m. r. vanner et al- pulsed quantum optomechanics

8/3/2019 M. R. Vanner et al- Pulsed Quantum Optomechanics

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Pulsed Quantum Optomechanics

M. R. Vanner,1, ∗ I. Pikovski,1 M. S. Kim,2 C. Brukner,1, 3 K. Hammerer,3, 4 G. J. Milburn,5 and M. Aspelmeyer1

1 Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria 2 QOLS, Blackett Laboratory, Imperial College London, London SW7 2BW, United Kingdom

3 Institute for Quantum Optics and Quantum Information (IQOQI) of theAustrian Academy of Sciences, A-1090 Vienna and A-6020 Innsbruck, Austria

4 Institute for Theoretical Physics and Albert Einstein Institute, University of Hannover,

Callinstr. 38, D-30167 Hannover, Germany.5 School of Mathematics and Physics, The University of Queensland, Australia 4072 (Dated: November 3, 2010)

Quantum state preparation and especially quantum state reconstruction of mechanical oscillatorsis currently a significant challenge. Here we propose a scheme that employs short optical pulses torealize quantum state tomography, squeezing and state purification of a mechanical resonator. Thescheme presented also allows observation of mechanical quantum noise despite preparation from athermal state and is shown to be experimentally feasible. The proposal thus provides a promisingmeans to explore the quantum nature of massive oscillators and can be used in other systems suchas trapped ions.

Utilizing the precision and control of quantum opticsto study mechanical resonators is a rich avenue for the

exploration of quantum mechanical behavior in a macro-scopic regime. Quantum state preparation (QSP) andthe ability to probe the dynamics of the mechanical os-cillator, the most rigorous method being quantum statereconstruction (QSR), are essential for such investiga-tions. These have been experimentally realized in manyfacets of quantum optics, e.g. light [1], atomic ensembles[2] and intra-cavity microwave fields [3]. By contrast, anexperiment realizing both QSP and QSR of a mechanicalresonator has not yet been achieved. Also, schemes thatcan probe quantum features without full QSR (e.g. [4])are similarly challenging. In nano-electromechanics theground state has been observed [5] but QSR [6] re-

mains outstanding. In cavity optomechanics [7] signifi-cant experimental progress has been made towards quan-tum state control over mechanical resonators which in-cludes classical phase-space monitoring [8], laser coolingclose to the ground state [9] and low noise measurementof mechanically induced phase fluctuations [10]. Still,QSP is technically challenging primarily due to thermalbath coupling and weak radiation pressure interactionstrength and QSR remains little explored. Thus far, acommon theme in proposals for optomechanical QSR isstate transfer to and then readout of an auxillary quan-tum system [11], which amounts to an implementationof a quantum memory. Additionally, for continuous dis-placement measurement the ultimate sensitivity achiev-able (i.e. the standard quantum limit (SQL) [12]) is lim-ited by the quantum phase noise of the impinging lightand the associated back-action noise and is equal to thewidth of the mechanical ground state. For QSR, how-ever, it is imperative that features below this scale beresolved.

The SQL can be overcome with back-action evadingposition measurements. This can be achieved by a varia-

FIG. 1: (a) Schematic of the optical setup: An incident pulse(in) drives an optomechanical cavity, where the intracavityfield a accumulates phase with the position of a mechanicaloscillator described by b. The field emerges from the cavity(out) and balanced homodyne detection is used to measure

the optical phase with a local oscillator pulse (LO) shaped tomaximize the measure of the position. (b) Scaled envelopesof the input pulse, intracavity field and local oscillator.

