Quantum Optomechanics

Annalen der Physik, 16 October 2012

A short walk through quantum optomechanicsP. Meystre

This paper gives an brief review of the basic physics ofquantum optomechanics and provides an overview ofsome of its recent developments and current areas of fo-cus. It first outlines the basic theory of cavity optomechan-ical cooling and gives a brief status report of the exper-imental state-of-the-art. It then turns to the deep quan-tum regime of operation of optomechanical oscillatorsand cover selected aspects of quantum state prepara-

tion, control and characterization, including mechanicalsqueezing and pulsed optomechanics. This is followed bya discussion of the bottom-up approach that exploits ul-tracold atomic samples instead of nanoscale systems. Itconcludes with an outlook that concentrates largely on thefunctionalization of quantum optomechanical systems andtheir promise in metrology applications.

1 Introduction

Broadly speaking, quantum optomechanics provides auniversal tool to achieve the quantum control of mechan-ical motion [1]. It does that in devices spanning a vastrange of parameters, with mechanical frequencies from afew Hertz to GHz, and with masses from 1020g to severalkilos. At a fundamental level, it offers a route to deter-mine and control the quantum state of truly macroscopicobjects and paves the way to experiments that may leadto a more profound understanding of quantum mechan-ics; and from the point of view of applications, quantumoptomechanical techniques in both the optical and mi-crowave regimes will provide motion and force detectionnear the fundamental limit imposed by quantum mechan-ics.

While many of the underlying ideas of quantum op-tomechanics can be traced back to the study of gravita-tional wave detectors in the 1970s and 1980s [2, 3], thespectacular developments of the last few years rely largelyon two additional developments: From the top down,it is the availability of advanced micromechanical andnanomechanical devices capable of probing extremelytiny forces, often with spatial resolution at the atomicscale. And from the bottom-up, we have gained a detailedunderstanding of the mechanical effects of light and howthey can be exploited in laser trapping and cooling. Thesedevelopments open a path to the realization of macro-scopic mechanical systems that operate deep in the quan-tum regime, with no significant thermal noise remaining.

As a result, they offer both knowledge and control of thequantum state of a macroscopic object, and increasedsensitivity, precision, and accuracy in the measurementof feeble forces and fields.

It was Arthur Ashkin [4] who first suggested anddemonstrated that small dielectric balls can be acceler-ated and trapped using the radiation-pressure forces as-sociated with focused laser beams. In later experimentsthese particles, weighting on the order of a microgram,were levitated against the Earth gravitational field. This ad-vance led to the realization of optical tweezers, whose ap-plications in biological science have become ubiquitous.In parallel, the use of the strong enhancement providedby resonant light scattering lead to the laser cooling ofions and of neutral atoms by D. Wineland, T. W. Hnsch, S.Chu, W. D. Phillips, C. Cohen-Tannoudji and many others,resulting in a wealth of extraordinary developments [5]culminating in 1995 with the invention of atomic Bose-Einstein condensates [6, 7], and the subsequent explosionin the study of quantum-degenerate atomic systems.

Non-resonant light-matter interactions present theconsiderable advantage of being largely wavelength in-dependent, providing one with the potential to achieveoptomechanical effects for a broad range of wavelengthsfrom the microwave to the optical regime. Resonant in-teractions, on the other hand, can result in a very largeenhancement of the interaction, but at the cost of be-ing limited to narrow ranges of wavelengths. Cavity op-tomechanics exploits the best of both worlds by achievingresonant enhancement through an engineered resonant

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P. Meystre: A short walk through quantum optomechanics

structure rather than via the internal structure of materi-als. This could be for example an optical resonator witha series of narrow resonances, or an electromagnetic res-onator such as a superconducting LC circuit. Indeed, nu-merous designs can achieve optomechanical control viaradiation pressure effects in high-quality resonators. Theyrange from nanometer-sized devices of as little as 107

atoms to micromechanical structures of 1014 atoms andto macroscopic centimeter-sized mirrors used in gravita-tional wave detectors.

That development first appeared at the horizon in the1960s, but more so in the late 1970s and 1980s. It wasinitially largely driven by the developments in opticalgravitational wave antennas spearheaded by V. Bragin-sky, K. Thorne, C. Caves, and others [2, 3, 8]. These anten-nas operate by coupling kilogram-size test masses to theend-mirrors of a large path length optical interferometer.Changes in the optical path length due to local changesin the curvature of space-time modulate the frequency ofthe cavity resonances and in turn, modulate the opticaltransmission through the interferometer. It is in this con-text that researchers understood fundamental quantumoptical effects on mechanics and mechanical detectionsuch as the standard quantum limit, and how the basiclight-matter interaction can generate non-classical statesof light.

Braginsky and colleagues demonstrated cavity op-tomechanical effects with microwaves [9] as early as 1967.In the optical regime, the first demonstration of theseeffects was the radiation-pressure induced optical bista-bility in the transmission of a Fabry-Prot interferometer,realized by Dorsel el al. in 1983 [10]. In addition to theseadiabatic effects, cooling or heating of the mechanical mo-tion is also possible, due to the finite time delay betweenthe mechanical motion and the response of the intracavityfield, see section 2.2. The cooling effect was first observedin the microwave domain by Blair et al. [11] in a Niobiumhigh-Q resonant mass gravitational radiation antenna,and 10 years later in the optical domain in several labo-ratories around the world: first via feedback cooling ofa mechanical mirror by Cohadon et al. [12], followed byphotothermal cooling by Karrai and coworkers [13], andshortly thereafter by radiation pressure cooling in severalgroups [1419]. Also worth mentioning is that as early as1998 Ritsch and coworkers proposed a related scheme tocool atoms inside a cavity [20].

This paper reviews the basic physics of quantum op-tomechanics and gives a brief overview of some of itsrecent developments and current areas of focus. Section2 outlines the basic theory of cavity optomechanical cool-ing and sketches a brief status report of the experimentalstate-of-the-art in ground state cooling of mechanical os-

Figure 1 (Color online) Generic cavity optomechanical system.The cavity consists of a highly reflective fixed input mirror anda small movable end mirror harmonically coupled to a supportthat acts as a thermal reservoir.

cillators, a snapshot of a situation likely to be rapidly out-dated. Of course ground state cooling is only the first stepin quantum optomechanics. Quantum state preparation,control and characterization are the next challenges ofthe field. Section 3 gives an overview of some of the majortrends in this area, and discusses topics of much currentinterest such as the so-called strong-coupling regime, me-chanical squeezing, and pulsed optomechanics. Section 4discusses a complementary bottom-up approach thatexploits ultracold atomic samples instead of nanoscalesystems to study quantum optomechanical effects. Finally,Section 5 is an outlook that concentrates largely on thefunctionalization of quantum optomechanical systemsand their promise in metrology applications.

2 Basic theory

To describe the basic physics underlying the main aspectsof cavity optomechanics it is sufficient to consider an op-tically driven Fabry-Prot resonator with one end mirrorfixed -and effectively assumed to be infinitely massive,and the other harmonically bound and allowed to oscil-late under the action of radiation pressure from the in-tracavity light field of frequency L , see Fig. 1. Braginskyrecognized as early as 1967 [9] that as radiation pressuredrives the mirror, it changes the cavity length, and hencethe intracavity light field intensity and phase. This resultsin two main effects: the optical spring effect, an opti-cally induced change in the oscillation frequency of themirror that can produce a significant stiffening of its effec-tive frequency; and optical damping, or cold damping,whereby the optical field acts effectively as a viscous fluidthat can damp the mirror motion and cool its center-of-mass motion.

