm @ = 1 , = 1 ms Retardation effects on photons are negligible!

54


of optomechanics

~ 3rd course ~

Antoine Heidmann





History: from the 70's to the 21th century… Radiation pressure and optomechanical sensors






Summary of the 1st and 2nd courses

Early hi hanics…

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~ 3rd course ~ Going to quantum:

Radiation pressure and quantum optics Control and cool a mechanical resonator

56


Resonator

Incident photons


Example of simple interpretation of optomechanical effects Corpuscular results validated by a complete quantum model Retardation effects are usually negligible!

Phys. Rev. A 50, 4237 (1994)

57

Change of optical path due to mirror motion: phase-shift and retardation effect

Semi-classical approach

Resonator

Phase measurement

Retardation effect Phase shift =

Input-output relation: = exp with = 2 /

Use a semi-classical description of noise (based on Wigner distribution) and determine its input-output relation (almost classical)

e

58

Linearization procedure: = + One gets: =

= +

For an intense beam, fluctuations are 'small' compared to the mean intensity .

Quadratures: = +

For a mean field = e , 2 specific quadratures: amplitude: = phase: = /

Intensity noise: = =

Linearization of quantum noise 59

Wigner quasi-probability distribution:

Field is described by semi-classical variables and randomly distributed according to the Wigner distribution .

Classical and quantum averages are equal when operators are in the symetrical order:

, = ,

Semi-classical description:

with : random classical variable /

For many states of light (vacuum, coherent states, squeezed states),

is always positive

Semi-classical representation 60

Minimum state for the uncertainty relation:

Vacuum, coherent state:

Squeezed state:

The Wigner distribution is a Gaussian

Coherent state Squeezed state

Vacuum, coherent and squeezed states…

, / = . / 1

Vacuum

Squeezed state

Coherent state

61

Input fields described by classical random variables

For laser coherent fields, noise spectra are: = = = 1 =

Input-output semi-classical formalism

Input fields

Output fields

Detection

System

Input-output relation for the fields

For quadratic Hamiltonians, the Wigner distribution obeys classical equations of evolution

Output field fluctuations are obtained from the incident ones using the linearized classical equations of the system

62

Usual equation for the mechanical system:

= + + +

with = 1/

Generalized Langevin force :

= coth

Phonons and generalized temperature for a high-Q resonator: = with

Semi-classical description of the resonator

= + and = 1/ / 1 = mean number of phonons

Frequency /

300 K +

100 mK 100 K

63

Change of optical path:

= exp with = 2 /

Phase shift (retardation effect neglected)

Mean fields:

Fluctuations (assuming = 0 and ):

Intensity is not perturbed

Phase sensitive to displacement


Resonator

Phase measurement

(

ee(

64

Orders of magnitude:

Spectral noise amplitude:

Measured spectrum:

Sensitivity:

Measurement noise Signal

Amplification by light

Sensitivity of the displacement measurement 65

Back-action on the resonator:

Resulting effect:

Radiation pressure:

with = , conjugate noise to the measurement noise

Resonator

Phase measurement

Back-action of radiation pressure 66

=

Perturbation :

Measure: Back-action:

Sensitivity:

Standard quantum limit:

Quantum limit in the measurement

Resonator

Phase measurement

Phase noise Radiation pressure

Quantum limit

Intensity

Noise

67


Semi-classical description of cavity optomechanics

Linearize equations around the mean values: = +

Inject quantum noises in the input fields and

Solve equations and compute the noise spectra!

Semi-classical approach:

Phase measurement

Cavity

Moving mirror

,

Classical optomechanical equations: = 2 + + ( ) + = = =

68


Semi-classical description of cavity optomechanics

Phase measurement

Cavity

Moving mirror

Linearized equations for the fluctuations:

2 = +

2 = + 2 + + = + = + +

General input-output relations for cavity optomechanics

69


Optomechanical sensor

Phase measurement

Cavity

Moving mirror

Dynamical equations for intracavity quadratures:

2 = 2 = 2 +

At resonance ( = 0), all mean fields can be taken as real

Amplitude and phase input-output relations:

2 = 2 +

2 = 2 + + 2

Measurement!

Intensity noise is left unchanged

70


Optomechanical sensor

Phase measurement

Cavity

Moving mirror

= + 4 +

Output noise spectrum:

Sensitivity: = 1 + / with =

As compared to a single mirror ( = / ), gain by the cavity finesse

Measurement is limited to the cavity bandwidth =

1

71


Back-action of radiation pressure

Phase measurement

Cavity

Moving mirror

= 2

Displacement due to radiation pressure:

Resulting contamination: = // =

Same compromise as for a single mirror between measurement sensitivity and back-action noise (but not at the same incident intensity!)