tional measurement scheme [13], which has also recentlybeen considered for the use of conditional state prepara-tion and state reconstruction in gravitational-wave detec-tors [14]. Alternatively the SQL can be surpassed usinga pulsed interaction [12], where the radiation-pressurenoise imparted to the mechanical momentum does notevolve into position fluctuations that for continuous mea-surement gives rise to the SQL. In this Letter, we build

upon this, and introduce an optomechanical frameworkwhich utilizes pulsed quantum measurement for QSP andQSR. We demonstrate that phase measurement of an op-tical pulse reflected from an optomechanical cavity re-motely prepares the mechanical resonator in a squeezedstate, and that the same process can be used to measurethe mechanical marginals for QSR. The proposed exper-imental setup is shown in Fig. 1: A pulse of durationmuch less than the mechanical period is incident uponan optomechanical Fabry-Perot cavity which we model

http://arxiv.org/abs/1011.0879v1






































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as being single-sided. The pulse accumulates phase de-pending upon the mechanical position and concurrentlyimparts momentum to the mechanical oscillator via radi-ation pressure. Homodyne detection is then used to de-termine the phase of the field emerging from the cavity,and thus to obtain a measure of the mechanical position.

Model.- The intracavity optomechanical Hamiltonianin the rotating frame at the cavity frequency is H =

ωM b†b− g0a†a(b + b†), where a (b) is the optical (me-chanical) field operator, ωM is the mechanical eigenfre-quency, g0 = ωc(x0/

√2L) is the coupling strength for

mean cavity frequency ωc, mean cavity length L and me-chanical ground-state size x0 =

/mωM with effective

mass m. The cavity field accumulates phase in propor-tion to the mechanical position and is driven by resonantradiation via the equation of motion

da

dt= ig0(b + b†)a − κa +

√2κain, (1)

where κ is the cavity decay rate and ain describes the op-

tical input including drive and vacuum. During a pulsedinteraction of timescale κ−1 ≪ ω−1M the mechanical posi-

tion is approximately constant. This decouples (1) fromthe corresponding mechanical equation of motion andduring the short interaction we have db

dt ≃ ig0a†a, wherewe neglect the mechanical harmonic motion, mechani-cal damping and noise processes. We write ain(t) =

N pαin(t) + ain(t), where αin(t) is the slowly varyingenvelope of the drive amplitude with

dt α2

in = 1 andN p is the mean photon number per pulse and similarlya =

N pα(t)+ a(t). Neglecting ig0(b+b†)a and approxi-

mating α as real, (1) becomes the pair of linear equations:

dαdt = √2κ αin − κα, (2)

da

dt= ig0

N p(b + b†)α +

√2κ ain − κa. (3)

After solving for a(t), the output field is then found by

using the input-output relation aout =√

2κa− ain.The mechanical position and momentum quadra-

tures are X = (b + b†)/√

2 and P = i(b† −b)/√

2, respectively, the cavity (and its input/output)

quadratures are similarly defined via a (ain/aout).The optical amplitude quadrature is unaffected bythe interaction, however, the phase quadrature con-

tains the phase dependendent upon the mechanicalposition. It has output emerging from the cavityP outL (t) = g0

κ

N p ϕ(t)X in + 2κe−κt

t−∞dt′ eκt

′

P inL (t′) −P inL (t), where ϕ(t) = (2κ)

3

2 e−κt t−∞dt′ eκt

′

α(t′) describes

the accumulation of phase, X in is the mechanical posi-tion prior to the interaction and the last two terms arethe input phase noise contributions. P outL is measured viahomodyne detection, i.e. P L =

√2

dt αLO(t)P outL (t). Tomaximize the measure of the mechanical position the lo-cal oscillator envelope is chosen as αLO(t) = N ϕ ϕ(t),

where N ϕ ensures normalization. The contribution of X in in P L scales with χ =

√2 1 N ϕ

g0κ

N p, which quanti-

fies the mechanical position measurement strength. ForGaussian optical input, the mean and variance of P L are

P L = χ

X in

, σ2P L = σ2

P inL

+ χ2σ2Xin , (4)

respectively, where σ2P inL

is the optical input phase noise.

The maximum χ is obtained for the input driveαin(t) =

√κe−κ|t|. This can be seen by noting that

N −2ϕ =

dt ϕ2(t), which in Fourier space is N −2ϕ ∝ dω (ω2 + κ2)−2 |αin(ω)|2. Hence, for such cavity-based

measurement schemes, the optimal drive has Lorentzianspectrum. This drive, α(t) obtained from (2) and thelocal oscillator are shown in Fig. 1(b). The resulting op-timal measurement strength is given by χ = 2

√5 g0

κ

N p.

