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Quantum optomechanics offers the potential to investigate quantum effects in macroscopic quantum systems in extremely well-controlled experiments. In this paper we discuss one such situation, the dynamic stabilization of a mechanical system such as an inverted pendulum. The specific example that we study is a “membrane-in-the-middle” mechanical oscillator coupled to a cavity field via a quadratic optomechanical interaction, with cavity damping the dominant source of dissipation. We show that the mechanical oscillator can be dynamically stabilized by a temporal modulation of the radiation pressure force. We investigate the system both in the classical and

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Dynamic stabilization of an optomechanical oscillator

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Quantum optomechanics offers the potential to investigate quantum effects in macroscopic quantum systems in extremely well-controlled experiments. In this paper we discuss one such situation, the dynamic stabilization of a mechanical system such as an inverted pendulum. The specific example that we study is a “membrane-in-the-middle” mechanical oscillator coupled to a cavity field via a quadratic optomechanical interaction, with cavity damping the dominant source of dissipation. We show that the mechanical oscillator can be dynamically stabilized by a temporal modulation of the radiation pressure force. We investigate the system both in the classical and quantum regimes highlighting similarities and differences.


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PHYSICAL REVIEW A 90, 043840 (2014)


H. Seok, E. M. Wright, and P. MeystreB2 Institute, Department of Physics and College of Optical Sciences The University of Arizona, Tucson, Arizona 85721, USA

(Received 12 August 2014; published 20 October 2014)

Quantum optomechanics offers the potential to investigate quantum effects in macroscopic quantum systemsin extremely well-controlled experiments. In this paper we discuss one such situation, the dynamic stabilizationof a mechanical system such as an inverted pendulum. The specific example that we study is a “membrane-in-the-middle” mechanical oscillator coupled to a cavity field via a quadratic optomechanical interaction, withcavity damping the dominant source of dissipation. We show that the mechanical oscillator can be dynamicallystabilized by a temporal modulation of the radiation pressure force. We investigate the system both in the classicaland quantum regimes highlighting similarities and differences.

DOI: 10.1103/PhysRevA.90.043840 PACS number(s): 42.50.Pq, 42.50.Ct, 42.50.Wk

I. INTRODUCTION

Dynamic stabilization is the process in which an object,unstable with its static potential, is trapped harmonically dueto the influence of a high-frequency oscillating force [1].Dynamic stabilization was first introduced by Kapitza for aninverted classical pendulum stabilized by rapidly oscillatingexternal modulations [2]. Specifically, Kapitza’s invertedpendulum was stabilized by an oscillating pivot in the verticaldirection. It can be described by Newton’s equation of motion,

θ = sin θ

[ω2

0 − A�2

�cos(�t)

], (1)

where θ is the polar angle from the vertical line and A and� are the amplitude and frequency of the vibration of thepivot, respectively. The proper frequency of the pendulum isω0 = √

g/�, where g is the gravitational acceleration and � isthe length of the pendulum.

The physics underlying Kapitza’s pendulum is that largeand rapid oscillations of the pivot compared to the properfrequency of the pendulum allow the force acting on thependulum to alternate between an attractive force and arepulsive force in time, resulting in a net stabilizing forcefor appropriate conditions [3].

Dynamic stabilization of a quantum system driven by arapidly oscillating perturbation has been extensively studiedin several papers [4–6]. It has been proposed for the controlof a quantum system in the context of atom optics, forexample, in novel optical trapping [7] and in the stabilizationof a Bose-Einstein condensate (BEC) [8–11]. Such stabilizingmechanisms have also found applications in trapping ions inelectromagnetic fields [12,13], focusing of charged particlesin a synchrotron [3,14], stabilizing spin-1 BEC [15], and thecontrol of the superfluid-Mott insulator phase transition [16].

Cavity optomechanics, a research area exploring mechan-ical degrees of freedom coupled to electromagnetic fieldsinside optical or microwave cavities, involves a variety ofexperimental setups in which the mass of a mechanicaloscillator ranges from several attograms to kilograms [17–23].Recent experimental progress has demonstrated cooling amacroscopic mechanical oscillator to its motional groundstate [24–27], allowing to explore the quantum nature ofmassive objects [28,29]. Cavity optomechanics thus paves theway to investigate dynamic stabilization of a mechanical object

in either classical or quantum regime as well as at the boundaryof the classical and quantum regimes.