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One can immediately understand how the opticalspring effect can result in a more quantum behavior of theoscillator by recalling that in the high temperature limitthe mean number of phonons nm in the center-of-massmotion of an oscillator of frequencym is given by

nm = kBT /m , (1)where kB is Boltzmans constant and T the temperature.For a given temperature, increasing m automaticallyreduces nm, allowing one to approach the quantumregime without having to reduce the temperature.

Cold damping, in contrast, does reduce the tempera-ture of the oscillating mirror by opening up a dissipationchannel to a reservoir that is effectively at zero temper-ature. To see how this work, we first remark that in theabsence of optical field the oscillating mirror is dissipa-tively coupled to a thermal bath at temperature T . Its av-erage center-of-mass energy, E, results from the balancebetween dissipation and heating,

dEdt

=E+kBT, (2)

where is the intrinsic mechanical damping rate. Whenan optical field is applied, an additional optomechanicaldamping channel with damping rate opt comes into playso that

dEdt

=E+kBT optE. (3)

Importantly, that channel does not come with an addi-tional (classical) thermal bath. Optical frequencies aremuch higher than mechanical frequencies, so that the op-tical field is effectively coupled to a reservoir at zero tem-perature. In steady state Eq. (3) gives E = kBT /(+opt),or

Teff =T

+opt. (4)

This simple phenomenological classical picture predictsthat the fundamental limit of cooling is T = 0. A moredetailed quantum mechanical analysis does yield a funda-mental limit given by quantum noise, see Section IIC, butin practice, this is usually not a major limitation to cool-ing the mechanical mode arbitrarily close to the quantumground state, nm = 0.

More quantitatively, we consider a single mode of theoptical resonator of nominal frequency c and assumethat radiation pressure results in a displacement x(t) ofthe harmonically bound end-mirror, and consequently ina change in the optical mode resonance frequency to

c =c Gx(t ), (5)

where

G =c/x. (6)

For a single-mode Fabry-Prot resonator of length L thisbecomes simply G =c/L.

Typical mechanical oscillator frequencies are in therange ofm/2pi=10Hz to 109Hz and the mechanical qual-ity factors of the mirrors are in the range of perhaps Qm 103107, so that typically the damping rate =m/Qmof the oscillating mirror is much slower than the damp-ing rate of the intracavity field. One can then gain con-siderable intuition by first neglecting mirror damping al-together and assuming that its motion is approximatelyharmonic,

x(t ) x0 sin(m t ). (7)

For a classical monochromatic pump of frequencyL andamplitude in the intracavity field obeys the equation ofmotion

d(t )

dt= [i (+Gx(t ))/2](t )+pin, (8)

with the steady-state solution

=pin

i (+Gx)+/2 . (9)

Here we have introduced the detuning

=L c (10)

and is the intracavity field amplitude, normalized sothat

||2 = (+Gx)2+ (/2)2

(P

L

)=

(+cx/L)2+ (/2)2(

P

L

)(11)

where

P =L |in|2 (12)

is the input laser power driving the cavity mode. This nor-malization allows for an easy generalization to the case ofquantized fields, in which casewill be interpreted as thesquare root of the mean number of intracavity photons,

=aa, with a and a the bosonic annihilation and

creation operators of the intracavity field. Note that |in|2has then the units of photons per second.

For oscillation amplitudes x0 small enough thatGx0/m 1, it can be shown that the mirror oscillations simply re-sult in the generation of two sidebands at frequencies



L m , see e.g. Refs. [21, 22]. The time-dependent com-plex field amplitude(t ) then takes the approximate form(t )'0(t )+1(t ) with

0(t ) 'pin

i+/2 , (13)

1(t ) '(Gx0

2

) pin

i+/2 (14)

(

eim t

i (+m)+/2 e

+im t

i (m)+/2)

.

The first sideband in Eq. (15) can be interpreted as an anti-Stokes line, with a resonance at L = c m , and thesecond one is a Stokes line. An important feature of thesesidebands is that their amplitudes can be vastly different,as they are determined by the cavity Lorentzian responsefunction evaluated at L m and L +m , respectively.

2.1 Static phenomena

Consider first a situation where the cavity damping rate is much faster than all other characteristic times of thesystem. One can then understand the mirror motion as re-sulting from the combined effects of the harmonic restor-ing force and the radiation pressure force Frp resultingfrom an adiabatic elimination of the intracavity field, seee.g. Ref. [23],

Frp =G||2 =cL||2, (15)

where ||2 is given by Eq. (11) and the second equalityholds for a simple Fabry-Prot. One can easily show thatthe force Frp can be derived from the potential

Vrp =||2

2arctan[2(+Gx)/] , (16)

the mirror of mass m being therefore subject to the totalpotential

V (x)= 12m2mx

2 ||2

2arctan[2(+Gx)/] . (17)

The potential Vrp slightly displaces the equilibrium po-sition of the mirror to a position x0 6= 0, as would be in-tuitively expected, and also changes its spring constantfrom its intrinsic value k =m2m to a new value

krp =m2m +d2Vrp(x)

dx2

x=x0

. (18)

The second term in this expression is the static opticalspring effect. For realistic parameters it can increase the

stiffness of the mechanical system by orders of magnitude.A third important static effect of radiation pressure is thatin general, there is a range of parameters for which thepotential V (x) can exhibit 3 extrema. Two of them corre-spond to stable local minima of V (x), and the third one toan unstable maximum. This results in radiation pressureinduced optical bistability [10], an effect that is physicallysimilar to the more familiar form of bistability that canoccur in a Kerr nonlinear medium. The difference is thatin one case, it is the optical length of the resonator thatis changed by a Kerr nonlinearity, with its physical lengthremaining unchanged, while in the other it is that physicallength that is intensity-dependent.

2.2 Effects of retardation

In general the optical field does not respond instantly tothe motion of the mechanical oscillator, therefore we needto account for the effects of retardation as well. We pro-ceed by assuming that he system is in equilibrium at somemirror position x0 with intracavity field 0, taken to bereal without loss of generality, and consider the linearizeddynamics of small displacements x(t) and (t) fromthat state under the effect of an external force F (t ),

x+x+2mx = G0(+) ,

= (i/2)+ iG0x. (19)These equations of motion can easily be solved, for in-stance in Fourier space, to give

()=(

iG0i (+)+/2

)x() (20)

where

=+Gx0, (21)resulting in a modification of the radiation pressure force

Frp()=G0[()+()] . (22)

Together with Eq. (20) this expression shows that the mir-ror motion exerts a dynamical back-action on the radi-ation pressure force, which acquires both a real and animaginary component, the physical origin of the imagi-nary component being the delayed response of the intra-cavity field. As a result the intracavity power acquires acomponent that oscillates out of phase with the mirrormotion, that is, with its velocity. It is through that frictionforce that the optical field acts as a viscous field for themirror.

The net effect of the real and imaginary components ofFrp can be conveniently cast in terms of the back-action



frequency shift opt and a damping rate opt, see e.g.Ref. [21, 22], with

opt =G2202mm

[+m

(+m)2+2/4+ +m

(m)2+2/4]

(23)

and

opt =G2202mm

[

(+m)2+2/4

(m)2+2/4]

.