We now construct the non-unitary operator Υ, whichincorporates both pulse interaction and measurementand yields the new mechanical state ρoutM ∝ ΥρinM Υ

†. Thisoperator can be determined from the probability density

of measurement outcomes (see for example [15]) which isPr(P L) = TrM

Υ†ΥρinM

. (5)

The classical component of the momentum transfer dueto α2 is parameterized by Ω = 3√

2

g0κ N p and we include

this in Υ as a displacement operator. Since the opera-tions considered here are Gaussian, Υ is thus constructedassuming pure Gaussian optical input:

Υ = (π2σ2P inL

)−1

4 exp

iΩX − (P L − χX )

2

4σ2P inL

. (6)

Υ can be readily understood by considering its oper-ation on a mechanical position wavefunction. It selec-tively narrows the wavefunction to a width scaling withχ−2 about a position which depends upon the measure-ment outcome. Moreover, Υ is back-action evading inX , i.e. the back-action noise imparted by the quantummeasurement process occurs in the P quadrature only.

Henceforth, we consider coherent drive, σ2P inL

= 1/2. In

this case, we note that the above result is equivalent toΥ = eiΩX P L| eiχXLX |0, though the non-unitary pro-cess of cavity decay is not explicit. We also remark thatthe construction of Υ can be readily generalized to in-clude non-Gaussian operations.

Squeezing.- We first consider QSP by Υ acting on amechanical coherent state |β. By casting the expo-nent of Υ in a normal ordered form, one can show thatthe resulting conditional mechanical state is N βΥ |β =S (r)D(µβ) |0. Here, N β is a β-dependent normalization,D is the displacement operator for µβ = (

√2β + iΩ +

χP L)/

2(χ2 + 1) and S is the squeezing operator whichyields the position width 2σ2

X = e−2r = (χ2 + 1)−1.We now consider Υ acting on a mechanical thermal

state ρn, quantified by its average phonon occupation n.



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For |X θ = e−iθb†b |X , the marginals of the state after

the action of Υ are

X θ|Υ ρnΥ† |X θ ∝ exp

− (X − X θ)2

2σ2θ

, (7)

where

X θ = Ω sin(θ) +

χ P L

χ2 + 11+2n

cos(θ),

σ2θ =

1

2

cos2(θ)

χ2 + 11+2n

+1

2(χ2 + 1 + 2n) sin2(θ)

(8)

are the mean and variance of the resulting conditionalstate, respectively. For large initial occupation (providedthermal fluctuations are negligible during the interac-tion), the resultant position quadrature of the mechan-ics has mean X θ=0 ≈ P L/χ and width 2σ2

θ=0 ≈ χ−2.Thus, squeezing in the X quadrature below the groundstate is obtained when χ > 1 and is independent of the

initial thermal occupation of the mechanics. By utiliz-

ing short optical pulses, we have thus shown how theremarkable behaviour of generalised quantum measure-ment (also used in Refs. [2, 14, 16, 17]) can be applied toa mechanical resonator for QSP.

Purification by measurement.- We quantify the result-

ing state purity via 1 + 2neff =

4σ2θ=0σ2

θ=π/2, where

neff is the effective occupation. When acting on an ini-tial thermal state, the measurement dramatically reducesuncertainty in the X quadrature, but leaves mixing inthe P quadrature unchanged: use of (8) for n ≫ 1 yieldsneff ≈

n/2χ2. The purity can be further improved by

a second pulse, which is maximized for pulse separationθ = ωM t = π/2, where the initial uncertainty in the

momentum becomes the uncertainty in position. Sucha sequence of pulses is represented in Fig. 2, where theresulting state was obtained akin to (7). The effectiveoccupation of the final state after two pulses is

n(2)eff ≈

1

2

1 +

1

χ4− 1

. (9)

For χ > 1, n(2)eff is well below unity and we therefore

suggest that this scheme can be used as an alternative to‘cooling via damping’ for mechanical state purification.