In this paper, we consider an optomechanical system inwhich a mechanical oscillator is coupled to a cavity field viaa quadratic optomechanical interaction. This situation can berealized, e.g., in an ensemble of ultracold atoms trapped at theextrema of an optical lattice in a high-finesse cavity [30], a BECtrapped in such a cavity [31], and the so-called membrane-in-the-middle geometry [32,33]; see Fig. 1. We have shown in aprevious paper [34] that a mechanical oscillator can be unstableif it is coupled to a cavity field via a quadratic optomechanicalinteraction with a negative coupling coefficient. Starting fromthis unstable configuration, we propose a stabilizing schemein which the radiation pressure force associated with thecavity field is modulated and explore features of dynamicstabilization of the mechanical motion in both classical andquantum regimes. In particular, we derive a time-averagedpotential and demonstrate that the mechanical oscillator can bestabilized within a certain parameter regime. We also show theclassical and quantum dynamics of the mechanical oscillatorunder the effects of the oscillating radiation pressure force aswell as dissipation.

Section II introduces our model system and the unstableconfiguration. A scheme for dynamic stabilization of the clas-sical system based on a time-averaged potential is proposed inSec. III, and Sec. IV provides simulations of the scheme usingthe full time-dependent potential for the mechanics. A master

FIG. 1. (Color online) Membrane-in-the-middle geometry. Twocavity fields tunnel through a membrane located at the center of afixed cavity and interact with the membrane in opposite directions,resulting in a quadratic optomechanical interaction.

1050-2947/2014/90(4)/043840(11) 043840-1 ©2014 American Physical Society

H. SEOK, E. M. WRIGHT, AND P. MEYSTRE PHYSICAL REVIEW A 90, 043840 (2014)

equation governing the quantum dynamics of the mechanicaloscillator is obtained in Sec. V. Section VI describes numericalsimulations to elucidate the features of quantum dynamicstabilization in comparison to the classical case. Summaryand outlook are presented in Sec. VII.

II. CLASSICAL DYNAMICS

We consider an optomechanical system in which a mechan-ical mode of effective mass m and frequency ωm is coupledto a single cavity field mode of frequency ωc via a quadraticoptomechanical interaction. The Hamiltonian describing thatsystem is

H = Ho + Hm + Hom + Hloss, (2)

where

Ho = �ωca†a + i�(ηe−iωLt a† − H.c.) (3)

describes the cavity field driven by an external field offrequency ωL with a rate η, and a denotes the bosonicannihilation operator for the cavity field, with [a,a†] = 1. Themechanical Hamiltonian is

Hm = p2

2m+ Um(x), (4)

where x and p are the position and momentum operators forthe mechanics with the commutation relation [x,p] = i�, and

Um(x) = mω2m

2x2 (5)

is the potential for the free mechanical oscillator. The quadraticoptomechanical interaction Hamiltonian is

Hom = �g(2)0 a†ax2, (6)

where g(2)0 is the quadratic optomechanical coupling constant.

It is assumed throughout this paper that g(2)0 is negative-valued

as is appropriate to trapping around a maximum of the cavityintensity [34]. Finally, Hloss represents the interaction ofthe optomechanical system with its reservoir and accountsfor cavity and mechanical dissipation with rates κ and γ ,respectively.

The classical equations of motion are found by replacingoperators by their c-number equivalent in the Heisenberg-Langevin equations derived from the Hamiltonian Eq. (2),neglecting noise sources for now. In a frame rotating at the laserfrequency ωL, this yields the following classical equations forthe position x, momentum p, and the dimensionless intracavityfield amplitude a,

x = p

m, (7)

p = −mω2mx − 2�g

(2)0 |a|2x − γp, (8)

a =[ic − ig

(2)0 x2 − κ

2

]a + η, (9)

where c = ωL − ωc is the pump detuning from the cavityresonance, κ is the phenomenological decay rate of the cavityfield, and γ is the mechanical damping rate.

In the physically relevant regime κ � ωm, the cavity fieldadiabatically follows the mechanical mode, and

|a|2 ≈ 2P0κ/(�ωL)[c − g

(2)0 x2

]2 + κ2/4, (10)

where P0 = �ωL|η|2/(2κ) is the continuous wave input powerof the cavity. Substituting this expression into Eq. (8) thengives

p = −mω2mx − 4g

(2)0 P0κ/ωL[

c − g(2)0 x2

]2 + κ2

4

x − γp. (11)

In the absence of mechanical dissipation, γ = 0, the aboveequations of motion for the mechanical system can be put inthe canonical form x = p/m,p = − ∂Hs

∂x, with Hamiltonian

Hs = p2

2m+ Us(x) (12)

and static mechanical potential

Us(x) = Um(x) − 4P0

ωL

arctan

[c − g

(2)0 x2

κ/2

]. (13)

This highlights the key lesson that adiabatic elimination ofthe cavity field involves concomitant replacement of the freemechanical potential Um(x) by the static mechanical potentialUs(x) in the dynamics of the mechanical mode.