Leads to the same standard quantum limit =

72

Experimental demonstration of back-action noise

Thin and high-Q membrane in a high-finesse cavity (membrane in the middle setup)

Intense signal beam generates radiation pressure effects

Measurement by a weaker meter beam on the wings of the Airy peak (measurement of transmitted intensity)

Meter beam also used for laser cooling (cryogenic 5 K 1.7 mK)

73 Experimental demonstration of back-action noise

Expected dependency with signal power:

@ 1.7 mK

Experimental results: At high signal power (orange spectrum), the displacement noise of the membrane is larger than at low power (blue spectrum)

Partly attributed to radiation pressure noise

74

Other optomechanical measurements

Avoid back-action effects due to radiation pressure (QND measurement)?

Measure one quadrature of the resonator motion, while back-action noise affects the conjugate quadrature

Usual back-action evading technique for harmonic oscillators

Achievable by sending a amplitude-modulated laser at mechanical frequency . Ideally allows noise-free measurement of a single quadrature

Reconstruction of the Wigner mechanical distribution (tomography), entanglement with light…

Measure the mechanical energy instead of the displacement

Use a membrane in the middle of a cavity, placed at a node: the frequency shift becomes sensitive to rather than

Would allow the observation of quantum jumps between phonon Fock states = 0, 1, 2…

75

Cavity detuning + becomes sensitive to the intracavity intensity Similar to a Kerr medium inserted in the cavity

Equivalence between changes of physical length and optical path induced by refractive index

Due to , the dynamics is often more complex

Radiation pressure is proportionnal to intracavity intensity and changes the cavity length

Radiation pressure = Displacement =

Quantum optics with optomechanics

Output measurement

Cavity

Moving mirror

,

L(I )

n(I)

Optomechanical bistability (1st course)

Quantum effects: squeezed states, QND measurement of light…

76

< 1

Generation of squeezed state of light

Interpretation of squeezed state generation: fields in phase space

Phase-shift of mean fields = Rotation in phase space

Each point in the incident Wigner distribution is a possible realization + of the field + with

Different rotation in phase space

Unitary transformation area of distribution is preserved

Generation of squeezed state

= 1

Reflected field Reflected

field

Incident field Incident

field

Phys. Rev. A 49, 1337 (1994)

77 Semi-classical description of ponderomotive squeezing

Linearized equations for the fluctuations :

2 = +

2 = + 2 + + = + = + +

Phys. Rev. A 49, 1337 (1994)

Write output fluctuations as a function of input noises , and various optomechanical parameters ( , , , …)

Compute the noise spectra of all outgoing quadratures

Find the minimum as a function of (optimal squeezing) or the amplitude quadrature…

78

Ponderomotive squeezing

Phys. Rev. A 49, 1337 (1994)

= 0 K

= 4 K

Low frequency Kerr effect due to residual low frequency motion ( 0 1/ )

No effect at zero frequency (photon conservation over long times)

Strong effects close to mechanical resonances

Degradation by non-zero resonator temperature

79 Experimental demonstration of squeezing

Ultracold atoms cloud trapped in the cavity: acts as a position-dependent refractive medium for the probe beam

Homodyne detection of the transmitted probe beam (local oscillator)

Cavity locked at a constant detuning

Dan Stamper-Kurn et al. (Berkeley), 2012

80

Other experimental demonstrations of squeezing

Oskar Painter et al. (Caltech), 2013: zipper cavity

Cindy Regal et al. (JILA), 2013: membrane in the middle of a high-finesse cavity

81


of optomechanics

~ 4th course ~

Antoine Heidmann



Collège de France, CNRS, France 82




Optomechanical sensors: going to zeptometer measurements High-sensitivity measurements from gravitational-wave interferometers to tabletop experiments, quantum effects on light




Summary of the firsts courses

Early hi ics…

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A short story of systems, devices and technologies


High-sensitivity measurements from gravitational-wave interferometers to tabletop experiments quantum ffeffects on light

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83

Optical spring, bistability

Binding or repulsive force: Dynamics due to the cavity storage time:

Viscous force:

Optical damping, cooling or heating of the mirror

In a detuned cavity, radiation pressure is sensitive to mirror displacements:

Dynamical effects in a detuned cavity 84

First experimental demonstration in the 70's

Torsional oscillator at the end of a microwave resonator

Optical measurement of oscillations

Observation of stiffness and damping changes under microwave radiation

V.B. Braginsky

85

Experimental demonstration with microwaves

100 × 50 mm high-Q (10 ) sapphire bar

Parametric coupling between acoustic (50 kHz) and microwave (10 GHz) modes in the crystal

Microwave readout with 1 W detuned pump

Observation of damping

86 Optical spring and damping effects

Linearized equations for the fluctuations (2nd course):

2 = +

2 = + 2 + + = with = + = + +

Mean fields ( assumed real): = and =

Calculation of radiation pressure force in the framework of semi-classical formalism