(24)

For detunings m the first term in Eq. (24) dom-inates over the second term, and the dynamical back-action results in an increase in the mechanical dampingof the mechanical oscillator and cooling, see Eq. (4). Itis therefore the asymmetry between the response func-tion of the Fabry-Prot at the frequencies of the two sidemodes that is responsible for cooling or anti-dampingif one changes the sign of and uses a blue-detuned in-stead of a red-detuned driving field. In particular, in theresolved sideband limitm we find

opt (

2

) G220mm

, (25)

which can in principle be increased arbitrarily (within thelimits of validity of the model) by increasing the incidentoptical power.

Together with Eq. (4) this analysis predicts that thecooling of the center-of-mass motion of the mirror canbe arbitrarily close to Teff = 0, a consequence of the factthat the optomechanical coupling between the intracav-ity field and the mirror results in the scattering of thethe driving field into an anti-Stokes line that is stronglydamped due to the high density of states at the cavity res-onance. Conversely, for the opposite detuning mit is the Stokes line that is strongly damped, resulting inanti-damping of the mirror motion. This can lead to para-metric oscillations and dynamical instabilities, a situationfurther discussed in section 3.5.

The quantum description of the next section will showthat cold damping and mirror cooling can also be inter-preted in terms of of the annihilation of phonons from thecenter-of-mass mode of oscillation when scattering thedriving laser field into the anti-Stokes sideband. Heatingcan similarly be understood as resulting from the creationof phonons associated with the scattering of the drivingfield into the lower frequency Stokes side mode.

Figure 2 (Color online) Schematic of sideband cooling: a co-herent light field driving the resonator acquires frequency side-bands due to the mirror oscillations. The origin of the highfrequency sideband is the parametric transfer of phonons fromthe mirror to the optical field and the lower sideband is due tothe reverse process, see section 2.2. Sideband cooling resultswhen the upper sideband frequency is resonant with the res-onator. The solid black curve depicts the resonator transmissionnear its mode of frequency c .

2.3 Quantum limit

The classical prediction that one can in principle reach anarbitrarily large degree of cooling needs to be revised toaccount for the effects of quantum and thermal noise. Asis well known, the open port of the interferometer usedto supply the optical drive of the oscillating mirror alsoallows for the coupling of vacuum fluctuations into theresonator, see e.g. Ref. [24]. This leads to a fundamen-tal limit to the degree of cooling that can be achieved. Aproper quantum description of the system must accountfor this effect as well as for the the bosonic nature of thephonons.

Ignoring in a first step the important effects of fluctua-tions and dissipation, and in case a single optical modeof the Fabry-Prot resonator and a single mode of oscilla-tion of the suspended mirror need to be considered, theoptomechanical Hamiltonian is simply

H =(x)aa+ p2

2m+ 1

2m2m x

2, (26)

where a and a are bosonic annihilation and creation op-erators for the cavity mode of frequency, and p and x arethe momentum and position of the oscillating mirror ofmass m and frequencym . In reality, though, this Hamil-tonian is more subtle than may appear at first. This isbecause the mode frequency (q) depends on the lengthof the resonator, which in turn depends on the intracavityintensity. Stated differently, the boundary conditions forthe quantization of the light field are changing in time,and do so in a fashion that depends on the state of that



field and its history. The rigorous quantization of this sys-tem is a far-from-trivial problem, but for most cases ofinterest in quantum optomechanics the situation is signif-icantly simplified since the transit time c/2L of the lightfield through the optical resonator is much faster than themechanical frequencym . The intracavity field thereforelearns about changes in its environment in times shortcompared to 1/m . Under these conditions one can as-sume that the cavity frequency follows adiabatically anychange in resonator length,

(x)= npicL+ x =c

(1

1+ x/L)c (1 x/L) (27)

where n is an integer that labels the mode of nominalfrequencyc and L is the nominal resonator length (in theabsence of light.) In the classical limit we recover the resultG =c/L valid for a simple Fabry-Prot. The Hamiltonian(26) reduces then to [2527]

H = c aa+ p2

2m+ 1

2m2m x

2Gaax (28)

= c aa+ p2

2m+ 1

2m2m x

2g0aa(b+ b).In the second line we have used the familiar relationshipbetween the position operator x and the annihilation andcreation operators b and b of the mechanical oscillator,

x = xzpt(b+ b) (29)with

xzpt =

2mm

. (30)

We also introduced the optomechanical coupling fre-quency

g0 = xzpfG =xzpfc/x, (31)which scales the optomechanical displacement to thezero-point motion of the mechanical oscillator. TheHamiltonian (28) is the starting point for most quantummechanical discussions of cavity optomechanics.

In order to establish the theoretical limit to cavity op-tomechanical cooling, it is necessary to expand the de-scription provided by the Hamiltonian (28) to account forthe optical drive of the resonator, cavity damping, and themechanical damping of the oscillator. This analysis wascarried out in Refs. [2830]. The main message of thesepapers is that at least for constant optomechanical cou-pling the best cooling can be achieved in the so-called re-solved sideband limit, m , with the minimum meanphonon number

nm =optn0m +nTm

+opt. (32)

Here n0m is the mean steady-state number of phonons inthe absence of mechanical damping, given by the detailedbalance expression

n0m +1n0m

= (+m)2+2/4

(m)2+2/4 exp

( mkBTeff

), (33)

nTm is the equilibrium phonon occupation determined bythe mechanical bath temperature, and

opt =g 20aa

2mm(34)

[

(+m)2+2/4

(m)2+2/4]

.

For nTm 0 one recovers the classical result of Eq. (4).If the optical damping opt dominates, opt , though,the mean phonon number is limited in the resolved side-band limitm to

n0m =(

4m

)2, (35)

which shows that the ground state can be approached, butnot reached, in that case. As expected from the classicalconsiderations of the preceding section, this is the bestpossible case. In practice, the theoretical limit (35) is diffi-cult to reach due to technical noise issues including lasernoise [31, 32], clamping noise [33], etc. but the discussionof these topics in beyond the scope of this brief review.Remarkably though, these experimental challenges havenow being overcome in several experiments, see section2.4. We also note that using pulsed optomechanical inter-actions may lead to improved cooling limits [34, 35]. Wereturn briefly to this point in section 3.7.

Importantly, we remark that optomechanical sidebandcooling is formally identical to the cooling of harmoni-cally trapped ions, or more generally of any harmonicallytrapped dipole, see Ref. [36] for a nice discussion of thispoint. In the case of trapped ions, the resolved sidebandcooling limit was understood as early as 1975 [37, 38], andthe ground state cooling of trapped ions was first demon-strated over 20 years ago [39, 40]. As already mentioned,the key new element contributed by cavity optomechan-ics is the use of engineered resonance-enhancing struc-tures.