Importantly, transient cases may also be treated in theappropriate regime. For example, an input power Pin(t)modulated at frequency � gives rise to a time-dependentpotential U (x,t), generalizing the static mechanical potential,provided that κ � �, so that damping of the cavity intensitytoward steady state happens on a much faster time scale thanthe applied modulation. Under these conditions, [Hm + Hom]may be replaced by the reduced mechanical Hamiltonian,

Hr =[

p2

2m+ U (x,t)

], (14)

following adiabatic elimination of the cavity field. We shalluse this in the next section and also in the quantum theory.

The static potential Us(x) exhibits a single minimum atx = 0 if P0 is less than the critical power,

Pc = mω2m

4∣∣g(2)

0

∣∣κ/ωL

[2

c + κ2

4

], (15)

whereas a symmetric double-well potential centered on x = 0results if the power is greater than Pc; see Fig. 2. In thatcase the repulsive radiation pressure acting on the oscillatorcentered at x = 0 is greater than the mechanical restoringforce. For small displacement from the origin x = 0, Us(x) canbe approximated as an inverted oscillator of frequency [34]

ω0 =√√√√∣∣∣∣∣ω2

m + 4P0/ωL

m

g(2)0 κ

2c + κ2/4

∣∣∣∣∣, (16)

rendering the origin unstable. This is the situation that weconsider in the following.

In Fig. 2 and in all following figures we measure theposition, momentum, and energy of the mechanical mode

043840-2

DYNAMIC STABILIZATION OF AN OPTOMECHANICAL . . . PHYSICAL REVIEW A 90, 043840 (2014)

40 20 20 40

xx0

2

2

4

Us E0

FIG. 2. (Color online) Static mechanical potential for an inputpower greater than the critical power Pc. The parameters are κ/ωm =200, c/ωm = 0, g

(2)0 x2

0/ωm = −0.01, P0/(E0ωL) = 1260. For ourparameters, the critical pumping power is Pc/(E0ωL) = 1250 andthe effective mechanical frequency is ω0/ωm = 0.09. Here and in allfollowing figures we measure the position, momentum, and energyof the mechanical mode in units of the natural length x0 = √

�/mωm,momentum p0 = √

m�ωm, and energy E0 = �ωm, respectively.

in units of the natural length x0 = √�/mωm, momentum

p0 = √m�ωm, and energy E0 = �ωm, respectively.

III. DYNAMIC STABILIZATION

In this section we investigate a scheme to stabilize amechanical oscillator at the unstable center illustrated in Fig. 2.Recalling that Kapitza’s pendulum can be stabilized by arapidly oscillating force, here we propose rapidly modulatingthe input power Pin(t) below and above the critical power: Thepotential at the center then concomitantly oscillates between amaximum and minimum and the force acting around the centeroscillates between repulsive and attractive.

We consider specifically a modulation of the form

Pin(t) = P0 − A sin (�t) , A < P0, (17)

where � and A are the frequency and amplitude of themodulation, respectively. Throughout this paper we chooseP0 > Pc so that the mean input power generates a symmetricdouble-well potential with unstable center for the mechanicsin the absence of the modulation. The amplitude of themodulation is chosen positive with the constraint A � P0 sothat the continuous wave input power remains nonnegative forall times.

The input power Eq. (17) yields a time-dependent potential,

U (x,t) = Us(x) + u(x,t)

= mω2m

2x2 − 4P0

ωL

arctan

[c − g

(2)0 x2

κ/2

]

+ 4A

ωL

arctan

[c − g

(2)0 x2

κ/2

]sin(�t), (18)

where the second line corresponds to the static portion Us(x)of the potential, which alone produces an unstable centeredmechanical mode, and the third line gives its oscillating portion

40 20 20 40

xx0

4

2

2

4

U E0

FIG. 3. (Color online) Time-dependent potential for the mechan-ics at t = 0, π/� (blue solid line), t = π/(2�) (green dashedline), and t = 3π/(2�) (red dotted line). Here, P0/(E0ωL) =1260, A/P0 = 1, �/ωm = 1.8 and the other parameters are the sameas those described in the legend of Fig. 2.

u(x,t). Figure 3 shows the static potential Us(x) (solid blueline) along with the time-dependent potential at the times atwhich it reaches the maximum and minimum powers Pin =P0 + A (red dotted line) and Pin = P0 − A (green dashed line).Note that the net force at the center x = 0 is attractive for themechanical potential corresponding to the minimum powerand repulsive for the maximum power. The alternating sign ofthe net force acting at the center is what raises the possibilityof dynamic stabilization of the mechanical motion.

To develop a physical understanding of how the mechanicalmode can be stabilized, we derive a time-averaged mechanicalpotential and identify the parameter regime for the modulationto realize dynamic stabilization of the mechanical motion.