87

Optical spring and damping effects

Effective susceptibility:

= + + + + Resulting motion:


= + (no effect at = 0)

Additional input noise (light noise)

Thermal Langevin force (unchanged)

88 Optical spring and damping effects


Effective susceptibility: = +

with =

Optical spring: + with = +

Optical damping: + with =

with = optomechanical strength in presence of an intense field

For a high-Q resonator ( ), still a Lorentzian shape with a frequency shift and an additional damping

Resonant for = ± : maximum damping = ±4 /

89

Damping described as collisions with a gas at temperature T

Fluctuating Langevin force

Additional gas at temperature T Fluctuation-dissipation theorem:

Does not reduce the temperature

No change of the spectrum area

Mirror cooling: increase the damping?

=

90

Increase damping by a viscous force without adding noise

Coupling to a thermal bath at zero temperature

Mirror cooling: cold damping

Coupling to 2 baths at different temperatures: = ×

Reduction of thermal noise spectrum =

91

Control applied by the radiation pressure of an auxiliary laser beam

Cold damping observed with a plano-convex mirror

Phys. Rev. Lett. 83, 3174 (1999)

Widening of the spectra

Cooling:

Former experimental demonstration at LKB 92

LKB (Arcizet et al., PRL 2006) D. Bouwmeester (Nature 2006)

Feedback by electrostatic actuation

D. Rugar (PRL 2007) 4.2 K down to 3 mK

Cavity finesse ~ 2000 Q ~ 100 000

Cold damping of micro-resonators 93

Cold damping: down to the ground state

Cold damping can in principle be used to reach the quantum ground state of the resonator

For a given cold damping gain , minimal contamination noise is reached for = /4

= + Phonon number reduced by 1 + , down to the ground state for large

= 10

( ) = 1 ( ) 0

Eur. Phys. J. D 17, 399 (2001)

= + + +

with = optomechanical cooperativity

= + Thermal equilibrium of a damped resonator ( = 1 + ) at a generalized temperature given by:

94 Optical cooling with a detuned cavity

= + + + + Motion of the resonator :


Thermal Langevin force (unchanged)

Effective susceptibility = + +

Optical spring: + with = +

Optical damping: + with =

Effective susceptibility (for a high-Q resonator):

95

Effective temperature:

Optical cooling with a detuned cavity

= Motion of the resonator :

Assuming the thermal regime, all quantum noises are neglected

Same Langevin fluctuations, but additional damping widened spectrum

Corresponds to a thermal equilibrium at a different temperature

+

+

= Cooling or heating depending on the sign of

96 Interpretation of optical damping

where = 4 / is the optomechanical cooperativity, and the Airy peak: = / /

Effect proportional to the dissymmetry of heights at ±

Optical damping: =

/ = +

No effect at resonance Positive damping for red detuning

Optimal damping: + = 0 and >

97

Energy transfer between optical and mechanical modes via radiation pressure:

Interpretation of optical cooling

Anti-Stokes diffusion (cooling)

Stokes diffusion (heating)

At resonance, the two processes are equiprobable

98

Negative detuning: anti-Stokes process becomes dominant cooling Stokes process negligible in the resolved sideband regime ( )

Energy transfer between optical and mechanical modes via radiation pressure:

Interpretation of optical cooling

k b d in

Anti-Stokes diffusion (cooling)

99

Laser heating and instability: resonance of the Stokes process

Laser cooling (LKB Paris and Vienna, 2006): effect limited by the residual Stokes process

At resonance: no effect Stokes anti-Stokes

Resolved sideband cooling (Kippenberg 2008): possibility to reach the ground state

Cooling and heating: different regimes 100

Khaled Karrai (Nature 2004)

First demonstration of (photothermal) cooling

KhKhKhKhKh llalalal ddededed KKKaKaKarrrrrr iiaiaiai ((N(N(N(N tatatatururureee 2020202020004040404)))))

Gold-coated micro-cantilever and coated fiber cavity of finesse 3

Photothermal effects in the cantilever induces a delayed observation of photothermal cooling down to 18 K

Radiation pressure effects negligible (1 %) and with an opposite sign

101

First demonstrations of optical cooling

2 demonstrations at the same time by LKB and Vienna (M. Aspelmeyer)

Micro-mirrors (~1 mm long, masses 20 ng and 100 μg), high-finesse cavities (500 and 30000)

2b

Vienna experiment: combination of radiation-pressure (40%) and photothermal (60%) coolings

Cooling down to 8 K

102

Silicon doubly-clamped beam:

Mirror coated by LMA

Typical size:

Experimental setup at LKB

High-finesse cavity: = 30 000

Compact and adjustable cavity

Pound-Drever-Hall detection (for frequency locking and displacement sensor)