2.4 Experimental status

Following the pioneering work on gravitational wave an-tennas, advances in material science and nanofabrica-tion in particular in microelectromechanical systems(MEMS), nanoelectromechanical systems (NEMS), and



Figure 3 (Color online) Artist conception of the microwaveoptomechanical circuit of Ref. [42]. Capacitor element of theLC circuit is formed by a 15 micrometer diameter membranelithographically suspended 50 nanometers above a lower elec-trode. Insert: cut through the capacitor showing the membraneoscillations. After Ref. [44].

ba

0.1

1

10

100

Occ

upan

cy

100 101 102 103 104 105

Drive Photons, nd

nm nc

10-32

10-31

10-30

10-29

S x (

m2 /

Hz)

10.55810.556Frequency (M Hz)c

n =27m

n =22m

n =8.5m

n =2.9m

n =0.93m

n =18d

n =71d

n =280d

n =1,100d

n =4,500 d

10-15

2

4

68

10-14

2

4

10-13

S/P o

(1/

Hz)

10.710.610.510.410.3Frequency (M Hz)

n =0.93m

n =0.55m

n =0.36m

n =0.34m

n =0.42mg /

n =4,500 d

n =11,000d

n =28,000d

n =89,000d

n =180,000d

Figure 4 (Color online) Phonon occupancy (blue) and intra-cavity photon occupancy (red) as a function of the drive photonnumber. In this example sideband cooling reduces the thermaloccupancy of the mechanical mode from nm=40 into the quan-tum regime, reaching a minimum of nm=0.34 0.05. FromRef. [42], with permission.

optical microcavities opened up the possibility to ex-tend these ideas in many new directions, leading to thedemonstration of significant cooling in a broad varietyof systems from 2006 on, see Refs. [1518], with the firstdemonstration of cooling in the resolved sideband regimereported in Ref. [36].

More recently these efforts have culminated in thecooling of the center-of-mass motion of at least three dif-ferent micromechanical systems with a mean phononnumber within a fraction of a phonon of their groundstate of vibrational motion, nm < 1 [4143]. We post-pone a discussion of Ref. [41] until the next section toconcentrate first on the two experiments [42, 43] thatutilized resolved sideband cooling to approach the me-

chanical ground state of center-of-mass motion. In onecase [42] the mechanical resonator was a suspended cir-cular aluminum membrane tightly coupled to a super-conducting lithographic microwave cavity. That cavitywas precooled to 20mK, corresponding to an initial occu-pation of 40 phonons and then further cooled by radia-tion pressure forces to an average phonon occupation ofnm 0.3. In contrast, Ref. [43] utilized an optomechan-ical structure with co-located photonic and phononicband gaps in a suspended on-chip waveguide. The struc-ture was precooled to 20K, corresponding to about 100thermal quanta, and then cooled via radiation pressureto nm 0.85. Shortly thereafter, that same group alsoobserved the motional sidebands generated on a secondprobe laser by a mechanical resonator cooled opticallyto near its vibrational ground state. They were able to de-tect the asymmetry in the sideband amplitudes betweenup-converted and down-converted photons, a smokinggun signature of the asymmetry between the quantumprocesses of emission and absorption of phonons [45].

3 Beyond the ground state

3.1 Strong coupling regime

Cooling mechanical resonators to their ground state ofmotion is an essential first step in eliminating the ther-mal fluctuations that normally mask quantum features.However, by itself that state is not particularly interest-ing, so the next challenge is to prepare, manipulate andcharacterize quantum states of the mechanical resonatorrequired for a specific science or engineering goal. Animportant first experimental step in that direction wasreported in Ref. [41]. In contrast to Refs. [42] and [43]this experiment did not rely on radiation pressure cool-ing to achieve the motional ground state. Because of itshigh frequency of about 6 GHZ, a conventional dilutionrefrigerator that can reach temperatures of about 25 mKwas sufficient to cool it to nm < 0.07. A key point of theexperiment is that it succeeded in coupling an acousticresonator to a two-state system, or qubit, that could detectthe presence of a single mechanical phonon. This is analo-gous to protocols that have been developed over the yearsin cavity quantum electrodynamics, see e.g. Ref. [46], withthe important distinction that photons are now replacedby phonons.

The coupling between a bosonic field mode and oneor more two-state systems paves the way to a numberof approaches to prepare and to observe genuine quan-tum features such as the energy quantization of the res-



onator, or to make controlled state manipulations at thefew phonons level. Many of those protocols have alreadybeen developed in quantum optics and can be readily ap-plied to phonon fields, at least in principle. In all casesdissipation and decoherence must be reduced to a mini-mum, as they rapidly lead to the destruction of the mostsalient quantum features of the state. In the experiment ofRef. [41] decoherence was just weak enough to observe afew coherent oscillations of a single quantum exchangedbetween the qubit and the mechanical structure. As suchit can be considered as the first demonstration of the capa-bility of coherent control of phonon fields in a microme-chanical resonator.

Generally speaking, and in complete analogy with thesituation in quantum optics and in cavity QED, the con-trol of the quantum state of a mechanical oscillator re-quires that one operates in the so-called strong couplingregime, where the energy exchange between the mechan-ical object and the system to which it is coupled anoptical field mode, a qubit, an electron, etc. is not nega-tively affected by dissipation and decoherence. Section 2.3showed that at the simplest level the optomechanical in-teraction takes the form (28),

H =c aa+ p2

2m+ 1

2m2m x

2g0aa(b+ b). (36)

At the single photon level, aa = 1, this interaction isusually much too weak for its coherent nature to domi-nate over the incoherent dynamics for realizable levels ofdecay and decoherence. Since for aa 1 the quantumnature of the optical field normally rapidly decreases inimportance, it is therefore challenging to reach situationswhere the full quantum nature of the interaction betweenthe photon and phonon fields is significant. There is a wayaround this difficulty, though, the trade-off being that theintrinsic nonlinear nature of the optomechanical inter-action (36) disappears in the process to be replaced by alinear effective interaction. As we shall see, this is not allbad, as that effective interaction offers itself a number ofnew opportunities.

Our starting point is the observation that strong in-tracavity optical fields can usually be decomposed as thesum of a classical, or mean-field part and a small quan-tum mechanical component. In terms of the mean fieldof the optical field mode = aa+ c (37)where c is again a photon annihilation operator. The op-tomechanical coupling term in the Hamiltonian (36) be-comes then

Hint =g0n(b+ b)g(c+ c

)(b+ b) (38)

where we have introduced the optomechanical couplingstrength

g = g0pn, (39)

n = ||2, and we have taken to be real for notational con-venience. The first term in the Hamiltonian (38) describesa simple Kerr effect, with a change in resonator length pro-portional to the classically intracavity intensity. This is theterm that leads to the radiation pressure induced opticalbistability observed e.g. in the experiments of Dorsel etal [10].

In a frame rotating at the driving field frequency, thecavity frequency and the mechanical frequency the sec-ond term in Eq. (38) can be reexpressed as

V = g[bcei (+m )t +h.c.

] g

[bcei (m )t +h.c.

](40)

This interaction describes the linear coupling betweenthe quantized component of the optical field and the me-chanical oscillator. The coupling g is enhanced from thesingle-photon optomechanical coupling frequency g0 bya factor

pn, which can be very substantial. Note however

that this enhancement comes at the cost of losing the non-linear character of the original interaction g0aa(b+ b).That nonlinear character is at the origin of a number ofquantum effects that are expected to appear when theradiation pressure of a single (or of very few) photonsdisplaces the mechanical oscillator by more than xzpf.These include two-photon blockade as well as quantita-tive changes in the output spectrum and cavity responseof the optomechanical system, leading for example to thepossible generation of non-Gaussian steady states of theoscillator [4749].