The potential Eq. (18) yields for the mechanical mode theNewton’s equation of motion

mx = Fs(x) + f (x,t) = −dUs(x)

dx− ∂u(x,t)

∂x. (19)

Here the static force Fs(x), which has only spatial dependence,is

Fs(x) = −mω2mx − 4g

(2)0 P0κ/ωL[

c − g(2)0 x2

]2 + κ2

4

x, (20)

and the radiation pressure force f (x,t) due to the modulationof the input power is

f (x,t) =(4g

(2)0 Aκ

/ωL

)x[

c − g(2)0 x2

]2 + κ2

4

sin(�t) = A(x) sin(�t) (21)

and is separable into spatial and temporal parts.Following the treatment in Refs. [1,2] we write the

mechanical coordinate as a sum of slow and fast varyingvariables,

x = x + ζ, (22)

where it is assumed that |x| � |ζ |, and the bar denotes atime-average over one oscillation cycle with a period of T =2π/�. Here, x is a slowly varying variable with respect toT and describes the macromotion of the mechanics, and ζ is

043840-3


the rapidly oscillating variable with zero mean that describesthe micromotion of the mechanics. Substituting Eq. (22) intoEq. (19) and expanding the right-hand side of Eq. (19) for therapidly oscillating component ζ to first order, we obtain

m ¨x + mζ = Fs(x) + ζdFs

dx

∣∣∣∣x=x

+ f (x,t) + ζ∂f

∂x

∣∣∣∣x=x

. (23)

We separate the fast varying portion of this equation as

mζ ≈ f (x,t). (24)

Substituting the approximate solution ζ (t) ≈ − 1m�2 f (t) of this

equation into Eq. (23) and taking the time average over theperiod T yields the equation of motion for the slowly varyingvariable

m ¨x = Fs(x) − 1

m�2T

∫ t

t−T

dt ′f (x,t ′)∂f (x,t ′)

∂x

= −dU (x)

dx. (25)

For simplicity in notation we hereafter replace x with x in thissection with the clear understanding that in the time-averagedtheory x refers to the slow portion of the mechanical motion.The time-averaged potential governing the macromotion of themechanics is then given by

U (x) = mω2m

2x2 − 4P0

ωL

arctan

[c − g

(2)0 x2

κ/2

]

+(

A2

m�2

) [2g

(2)0 κ/ωL(

c − g(2)0 x2

)2 + κ2

4

x

]2

, (26)

where the first two terms on the right-hand side coincide withthe static potential Us(x) and the last term arises from thesecond term on the right-hand side of Eq. (25) and accountsfor the effects of the modulation on the macromotion.

The factor A2/(m�2) multiplying the last term in Eq. (26)implies that a large amplitude of the modulation A can leadto an enhanced effect on the macromotion, while a highfrequency � tends to diminish the effect of the modulationon the macromotion. For this reason one might be tempted todecrease the modulation frequency to enhance the effect of themodulation. However, in the derivation of the time-averagedpotential we assume that the macromotion is much slowerthan the micromotion. The essence of this assumption isbasically the same as the adiabatic elimination of a fastvariable in quantum optics [35]. This assumption allows one toadiabatically eliminate the rapidly varying variable ζ on a timescale of T = 2π/�, resulting in the time-averaged potentialU (x) governing only the macromotion of the mechanics.Therefore, the modulation frequency � must exceed ω0, theeffective frequency of the mechanical motion at the center. Ifthis is not the case the micromotion cannot be separated fromthe macromotion, and hence the description of the mechanicalmotion in terms of the time-averaged potential breaks down.Thus, A2/(m�2) must be large enough that the modulation hassignificant contributions on the macromotion of the mechanicseven in the high-frequency regime.

Figure 4 shows the static potential (solid blue line) as wellas the time-averaged potentials for three different values of

40 20 20 40

xx0

2

1

1

2

3

4U E0

FIG. 4. (Color online) Time-averaged potential for several valuesof the modulation amplitude A with a fixed modulation frequency�/ωm = 1.8, A/P0 = 0.20 (green dashed line), A/P0 = 0.26 (or-ange dot-dashed line), A/P0 = 1 (red dotted line), along with thestatic potential (blue solid line). Other parameters are as described inthe legend of Fig 3.

the modulation amplitude A with fixed modulation frequency�. It illustrates that for a small modulation amplitude,the time-averaged potential remains a double-well potentialof reduced depth, which retains a local maximum at thecenter (green dashed line). For large enough modulationamplitude, however, the time-averaged potential develops alocal minimum at the center (orange dot-dashed and red dottedlines), indicative dynamic stabilization of the mechanical os-cillator, the frequency of the time-averaged stabilized potentialincreasing with A.