Mechanical quantum state tomography.- Our schemeallows precision measurement of the mechanical quadra-ture marginals, which can be used to uniquely determinethe quantum state [18]. Specifically, by utilizing free har-monic evolution of angle θ = ωM t preceding a pulsedmeasurement one can obtain access to all quadratureswith respect to an initial preparation. Thus reconstruc-tion of any mechanical quantum state can be performed.The optical phase distribution (5), including θ, becomes

Pr(P L) =

dX √

πe−(P L−χX)2X | e−iθb

†bρinM eiθb†b |X , (10)

FIG. 2: Wigner functions of the mechanical state (above)at different times (indicated by arrows) during the experi-mental protocol (below). From left: Starting with an initialthermal state (n = 10), a pulsed measurement is made with

χ = 1.5 and outcome P (1)L

= 4χ, which yields an X quadra-ture squeezed state. The mechanical state evolves into a P quadrature squeezed state following free harmonic evolutionof one quater of a mechanical period prior to a second pulse

with outcome P (2)L

= −3χ yielding the high purity mechanicalsqueezed state. Dashed lines indicate the 2σ-widths and thedotted lines show the ground state (n = 0) for comparativepurposes. The displacement Ω is not shown.

which is a convolution between the mechanical marginalof interest and a kernel dependent upon χ and the quan-tum phase noise of light. The effect of the convolutionis to broaden the marginals and smooth any featurespresent. For example, the X marginal of the coherentstate superposition |iδ + |−iδ contains oscillations ona scale smaller than the ground state. The convolution

scales the amplitude of these oscillations by exp(− 2δ2

χ2+1)and thus they become difficult to resolve for small χ.

To extract the mechanical marginals, it is neccessaryto perform deconvolution of the observed optical distri-bution in accordance with (10). The result obtained cor-responds to the mechanical position distribution for thevariable P L/χ. On the other hand, when using this vari-able, even assuming a direct correspondence between theoptical phase and the mechanical position distributionsmay be sufficient to obtain an approximation of the me-chanical marginals without the need for deconvolution.In practice, this offers a simpler experimental approachbut amounts to a sacrifice of precision. The limiting caseof infinite χ corresponds to an ideal projective measrue-ment of the mechanics, where the distribution obtainedfor P L/χ becomes identical to the mechanical marginals.

Following squeezed state preparation as discussed

above, one can use a subsequent ‘read-out’ pulse af-ter free evolution θ to perform tomography. Duringstate preparation however, the random measurementoutcomes will result in random mechanical means (8).This can be overcome by recording and utilizing themeasurement outcomes: one can achieve unconditionalstate preparation with use of appropriate displacementprior to the read-out pulse; use post-selection to ana-lyze the statistics of the read-out for states preparedwithin a certain window or; compensate by appropri-



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ately adjusting each measurement outcome obtained dur-ing read-out. Looking at the latter option, for a Gaus-sian mechanical position distribution, prepared with ran-dom mean

X ( p)

and variance σ2 to be characterized,

the read-out pulse will have the distribution Pr(P L) ∝exp

(−(P L − χ

X ( p)

)2)/(1 + χ22σ2)

. For each read-

out pulse, by taking P L| p = P L+χ

X ( p)

one can obtain

the conditional variance σ2P L

|p

for all θ to characterize the

noise of the prepared Gaussian state. We note that thisconcept of compensating for a random but known meancan also be used to characterize non-Gaussian states.