The linear coupling of Eq. (40) provides exciting oppor-tunities as well, and these are significantly less challengingto realize experimentally. On the red-detuned side of theFabry-Prot resonance, =m , we have after invokingthe rotating wave approximation

V 'g(bc+h.c.

), (41)

the so-called beam-splitter Hamiltonian of quantum op-tics. In contrast, in the blue-detuned side of the resonance,=+m , we haveV 'g

(bc+h.c.

), (42)

which describes the parametric amplification of thephonon mode and the optical field.

This approach has enabled experiments to reach theregime of strong phonon-photon optomechanical cou-pling in several micromechanical devices [5052]. A fa-miliar characteristic of strongly coupled systems is the



occurrence of normal mode splitting. For the Hamilto-nian (40) the normal mode frequencies are

= 12

[2+2m

(22m

)2+4g 2m]1/2 . (43)The first demonstration of normal mode coupling in anoptomechanical situation was realized by Grblacherand coworkers [50]. As pointed out by these authors theoptomechanical modes can be interpreted in a dressedstate approach as excitations of mechanical states thatare dressed by the cavity radiation field. Alternatively,they can also be interpreted as optomechanical polari-ton modes. Teufel and coworkers [51] carried a seriesof experiments in the strong coupling regime of quan-tum optomechanics. They measured the dressed cavitystates as a function of the pump-probe experiment wherethe coupling (39) was controlled by a pump field andthe resonator transmission measured by a weak probefield. Increasing the strength of g allowed them to mon-itor the change in cavity transmission as the strong cou-pling regime was reached, with an intermediate regimewhere the interference between the pump and probe fieldresults in an effect analogous to electromagnetically in-duced transparenty [53, 54].

It should be emphasized that by itself, the observationof normal mode splitting, which is both a classical andquantum feature of coupled systems, does not prove theexistence of coherent exchange of excitations between themechanical and optical field modes. An important steptoward the demonstration of quantum coherent couplingwas recently achieved by Kippenberg and coworkers [52]in a system where the optomechanical coupling is de-scribed by the beam splitter interaction (41). This exper-iment considered a micro-mechanical oscillator cooledto a mean phonon number of the order of nm 1.7, andin addition excited the system with a weak classical lightpulses to achieve coherent coupling between the opticalfield and the micromechanical oscillator and the level ofless than one quantum on average. These results, whilestill preliminary in many ways, open up a promising routetowards the use of mechanical oscillators as quantumtransducers, as well as in microwave-to-optical quantumlinks as we now discuss.

3.2 State transfer

The beam-splitter Hamiltonian (41) describes the coher-ent exchange of cavity photons and mechanical phonons.One of its remarkable properties is that it offers the poten-tial to precisely transfer the quantum state of the mechan-ical oscillator to the electromagnetic field, and conversely.

0

1

Pro

be tr

ansm

issi

on, |

T|2

7.47407.47357.47307.4725Probe frequency,

102

103

104

105

106

100 102 104 106

7.4740

7.4735

7.4730

7.4725

a

c

bnd=5x10

-1

/2 (GHz) p

/

2 (G

Hz)

p g

/ (H

z)

n d

nd=5x100

nd=5x101

nd=5x102

nd=5x103

nd=5x104

nd=5x105

nd=5x106

Cou

plin

g ra

te,

g /

Drive photons,

Figure 5 (Color online) Normalized cavity transmission forincreasing resonator drive intensity nd = ||2. For moderatedrive intensities the interference between the drive and probephotons results in a narrow peak in the cavity spectrum, theonset of electromechanically induced transparency. For higherintensities the cavity resonance then splits into normal modes.From Ref. [51], with permission.

This is seen easily by considering the Heisenberg equa-tions of motion for the annihilation operators b and c inthe absence of decay,

b(t ) = b(0)cos(g t )+ i c(0)sin(g t ),c(t ) = c(0)cos(g t )+ i b(0)sin(g t ). (44)The optomechanical interaction g can easily be madetime dependent by pulsing the classical driving laser fieldintensity, n n(t ). For an interaction time tint and a driv-ing laser pulse intensity such that

g0 tint

0dtn(t )t =pi/2



we then have that b(tint)= c(0) and c(tint)= i b(0), indica-tive of a perfect state transfer between the optical andphonon modes assuming of course that dissipation anddecoherence can be ignored during that time interval.

The interest in devices capable of high-fidelity statetransfer between optical and acoustical fields is largelymotivated by its potential for quantum information ap-plications. This is because due to their potentially slowdecoherence rate motional states of mechanical systemsare well suited for information storage. However mechan-ics does not permit fast information transfer, while op-tical fields are ideal as information carriers, but are typ-ically subject to fast decoherence that limits their inter-est for storage [55]. The coherent quantum mapping ofphonon fields to optical modes also promises to be use-ful in quantum sensing applications, by combining theremarkable sensitivity of nanoscale cantilevers to feebleforces and fields with reliable and high-efficiency opti-cal detection schemes. And in addition to standard statetransfer between motional and optical states, phononfields could also serve as convenient transducers betweenoptical fields of different wavelengths, or between opticaland microwave fields.

The first theoretical proposal that analyzed a schemeto transfer quantum states from a propagating light fieldto the vibrational state of a movable mirror by exploit-ing radiation pressure effects is due to Jin Zhang andcoworkers [56]. This work was then expanded in severaldirections, especially in the context of quantum optome-chanics. For instance Tian and Wang [57] proposed anoptomechanical interface that converts quantum statesbetween optical field of distinct wavelengths through asequence of optomechanical pi/2 pulses. In another re-cent proposal, Didier et al. [58] considered exploiting thebeam splitter coupling of a mechanical oscillator and amicrowave resonator to measure and synthesize quantumphonon states, and also to generate and detect entangle-ment between phonons and photons. They also proposedgenerating the entanglement of two mechanical oscilla-tors and its detection by the cavity field after entangle-ment swapping. The first experimental demonstration ofstate transfer between a microwave field and a mechani-cal oscillator with amplitude at the single quantum levelwas recently achieved by Palomaki et al. [59].

3.3 Two-mode squeezing

The Hamiltonian (42) is essentially the familiar two-modesqueezing Hamiltonian of quantum optics. This becomesreadily apparent if one accounts for the (controllable)

phase of the classical driving field, so that

g = g0pn i g0

pn exp(i)

and

V =i[g bc gbc

](45)

with the associated evolution operator

Sab(t )= exp[(gbc g bc)t ], (46)the well-known unitary two-mode squeezing operator. In-troducing the generalized two-mode quadrature operator

Xab =1

23/2(c+ c+ b+ b) (47)

one finds that the variance of a system initially in a two-mode vacuum state is given by [60]

(X )2 = 14

[e2|g |t cos2(/2)+e2|g |t sin2(/2)] . (48)

That same result also holds if the two modes are initiallyin coherent states. For the choice = pi/2 one finds im-mediately that (X )2 can be well below the standardquantum limit of 1/4, a signature of two-mode squeez-ing. Two-mode squeezed states are known to be entan-gled, indicating that this form of interaction can result inquantum entanglement between the photon and phononmodes. As such this configuration represents a useful re-source for demonstrating fundamental quantum mechan-ical effects as well as for exploiting cavity optomechanicaldevices in a quantum information context.