FIG. 5. (Color online) Stability domain of an optomechanicaloscillator located at the center, for an input power modulatedaccording to Eq. (17), for the parameters described in the legendof Fig. 3. The oscillator is dynamically stable in the region abovethe dashed blue line (blue-colored region). The black dots denote thepoints used in the simulations described in the legends of Figs. 7–10,and label these points in the figures. In all cases �/ωm = 1.8, and(A/P0) = (a) 0, (b) 0.10, (c) 0.20, (d) 0.26, and (e) 1.

043840-4


The dynamic stability at the equilibrium position x = 0 canbe evaluated using the curvature of the potential

D = d2U (x)

dx2

∣∣∣∣x=0

, (27)

where U (x) is the time-averaged potential. Positive D ensuresthat small mechanical oscillations around x = 0 are confinedto the trap leading to stability. In contrast, negative D indicatesthat the time-averaged potential acquires a local maximum atx = 0, rendering the mechanical mode unstable. Based on thiscriterion, Fig. 5 shows the boundary between the unstable andstable regimes (blue dashed line) in the (�/ωm,A/P0) plane.The mechanical mode at the center is unstable in the regimebelow the boundary (unshaded region) and becomes stable inthe regime above the boundary (blue-colored region).

IV. CLASSICAL SIMULATIONS

This section presents selected simulations of the classicaldynamics of the system for the driving frequency �/ωm = 1.8(�/ω0 = 20), for which the adiabaticity condition � � ω0 isfulfilled and (A/P0) = (a) 0, (b) 0.10, (c) 0.20, (d) 0.26, and(e) 1; see Fig. 5. For the parameters of that figure time-averagedpotential develops a minimum at x = 0 for (A/P0) > 0.20,so that cases (a)–(c) are expected to be classically unstable,whereas cases (d) and (e) are stable according to the time-averaged potential. However, by construction that potentialcaptures only the low-frequency mechanical macromotionresulting from the modulation, but not the high-frequencymicromotion. To explore the full classical dynamics we nowinclude the time-dependent potential U (x,t) with and withoutmechanical dissipation and compare with expectations fordynamic stabilization based on the time-averaged potential.

We start from Newton’s equation of motion for the mechan-ics including both the time-dependent forcing, dissipation, andassociated thermal noise,

mx = −mω2mx − 4g

(2)0 Pin(t)κ/ωL[

c − g(2)0 x2

]2 + κ2

4

x − γp + ξ, (28)

where Pin(t) is given by Eq. (17), the classical thermalfluctuations possess a two-time correlation function [36],

〈ξ (t)ξ (t ′)〉 = 2mγkBT δ(t − t ′), (29)

where kB is the Boltzmann constant, and T is the temperatureof the heat bath of the mechanical oscillator. We note that herex is the full mechanical coordinate as opposed to the time-averaged value x. In order to explore the full range of dynamicsof the mechanical oscillator, we consider an ensemble ofinitial conditions for the position and momentum chosen fromGaussian probability distributions with standard deviationsσx = x0/

√2 and σp = p0/

√2, respectively. The joint proba-

bility density for the initial positions and momenta is given by

P (x,p,t = 0) = 1

πx0p0e−x2/x2

0 e−p2/p20 , (30)

and the associated energy distribution reads

P (E/E0) = 2e−2E/E0 , (31)

with mean energy E = E0/2, as shown in Fig. 6.

1 2 3 4EE0

0.5

1.0

1.5

2.0P E E0

FIG. 6. (Color online) Initial energy distribution of the mechan-ical oscillator (blue solid line) with the mean energy of E = E0/2(gray dashed line).

A. Undamped case

We first consider the situation where the mechanicaloscillator has a sufficiently high quality factor Qm = ωm/γ

that dissipation can be neglected over time scales of interest.This situation allows the effects of the modulation of theradiation pressure force on the classical mechanical oscillatorto be highlighted. For a given set of parameters, we generated1000 trajectories from a random sample of initial positionsand momenta generated from the Gaussian probability densityEq. (30). We display them in the same color-coded two-dimensional plot in such a way that the darker a region,the more trajectories cross that region: The resulting plotsmay then be viewed as spatial probability densities (withappropriate normalization).

Figures 7(a)–7(e) summarize results of such simulationsfor the parameters marked by black dots in Fig. 5 and labeled(a)–(e) in the corresponding figure caption. Figure 7(a) showsthe dynamics of the mechanics in the static double-wellpotential, indicating that the mechanics is neither localized atthe center nor bounded in one of the local potential wells. Thisarises since the initial mean energy E0/2 of the mechanicsis higher than the depth of the static double-well potential.The relatively high probability density at the center arises dueto the fact that there is a local potential maximum there andthe trajectories therefore tend to slow down and linger in thevicinity of the center.