103

Negative detuning: Enhancement of anti-Stokes process

Positive detuning: Enhancement of Stokes process

Airy peak of the cavity

Temperature reduced by 30, or increased by 10 Limited by parametric instability

Thermal noise of the micromirror:

Laser cooling and heating

Nature 444, 71 (2006)

104

Damping is only due to radiation pressure

Optical damping for different incident powers

3.2 mW 2.2 mW 1.6 mW 0.9 mW 0.5 mW

Optical damping

Adjusted intracavity power as a function of the incident power

Dashed curve corresponds to the actual cavity damping

Incident power (mW)

Intra

cavi

ty p

ower

(W

)

105

Spontaneous oscillation in the unstable domain + < 0 with amplitude ~ 10 pm

Airy peaks in P plane:

Cooling, heating, and parametric instability 106

T. Kippenberg (Garching, Nature 2008): cooling of a toroid microcavity

First resolved sideband cooling Needed to reach the quantum regime

Optical cooling is neither limited to micromirrors…

Mechanical breathing mode at 73 MHz, high Q (30 000)

Ultra-high finesse toroidal optical micro cavity, bandwidth 3 MHz

107

K.R. Brown, D.J. Wineland (PRL 2007)

K.C. Schwab (Nature 2006)

Self-cooling of a micro- cantilever by capacitive coupling to a radio-frequency resonant circuit

Self-cooling by the electrostatic force resulting from the coupling with the SSET

)

… nor to optomechanical devices!

And many others since then!

108 Optical cooling: down to the ground state

= + + + + Motion of the resonator :


Thermal Langevin force = = +

Effective susceptibility

where = is the cooperativity and the Airy peak: = / /

Optical damping: / = +

Thermal equilibrium of a damped resonator ( + ) at a generalized temperature given by: = 1 + / + 1 + + +1 + +

Red-detuned operation in the resolved sideband regime with >

109

Hamiltonian interpretation of optical cooling

= + = + + + = =

= d

+ +++

Linearization

110 Hamiltonian interpretation of optical cooling

= + =

= += = + =

= +=

Beam-splitter Hamiltonian Parametric Down-Conversion Hamiltonian

= d

111

Beam-splitter Hamiltonian (sideband cooling)

Optical mode Mechanical mode

Environment Coherent dynamics: periodic swap of optical and mechanical quantum states In presence of fast optical damping: Cooling of the mechanical mode via « optical pumping » into the quantum ground state

d Environment

= + =

112

Micro-toroid

T. Kippenberg (Lausanne), 2011

1.7 phonons

Microresonator coupled to a microwave

0.3 phonon

K. Lehnert (Boulder), 2011

Piezo-electric mechanical resonator

A.N. Cleland (Santa Barbara), 2010

0.07 phonon No active cooling!

Silicon nanobeam

0. Painter (Caltech), 2011

0.8 phonon

0000.0. PPPain

0

Experimental demonstration of quantum ground state 113

Recent developments and perspectives

Thermometry of mechanical resonators

Quantum optomechanical entanglement (Lehnert, Science 2013)

Study of decoherence of a massive mechanical object and fundamental tests of quantum mechanics, EPR experiments…

Quantum control of a resonator:

QuBit-resonator swap oscillations (Cleland, Nature 2010)

Coherent transfer between propagating microwave fields and a resonator (Lehnert, Nature 2013)

Mechanics as a bus connecting hybrid components

Optomechanical circuits for quantum memory and quantum information processing

Quantum coherent state transfer:

Proposal for single-phonon preparation by single photon detection from an entangled state (Kippenberg, PRL 2014)

Nonlinear quantum effects, generation of non Gaussian states…

Single photon optomechanical coupling:

114








Outline of the courses

Control and cooling of a mechanical resonatorDowDown tn to to thehe quaquantuntum gm grouroundnd stastatete

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DowDown tn to to thehe quaquantuntum gm grouroundnd stastatete

Resonators, cavities and other devicesA short story of systems, devices and technologies

TTo a drds recent e oll itions


115

Different kinds and sizes of resonator…

The quest for perfect device: High frequency High mechanical Q Low mass High finesse cavity Strong frequency dependency on resonator motion

From Aspelmeyer, Kippenberg, Marquardt, Rev. Mod. Phys. (2014)

Cold atoms

Microtoroid Nano-object

in a cavity

Photonic crystal nano-beam

Micromirror

Capacitor and microwave

116 Different kinds and sizes of resonator…

Aspelmeyer (Vienna) Heidmann (Paris)

Kippenberg (EPFL) Weig (Konstanz)

Harris (Yale) Rigal (JILA)

Painter (Caltech) Tang (Yale)

Stamper-Kurn (Berkeley) Treutlein (Basel)

Lehnert (JILA) Cleland (UCSB)

117

early history and fundamentals of optomechanics...maxwell’s theory of electromagnetism: in terms...

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