We note for completeness that in early work, Fabreand coworkers [61], and independently Mancini andTombesi [62] exploited the analogy between the situa-tion of an optical resonator and a cavity filled by a Kerrmedium to predict single mode squeezing of the reflectedoptical field in situations where the motion of the mirroris dominated by thermal fluctuations and can be treatedclassically.

3.4 Squeezing via back-action evadingmeasurements

As shown by Braginsky et al. [63] and further analyzed byClerk et al. [64] it is possible to implement back-actionevading measurements of the membrane position whendriving it with an input field resonant with the cavity fre-quency c , but modulated at the mirror frequency m .The mean-field amplitude of the intracavity field is then

(t )=pn cos(m t ), (49)



where

=pin

im +/2. (50)

Introducing the quadratures

X (t ) = 1p2

(ce im t + ceim t

),

Y (t ) = ip2

(ce im t ceim t

)(51)

of the motional mode, with [X (t ), Y (t )]= i andx(t )=p2xzpt

(X (t )cosm t + Y (t )sinm t

), (52)

and keeping as before only linear terms in the quantumcomponent of the field, the optomechanical interactionHamiltonian reduces then to

V =p2g [X (1+cos(2m t ))+ Y sin(2m t )] (b+ b)(53)

where g = g0= g0pn as before.

In a time-averaged sense the interaction Hamilto-nian (53) reduces to

V p2g X (b+ b) (54)and commutes with X , thus giving rise to the possibil-ity of performing a back action evading measurement ofthe X quadrature of mirror motion. This was verified ex-perimentally in the classical regime by J. B. Hertzberg etal. [68], but not yet in the quantum regime so far.

Since the interaction (54) is linear in X it is perhapsless evident that it can also lead to quadrature squeezing.This can be achieved by first performing a precise mea-surement of X , following which its quadrature can clearlybe below the standard quantum limit. Following that mea-surement the system would normally rapidly relax back toa classical state, but by applying an appropriate feedback,the measurement induced squeezing can be turned intoreal squeezing. This is discussed in detail in Ref. [64].

3.5 Parametric instability

We have seen that for a driving laser red-detuned from thecavity frequency c the upper sideband is resonantly en-hanced by the cavity, which leads to preferred extractionof mechanical energy, i.e. cavity cooling. For blue-detunedlight, in contrast, it is the lower sideband that is reso-nantly enhanced by the cavity, resulting in the preferreddeposition of mechanical energy, i.e. the optical amplifi-cation of mechanical motion. Invoking the rotating-wave

approximation for +m one finds that this processis described at the simplest level by the 2-mode squeez-ing interaction (42) instead of the beam-splitter Hamilto-nian (41) of section 3.2.

In that regime the optomechanical system can displaydynamical instabilities. For appropriate parameters theyresult in stable mechanical oscillations somewhat remi-niscent of laser action, but for a phononic field [65, 66],or even in unstable dynamics and chaos [67]. That thiscan be the case is already apparent at the classical levelfrom the fact that opt can become negative for blue de-tuning, see Eq. (34). If the laser intensity is strong enoughthat the total damping rate +opt is itself negative, thenany amplitude oscillation will grow exponentially until itsaturates due to the onset of nonlinear effects.

Following Ludwig et al. [69] we assume that the motionof the cantilever is approximately sinusoidal,

x(t ) x+ A cos(m t ), (55)

with the average position x given by the radiation pressureforce,

x = 1m2m

Frad =G

m2m|(t )|2 (56)

where (t )|2 is the intracavity light intensity and A is theamplitude of oscillations of the mirror. Marquardt andcoworkers [71] showed that with Eq. (55), Eq. (8) yields forthe intracavity field

(t )= e i(t )nne

inm t , (57)

with

n =(max

2

) Jn(GA/m)inm/+ i (Gx)/+1/2

. (58)

Here (t) = (GA/m)sin(m t) and Jn are Bessel func-tions of the first kind. The stability of the system can bedetermined simply comparing the mechanical power Praddue to radiation pressure to the dissipated power Pfric dueto friction. When their ratio increases above unity the sys-tem starts to undergo self-induced oscillations [69].

Figure 6 is an example of a stability diagram deter-mined from such an analysis. It shows the ratio Prad/Pfricas a function of the detuning and the square of the(dimensionless) mechanical energy A2. Regions withPrad/Pfric > 1 are unstable, and the solid line defines an at-tractor where there is an exact power balance between am-plification and damping. In general the parameter space(,A) is characterized by the presence of a number of suchattractors. For relatively weak amplitudes A, nonlineareffects tend to stabilize the oscillations of the cantilever,



Figure 6 (Color online) Attractor diagram obtained from therequirement that the optical power fed into mechanical oscil-lations is balanced by the power lost to friction. Adapted fromFrom Ref. [69], with permission.

leading to "laser-like" oscillations [65,66], but for larger os-cillations amplitudes the system can become chaotic [67].Quantum mechanically fluctuations are strongly ampli-fied just below threshold, so that the attractor is no longersharp [70].

3.6 Quadratic coupling

So far we have considered geometries where the optome-chanical coupling is linear in the oscillator displacement.Other forms of coupling can however be considered, mostinterestingly perhaps a coupling quadratic in the displace-ment. This can be realized in so-called membrane-in-the-middle geometries, as first demonstrated by J. Harrisand his collaborators at Yale [72, 73], see also Refs. [74, 75].As implied by its name, this geometry involves an oscil-lating mechanical membrane placed inside a Fabry-Protwith fixed end-mirrors.

An attractive feature of membrane-in-the-middle con-figurations is the ability to realize relatively easily eitherlinear or quadratic optomechanical couplings, dependingon the precise equilibrium position of the membrane. Incase the membrane is located at an extremum ofc (x), sothatG =c/x = 0, see Eq. (6), we have to lowest order

c (x)c +1

2

2c

x2(59)

so that the optomechanical Hamiltonian becomes

H =c aa+M bb+ 12

2c

x2x2zpt(b+ b)2aa. (60)

In the rotating wave approximation this reduces to

H = c aa+M bb+x2zpf2c

x2

(bb+1/2

)aa

= c aa+M bb+g (2)0(bb+1/2

)aa (61)

where

g (2)0 x2zpf2c

x2. (62)

Quadratic coupling opens up the way to a number of in-teresting possibilities, including the direct measurementof energy eigenstates of the mechanical element, ratherthan the position detection characteristic of linear cou-pling. J. Harris and coworkers estimate that it may be pos-sible in the future to use this scheme to observe quantumjumps of a mechanical system [73]. In another theoreti-cal study, Nunnenkamp and coworkers [76] consideredoptomechanical cooling and squeezing via quadratic op-tomechanical coupling. They showed that for high tem-peratures and weak coupling, the steady-state phononnumber distribution is nonthermal, and demonstratedhow to achieve mechanical squeezing by driving the cav-ity with two optical fields.

Another possibility offered by that geometry is to ob-serve the quantum tunneling of an optomechanical sys-tem operating deep in the quantum regime through aclassically forbidden potential barrier. One proposed ap-proach [77] relies on adiabatically raising a potential bar-rier, whose parameters can be widely tuned, at the lo-cation of a mechanical element. For the right choice ofparameters the optomechanical potential is a double-wellpotential, and it is estimated that quantum tunneling be-tween its wells can occur at rates several orders of magni-tude larger than the decoherence rate of the mechanicalmembrane. Besides tunneling, that scheme may also al-low for the study of the quantum Zeno effect in a mechan-ical context and provide a comparatively simple schemefor the preparation and characterization of non-classicalmechanical states of interest for quantum metrology andsensing.