As indicated in Fig. 5, dynamic stabilization arises formodulation amplitudes (A/P0) > 0.20. This is borne out, toan extent limited by the impact of micromotion as discussedbelow, by a comparison of Figs. 7(b), 7(c) and 7(d), 7(e).In the first two cases, the modulation amplitude is not largeenough to trap the mechanics close to the center. Instead,the trajectories explore the spatial extent of the double-wellpotential spanning the energy range from the potentialminimum up to an additional energy of E0/2. In contrast, inFigs. 7(d) and 7(e), which are for (A/P0) > 0.20, one candiscern the onset of the predicted dynamic stabilization of theprobability around the center.

Importantly, however, micromotion makes the sharpunstable-to-stable transition predicted by the static potential

043840-5


FIG. 7. (Color online) Trajectories of the classical mechanicaloscillator with initial conditions generated at random from theGaussian distribution function Eq. (30) with σx = x0/

√2, σp =

p0/√

2, and �/ωm = 1.8. The curves follow the labeling ofFig. 5 with values of (A/P0) given by (a) 0, (b) 0.10,(c) 0.20, (d) 0.26, and (e) 1. Here, γ /ωm = 10−6, T = 0, κ/ωm =200, c/ωm = 0, g

(2)0 x2

0/ωm = −0.01, P0/(E0ωL) = 1260.

much less evident. The transition to stability is now muchmore progressive, with the trapping becoming gradually morepronounced as the modulation amplitude is increased past(A/P0) = 0.20. Still with this important caveat these resultsvalidate the concept of dynamic stabilization of an optome-chanical oscillator in the absence of mechanical dissipation.

B. Damped case

To explore the effects of mechanical dissipation, Fig. 8repeats the same simulations as in Fig. 7 but now includingdamping of the mechanical oscillator via coupling toa reservoir at zero temperature. For the simulations inFigs. 8(a)–8(c) the modulation amplitude (A/P0) � 0.20, andeach trajectory ultimately gets trapped in one or the otherof the wells of the time-averaged double-well potential, plussome high-frequency micromotion. The amplitude of the mi-cromotion is quite large due to the fact that the amplitude of thetime-dependent radiation pressure force appearing in Eq. (21),

A(x) = 4g(2)0 Aκ/ωL[

c − g(2)0 x2

]2 + κ2/4x, (32)

depends on the mechanical displacement x. It is small nearthe center but can become significant around the minima ofthe double-well potential.

Turning next to the cases with (A/P0) > 0.20 shown inFigs. 8(d) and 8(e), the trajectories damp into the centerconsistent with the idea of dynamic stabilization. Theseresults show that the concept of dynamic stabilization survives

FIG. 8. (Color online) Trajectories of the classical mechanicaloscillator in the presence of a viscous damping force for a reservoir atzero temperature. Here, γ /ωm = 2 × 10−2, T = 0. Other parametersas described in the legend of Fig. 7.

the inclusion of mechanical dissipation, the stable-unstabletransition following closely the predictions based on thetime-averaged potential.

V. QUANTUM DYNAMICS

To properly account for fluctuations and noise in thequantum regime, we find it convenient to work in theSchrodinger picture, where the combined field-mechanicssystem is described by the master equation

ρ(t) = − i

�[H,ρ(t)] + (Lm + Lo)ρ(t), (33)

where H = Ho + Hm + Hom [see Eqs. (3)–(6)], and Lm andLo are standard Lindblad forms that describe the dissipationof the mechanics and the cavity field due to the coupling totheir respective reservoirs, which are assumed for simplicityto be at zero temperature T = 0.

For fast dissipation of the optical field, κ � ωm, we assumethat decoherence prohibits the build up of quantum correlationbetween the two subsystems, so that the total density operatorcan be factorized as

ρ(t) ≈ ρm(t) ⊗ ρo(t). (34)

By taking partial traces over the mechanics and the opticalfield, it is then possible to get reduced master equations for thetwo subsystems:

ρm = − i

�

[Hm + �g

(2)0 〈a†a〉x2,ρm

] + Lmρm, (35)

ρo = − i

�

[Ho + �g

(2)0 a†a〈x2〉,ρo

] + Loρo. (36)

043840-6


Note that because of the approximate absence of correlationsbetween the two subsystems, it is only the mean photonnumber that appears in the master equation for the reduceddensity operator of the mechanics, and likewise only theexpectation value 〈x2〉 that appears in the master equation forthe field mode. This indicates that the frequency of the fieldmode is shifted by the optomechanical interaction from ωc toωc + g

(2)0 〈x2〉, in keeping with expectations from the classical

analysis.The next step is to adiabatically eliminate the optical

field. While we make use of a master equation approach fornumerical convenience, it is particularly instructive to derivethe approximate quantum reduced mechanical potential usinga Wigner representation. This is outlined in Appendix, whichshows that the adiabatic elimination of the cavity field inthe regime where κ � ωm allows to replace [Hm + Hom] =[Hm + �g