Experimental considerations.- We now discuss an ex-perimentally feasible parameter regime for the generationand characterization of a squeezed mechanical state. Inorder to observe mechanical squeezing, i.e. σ2

X < 1/2,the conditional variance must satisfy σ2

P L|p < σ2P inL

+χ2/2,

where additional noise sources that do not affect the me-chanical state, e.g. detector noise, can be subsumed intoσ2P inL

. It is therefore necessary to have an accurate ex-

perimental calibration of χ to quantify the mechanical

width. (Similarly, Ω must also be accurately known.)To ensure that the interaction time be much less than

mechanical time-scales the cavity bandwidth must belarge. To this end, we consider the use of an opti-cal microcavity operating at λ = 1064 nm, length 4λand finesse of 7000, which has an amplitude decay rateκ/2π ≃ 2.5 GHz. For a mechanical resonator withωM /2π = 500 kHz and effective mass m = 10 ng, whichhas x0 = 1.8 fm, optomechanical coupling proceeds atg0/2π ≃ 86 kHz. Using pulses of mean photon numberN p = 108, which are readily homodyned, yield Ω ≃ 104

[20] and χ ≃ 1.5. For this χ, the action of a single pulseon a large thermal state reduces the mechanical varianceto σ2

X ≃ 0.2, i.e. less than half the width of the ground-state. With a second pulse after mechanical evolution

the effective occupation (9) is n(2)eff ≃ 0.05.

Finite mechanical evolution during the interaction de-creases the back-action-evading nature of the measure-ment. This will increase the conditional variance of themechanics by a quantity of order χ2ω2

M /κ2 (to lowest or-der in ωM /κ). Experimentally such evolution will notbe a severe limitation, for our parameters ωM /κ ≃ 10−4.The main limitation of the measurement strength is theoptical intensity that can be homodyned before photode-tection begins to saturate. The optical measurement effi-

ciency η, affected by optical loss, inefficient detection andmode-mismatch, yields a reduced measurement strengthχ → √

ηχ. Additionally, in many situations coupling toother mechanical vibrational modes is expected. Thiscontributes to the measurement outcomes and yields aspurious broadening of the tomographic results for themode of interest. In practice however, one can minimizethese contributions by engineering mechanical deviceswith high effective masses for the undesired modes andtailoring the intensity profile of the optical spot to have

good overlap with a particular vibrational profile [19].

For our tomography scheme the mechanical quantumstate must not be significantly perturbed during the time-scale ω−1

M . To estimate the effect of the thermal bathfollowing state preparation we consider weak and linearcoupling to a Markovian bath of harmonic oscillators.For this model, assuming no initial correlations betweenthe mechanics and the bath, the thermal contributionsscale with nγ M , where γ M is the mechanical dampingrate. It follows that an initially squeezed variance willincrease to 1/2 on a time-scale Q

nωM12(1 − e−2r). Thus,

for the parameters above and Q ≃ 105 a temperatureT 1 K is required. Moreover, we note that the positionmeasurements of this scheme can be used to probe opensystem dynamics and thus provide an empirical means toexplore decoherence and bath coupling models.

Conclusions.- We have discussed how a pulsed interac-tion allows for a back-action-evading position measure-ment of a mechanical oscillator below the SQL. Here wehave demonstrated how this can be used for quantum

state engineering and reconstruction of the resonator.The scheme can be readily implemented and we discussan experimentally feasible parameter regime. Our proto-col allows for state purification, remote preparation of amechanical squeezed state and quantum state tomogra-phy via measurement of the mechanical marginals. Theoptomechanical entanglement generated by the pulsedinteraction may also be a useful resource for other ap-plications, e.g. quantum information processing. More-over, the scheme can be generalized to other systems,e.g. nano-electromechanical systems, or used with dis-persive interaction to study the motional state of me-chanical membranes, trapped ions or particles in a cavity.

We thank the ARC (grant FF0776191), EPSRC, ERC(StG QOM), European Commission (MINOS), FQXi,FWF (L426, P19570, SFB FoQuS, START), QUEST,and an OAD/MNiSW program for support. M.R.V. andI.P. are members of the FWF Doctoral Programme Co-QuS (W 1210). The kind hospitality provided by theUniversity of Gdansk (I.P.), the University of Queens-land (M.R.V.) and the University of Vienna (G.J.M.) isacknowledged. We thank G.D. Cole, S. Hofer, N. Kieseland M. Zukowski for useful discussion.

∗ E-mail: [email protected]

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m. r. vanner et al- pulsed quantum optomechanics

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