3.7 Pulsed optomechanics

So far we have largely limited our discussion to situationswhere the optomechanical coupling is either constant orslowly varying in time. One notable exception was the



quantum state transfer protocol outlined in Section III.B,which requires that the interaction g (t) be turned off atthe precise time when the state transfer has been com-pleted. However there are a number of situations wherepulsed interactions are desirable, as already realized byBraginsky [8, 78] in his proposal for a back-action evadingposition measurement scheme. In a recent paper, Vannerand coworkers [79] proposed to use a pulsed interactionof duration short compared to the period of the mechani-cal oscillator to generate and fully reconstruct quantumstates of mechanical motion: As a result of the interac-tion the phase of the pulsed driving optical field becomescorrelated with the position of the mechanical oscillator,while its intensity imparts it a momentum boost. A timedomain homodyne detection scheme can then be used tomeasure the phase of the field emerging from the cavity,thereby providing a measurement of the mechanical posi-tion. This scheme can also be used to achieve squeezingand state purification of the mechanical resonator. It hasalso recently been proposed that pulsed optomechanicscould be used, at least in principle, to surpass the limitsof conventional sideband cooling by using an optimizedsequence of driving optical pulses [34, 35].

In an intriguing potential application of pulsed op-tomechanics, Pikovski and colleagues [80] considered ascheme to measure the canonical commutator of a mas-sive mechanical oscillator and by doing so to detect pos-sible commutator deformations due to quantum grav-ity: there are speculations that the existence of a min-imum length scale where space-time is assumed to bequantized, possibly of the order of the Planck lengthLP = 1.61035m, could result in such deformations. Inthis proposal a sequence of optomechanical interactionswould be used to map the commutator of the mechanicalresonator onto an optical pulse. Remarkably the analysisof Ref. [80] suggests that as a result Planck-scale physicsmight be observable in a relatively mundane quantumoptics experiment.

4 Cold atoms

In a development complementary to the research onnanoscale mechanical systems, recent quantum optome-chanics experiments have also manipulated and con-trolled at the quantum level the center-of-mass degreesof freedom of ultracold atomic ensembles [8184]. In thefollowing we restrict our discussion to the case of a neutralatomic sample cooled well below its recoil temperatureand trapped inside a single-mode Fabry-Prot resonator.This could be for example a nearly homogeneous and col-

lisionless Bose-Einstein condensate (BEC) at T 0 or asample cooled near the vibrational ground state of one ora few wells of the optical lattice formed by the optical field.Side mode excitations of the condensate in the first case,and the vibrational motion of thermal atoms in the sec-ond case, provide formal analogs of one or several movingmirrors.

To see how this works we consider first a generic modelconsisting of a BEC at T = 0 trapped inside a Fabry-Protcavity of length L and mode frequency c . The atoms ofmass M are driven by a pump laser of frequency L andwave number k. When L is far detuned from the atomictransition frequency a the excited electronic state of theatoms can be adiabatically eliminated and the atoms in-teract dispersively with the cavity field. In the dipole androtating-wave approximations, the Hamiltonian describ-ing the interaction between the atoms and the opticalfield is

H =Hatom+Hfield, (63)

where

Hfield =c aa (64)

and

Hbec =dx(x)

[p2x2M

+U0 cos2(kx)aa](x). (65)

Here (x) is the bosonic Schrdinger field operator forthe atoms, a is the photon annihilation operator as before,and the atoms interact with the light field via the familiaroff-resonant coupling

U0 = g 2R/(L a), (66)

where gR is the single-photon Rabi frequency. As alwaysH should be complemented by contributions describingthe external driving of the cavity field, dissipation andcollisions.

When the light field can be approximated as a planewave the atomic field operator can likewise be expandedin terms of plane waves as

(x)= (1/pL)qbke

i qx , (67)

where bq and bq are annihilation and creation opera-

tors for atomic bosons with the momentum k, satisfyingthe bosonic commutation relations [bq , b

q ] = q,q and

[bq , bq ]= 0.Consider for simplicity the case of scalar bosonic

atoms: In the absence of light field and atT = 0 the ground



state of the sample would be a condensate with zero mo-mentum,

|0 = (b0)N |0, (68)

but as a result of virtual transitions the atoms can acquirea recoil momentum 2`k, where ` = 0,1,2, . . . In thelimit of low photon numbers it is sufficient to considerthe lowest diffraction order, ` = 1 and the atomic fieldoperator can be conveniently expressed in terms of a zero-momentum component and a sine mode,

b00(x)+ b22(x) (69)where 0(x) is the condensate wave function and 2(x)=p

2cos(2kx). For very weak optical fields the occupationof the sine mode remains much smaller the the zero-momentum mode, so that b0 '

pN and b2b2N . Sub-

stituting then Eq. (69) into the Hamiltonian (65) and in aframe rotating at the pump laser frequency the Hamilto-nian H becomes

Hom,bec = 4recb2b2+aa[+ g2(b2+ b2)

], (70)

where

g2 = (U0/2)pN/2 (71)

is the effective atom-field coupling constant, rec =K 2/2M the recoil frequency, and =c L+U0N/2 isan effective Stark-shifted detuning .

The reduced Hamiltonian (70) describes the couplingof two oscillators, the cavity mode a and the momentumside mode b2 via the optomechanical coupling g2aa(b2+b2). This shows that the condensate momentum sidemode behaves formally like a moving mirror driven bythe radiation pressure of the intracavity field, see Eq. (28)for comparison.

A similar analogy can be established when consider-ing a sample of ultracold atoms tightly confined to anharmonic trap of frequency z centered at some locationz0 along the resonator axis. The position of atom i is thenzi = z0+zi , and the vacuum Rabi frequency with whichit interacts with the field is

gR (zi )= gR sin(0+2kzi ), (72)

where 0 = kz0 so that Eq. (66) becomes

U0 = g (zi )2

L a. (73)

Summing over all atoms in the sample and expandingthen the far off-resonant atom-field interaction to lowest

order in Kzi one finds for L =c [85]H (c +NU0 sin20)aa+z

ibi bi

+ U0 sin(20)aa[

ikzi

](74)

where N is the number of atoms and the operator bi de-scribes the annihilation of a phonon from the center-of-mass motion of atom i .

The second line of the Hamiltonian (74) describes theoptomechanical coupling of the intracavity optical fieldto the collective atomic variable

kizi = kNZcm (75)

which is nothing but a measure of the normal mode ofthe sample, its center of mass Zcm =N1i zi . For smalldisplacements that mode can be described as a harmonicoscillator of frequency z and mass NM . In this picture,the atom-field system is therefore modeled by the optome-chanical Hamiltonian

Hom,at =c aa+z bb+gN (b+ b)aa, (76)

where b and b are bosonic annihilation and creationoperators for the center-of-mass mode of motion of theatomic ensemble, zzpf =

/2Nmz andgN =NU0(K zzpf)sin20 (77)

with scales aspN . Quantum optomechanics experiments

with non-degenerate ultracold atoms samples have so farbeen carried out principally in the group of D. Stamper-Kurn at UC Berkeley, while T. Esslinger and coworkers atETH Zrich have concentrated on the use of Bose conden-sates [86]. In a trailblazing experiment [85] Purdy et al po-sitioned a sample of cold atoms with sub-wavelength ac-curacy in a Fabry-Prot cavity to demonstrate the tuningfrom linear to quadratic optomechanical coupling fromthe linear to the quadratic coupling regime. The Berke-ley group also observed the measurement back-actionresulting from the quantum fluctuations of the opticalfield by measuring the cavity-light-induced heating of theatomic ensemble [81], the first observation of quantumback-action on a macroscopic mechanical resonator atthe standard quantum limit. More recent work [87] de-tected the asymmetric coherent scattering of light by acollective mode of motion of a trapped ultracold gas with0.5 phonons of average excitation, a result that comple-ments the work of Safavi-Naeini et al. [45] on the asymmet-ric absorption of light by a nanomechanical solid-stateresonator, see section 2.4.