(2)0 〈a†a〉x2] in Eq. (35) with the reduced Hamiltonian

Hr =[

p2

2m+ U (x,t)

]. (37)

To gain some insight into how this replacement manifests itselfin the quantum theory it is useful to consider the quantumaveraged reduced potential 〈U (x,t)〉, with U (x,t) given byEq. (18) with x → x. Then, consistent with the fact that only〈x2〉 appears in the master equation (36) for the field mode,we factorize products 〈x2n〉 = 〈x2〉n, n = 0,1,2, . . ., yieldingthe result 〈U (x,t)〉 = U (

√〈x2〉, t). Given that we consider

a potential that is symmetric around the origin, and taking〈x〉 = 0 for a symmetric initial condition, then x =

√〈x2〉,

and the quantum averaged reduced potential U (x,t) isthe same as the classical one with the classical mechanicaldisplacement replaced by the root-mean-square displacement.In this way the properties of the classical reduced potentialalso manifest themselves in the quantum theory, meaning thatquantum dynamic stabilization is also a possibility.

Bringing the above results together yields the effectivemaster equation for the mechanics

ρm = − i

�

[p2

2m+ U (x,t),ρm

]+ Lmρm. (38)

Then expanding the density matrix for the mechanics in theposition representation as

ρm(t) =∫

dx

∫dx ′ρm(x,x ′,t)|x〉〈x ′|, (39)

and substituting into the master equation (38) yields theequation of motion

∂

∂tρm(x,x ′,t)

={

i�

2m

(∂2

∂x2− ∂2

∂x ′2

)− i

�[U (x,t) − U (x ′,t)]

+ γ

2

(x

∂

∂x ′ + x ′ ∂

∂x+ 1

)+ γ �

4mωm

(∂

∂x+ ∂

∂x ′

)2

− γmωm

4�(x − x ′)2

}ρm(x,x ′,t). (40)

In future work we plan to go beyond the approximationsunderlying this equation, namely the decorrelation approxima-tion Eq. (34) and the adiabatic approximation resulting in thereduced Hamiltonian Eq. (37), but for this proof-of-principlestudy, Eq. (40) is the basis of our study of quantum dynamicstabilization.

VI. QUANTUM SIMULATIONS

We used a finite difference method to solve the second-orderpartial differential equation, Eq. (40), on a finite spatial grid,making sure that the density matrix is negligible at the edgesof the grid and allowing the norm of the density matrix to beconserved to a high degree of accuracy. For all the followingsimulations it is assumed that the mechanical oscillator isinitially prepared in the quantum mechanical ground stateof the bare harmonic trapping potential with frequency ωm

and average energy E0/2, thus allowing comparison with theclassical simulations.

A. Undamped case

As in the classical case we first consider the case wherethe mechanical damping rate is small enough compared to themechanical frequency that it may be neglected over the timescale of our simulations, and the evolution of the system isHamiltonian. Figure 9 shows plots of the spatial probabilitydensity

P (x,t) = ρm(x,x,t), (41)

FIG. 9. (Color online) Time evolution of the spatial probabilitydistribution of the quantum mechanical oscillator initially prepared inthe ground state of the bare harmonic trapping potential of frequencyωm. Here, the parameters employed and plot labels are identical tothose described in the legend of Fig. 7.

043840-7


that are in one-to-one correspondence with the classical resultsshown in Fig. 7. Recalling that for the chosen parametersclassical dynamic stabilization arises for modulation ampli-tudes (A/P0) > 0.20, classical stabilization is expected in theplots in Figs. 9(d) and 9(e).

Figure 9(a) shows the quantum dynamics for the case of thestatic double-well potential. The main distinctions with respectto the classical result of Fig. 7(a) are the pronounced quantuminterferences. Such “quantum carpet” patterns, characteristicof quantum interferences of mechanical wave packets in boundpotentials [37], are a distinct and expected feature in all of theabove cases in comparison to the classical case. For modulationamplitudes (A/P0) � 0.20, Figs. 9(b) and 9(c), the probabilitydensity is spatially extended and, similarly to the classical case,bounded by the potential barriers given by the time-averageddouble-well potential evaluated at the average energy E0/2of the initial condition. Furthermore, similar to the classicalcase, the micromotion is largest at the boundary of the spatialprobability distribution of the mechanics since the amplitudeof the time-dependent radiation pressure forceA(x) in Eq. (32)is the largest at that point.

For modulation amplitudes above the threshold for dynamicstabilization, Figs. 9(d) and 9(e), the spatial width of P (x,t)decreases. As in the classical case, the micromotion softensthe transition to strong dynamic stabilization about x = 0.Except for the quantum interferences characteristic of wavepacket dynamics in a potential well, the quantum and classicalcases yield therefore quite similar probability densities. Assuch our results validate the concept of dynamic stabilizationof a quantum optomechanical oscillator in the absence ofdamping.