150

100

50

0

15

10

5

0

1

0

-130 -120 -110 -100 -90 13012011010090Frequency (kHz)

Phot

on s

pect

rum

(pho

tons

)A)

B)

0.5

1.5

0.40.6

1

2

46

10

Phon

on o

ccup

atio

n

0.4 0.6 1 2 4 6 10 20Cooperativity

Figure 7 (Color online) (a) Asymmetric optical scattering fromquantum collective motion, with the measured Stokes side-bands [left panels, (red) circles] and anti- Stokes sidebands[right panels, (blue) circles] at various mean phonon numbers,characterized by the so-called cooperatively coefficientC . Fromtop to bottomC= 9.6; 1.9; 0.4. (b) Measured phonon occupationvs cooperativity. From Ref. [87], with permission.

Turning now to quantum degenerate gases, Brennekeet al. [82] studied the dynamics of a Bose condensate of87Rb atoms trapped inside a high-finesse Fabry-Prot anddriven by a feeble optical field. This experiment demon-strated the optomechanical coupling of a collective den-sity excitation of the condensate, showing that it behavesprecisely as a mechanical oscillator coupled to the cavityfield, in quantitative agreement with a cavity optomechan-ical model of Eq. (70). These authors also succeeded inapproaching the strong coupling regime of cavity optome-chanics, where a single excitation of the mechanical os-

cillator substantially influences the cavity field. In subse-quent work, the Bose condensate was irradiated from theside of the optical resonator, resulting in the demonstra-tion of a second-order quantum phase transition wherethe condensed atoms enter a self-organized super-solidphase, a process mathematically described by the Dickemodel of an ensemble of two-state systems coupled toa single-mode electromagnetic field. In contrast to thesituation in the usual Dicke model, where the two statesof interest are two atomic electronic levels coupled by adipole optical transition, in the present case the relevantstates are two different momentum states coupled to thecavity field mode [88, 89].

5 Outlook Functionalization and hybridsystems

The rapid progress witnessed by quantum optomechan-ics makes it increasingly realistic to consider the use ofmechanical systems operating in the quantum regimeto make precise and accurate measurements of feebleforces and fields [90]. In many cases, these measurementsamount to the detection of exceedingly small displace-ments, and in that context the remarkable potential forfunctionalization of opto and electromechanical devicesis particularly attractive. Their motional degree(s) of free-dom can be coupled to a broad range of other physicalsystems, including photons via radiation pressure froma reflecting surface, spin(s) via coupling to a magneticmaterial, electric charges via the interaction with a con-ducting surface, etc. In that way, the mechanical elementcan serve as a universal transducer or intermediary thatenables the coupling between otherwise incompatiblesystems. This potential for functionalization also suggeststhat quantum optomechanical systems have the potentialto play an important role in classical and quantum infor-mation processing, where transduction between differentinformation carrying physical systems is crucial.

Much potential for the functionalization of optome-chanical devices is offered by interfacing them with a sin-gle quantum object. This could be an atom or a molecule,but also an artificial atom such as a nitrogen vacancycenter (NV center) in diamond [91], a superconductingqubit [41, 92, 93] or a Bose-Einstein condensate [94]. Sev-eral theoretical proposals [91, 94100] and more recentlyexperimental realizations [101, 102] involving atomic sys-tems have been reported. For example, a recent experi-ment [103] realized a hybrid optomechanical system bycoupling ultracold atoms trapped in an optical lattice to amicromechanical membrane, their coupling being medi-



ated by the light field. Both the effect of the membrane mo-tion on the atoms and the back-action of the atomic mo-tion on the membrane were observed. Singh and cowork-ers [104] considered a variation on that scheme where aBose condensate is trapped inside a Fabry-Prot with amoveable end mirror driven by a feeble optical field. Theyshowed that under conditions where the optical field canbe adiabatically eliminated one can achieve high fidelityquantum state transfer between a momentum side modeof the condensate, see Eq. (69), and the oscillating end-mirror.

Artificial atoms such as NV centers are of much inter-est for hybrid optomechanical systems [91] due to theattractive combination of their optical and electronic spinproperties. Their ground state is a spin triplet [105] thatcan be optically initialized, manipulated and read-out bya combination of optical and microwave fields, and theyare characterized by remarkably long room-temperaturecoherence times for solid-state systems. As such, theyoffer much promise for applications e.g. in quantum in-formation processing and ultrasensitive magnetometry,where the spin is used as an atomic-sized magnetic sen-sor [106108]. In this context, a spin-oscillator systemof particular interest consists of a magnetized cantilevercoupled to the electronic spin of the NV center. A recentexperiment by Arcizet and colleagues demonstrated thecoupling of a nanomechanical oscillator to such a defectin a diamond nanocrystal attached to its extremity [109].

In two further recent demonstrations of the potentialof hybrid optomechanical systems, a mechanical oscil-lator was used to achieve the coherent quantum controlof the spin of a single NV center [111], and the coherentevolution of the spin of an NV center was coupled to themotion of a magnetized mechanical resonator to senseits motion with a precision below 6 picometers [110]. Theauthors of that experiment comment that it may soonbecome possible to detect the mechanical zero-point fluc-tuations of the oscillator.

More speculatively perhaps, micromechanical oscil-lators in the quantum regime offer a route toward newtests of quantum theory at unprecedented sizes and massscales. For instance, spatial quantum superpositions ofmassive objects could be used to probe various theo-ries of decoherence and shed new light on the transi-tion from quantum to classical behavior: In contrast tothe generally accepted view that it is technical issuessuch as environmental decoherence that rapidly destroysuch superpositions in massive objects and establishthe transition from the quantum to the classical world,some authors [112116] have proposed collapse mod-els that are associated with more fundamental mecha-nisms and the appearance of new physical principles.

Bouwmeester [117] has pioneered the idea that quantumoptomechanics experiments may shed light on this issueand on possible unconventional decoherence processes,and in recent work Romero-Isart has analyzed the require-ments to test some of these models and discussed thefeasibility of a quantum optomechanical implementationusing levitating dielectric nanospheres [118, 119].

Acknowledgements. This work is supported by the US Na-tional Science Foundation, by the DARPA ORCHID andQuASAR programs through grants from AFOSR and ARO,and by the US Army Research Office. We acknowledge en-lightening discussions with numerous colleagues, in particularM. Aspelmeyer, L. Buchmann, A. Clerk, H. Jing, M. Lukin, R.Kanamoto, K. Schwab, H. Seok, S. Singh, S. Steinke, M. Ven-galattore, E. M. Wright and K. Zhang.

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