B. Damped case

The contrast between the classical and quantum casesis more pronounced in the presence of damping, as shownin Fig. 10, which is in one-to-one correspondence with theclassical results in Fig. 8. For the case of a static mechanicalpotential with no applied modulation, see Fig. 10(a), P (x,t)is asymptotically split with dual peaks at the local minimaof the underlying double-well potential, in agreement withexpectations from the classical theory. This splitting persistsfor lower values of the modulation amplitude, see Fig. 10(b).In Sec. IV, Figs. 8(c) and 8(d) illustrated that in the caseof the classical oscillator damped by a reservoir at zerotemperature, a sharp transition occurs from the unstable tostable regime at (A/P0) = 0.20. In contrast, the transition todynamic stabilization is much more gradual in the quantumtheory, as illustrated by Figs. 10(c) and 10(d), which straddlethe threshold with little change in features. This is a purelyquantum effect: while a damped classical oscillator at zerotemperature does not experience any noise, the correspondingquantum oscillator experiences quantum noise, which blurs theclassical stability transition. Dynamic stabilization still occursfor sufficiently large modulation amplitudes, as illustratedin Fig. 10(e) for (A/P0) = 1. Once again we see thatfor sufficiently large modulation amplitudes the probabilitydensities showing dynamic stabilization from the quantumand classical theories are quite similar, modulo the expectedquantum interferences.

FIG. 10. (Color online) Time evolution of the spatial probabilitydistribution of the damped quantum mechanical oscillator initiallyprepared in the ground state of the harmonic potential of frequencyωm. Here, the parameters employed and plot labels are identical tothose described in the legend of Fig. 8.

C. Phase-space distributions

Further information and insight regarding dynamic stabi-lization in the classical and quantum domains can be obtainedfrom the corresponding phase-space distributions. For theclassical case this is constructed by plotting the ensemble oftrajectories in the (p,x) plane and interpreting the density oftrajectories as the probability density. For example, Fig. 11(a)shows the classical phase-space distribution corresponding tothe results in Fig. 7(a) for a time ωmt = 100. For the quantumcase the phase-space distribution is obtained from the Wignerquasiprobability distribution,

W (x,p,t) = 1

π�

∫ ∞

−∞〈x + y|ρm(t)|x − y〉e−2ipy/�dy, (42)

and Fig. 11(c) shows the quantum phase-space distributioncorresponding to the results in Fig. 9(a) for a time ωmt = 100.The quantum Wigner distribution displays negative regions,seen as white, these being signatures of nonclassicality. Wepoint to these regions as they appear in our numerics butdo not want to overemphasize their significance given theapproximations underlying Eq. (40). Rather our goal is todemonstrate that dynamic stabilization is also possible in thequantum domain.

In Fig. 11 the upper row shows the classical phase-space distributions for (a) A/P0 = 0, the case of a staticmechanical potential, (b) A/P0 = 0.20, and (c) A/P0 = 1,for a time ωmt = 100, all other parameters being the sameas before. The lower row of plots labeled Figs. 11(d)–11(f)are the corresponding quantum phase-space distributions.Comparing upper and lower rows, we see that the classical and

043840-8


FIG. 11. Phase-space distributions of the classical oscillator (upper row) and corresponding Wigner quasiprobability distributions of thequantum oscillator (bottom tow) at time ωmt = 100 in the absence of dissipation. Here, γ /ωm = 10−6, �/ωm = 1.8, and (a, d) A/P0 = 0,(b, e) A/P0 = 0.20, (c, f) A/P0 = 1, κ/ωm = 200, c/ωm = 0, g

(2)0 x2

0/ωm = −0.01, P0/(E0ωL) = 1260. The regions lighter than neutral gray(see scales on the side of the plots) correspond to negative values of the Wigner function.

quantum plots share broad structural features while displayingmarked differences in detail. For example, Fig. 11(a) forthe static mechanical potential shows the classic figure-eightphase-space portrait characteristic of a double-well, whileFig. 11(d) reflects similar structure plus oscillatory structuresand negative regions that are uniquely quantum. The same

comments apply to Figs. 11(b) and 11(e) for A/P0 = 0.20,which is below the threshold for dynamic stabilization. Forthe results shown in Figs. 11(c) and 11(f) for A/P0 = 1, wesee that fluctuations in the displacement x around the origin arereduced with respect to the other examples, which is consistentwith the fact that dynamic stabilization is expected in this case.

FIG. 12. Phase-space distributions of the classical oscillator (upper row) and corresponding Wigner quasiprobability distributions of thequantum oscillator (lower row) at time ωmt = 300. Both classical and quantum oscillators are damped via a reservoir at zero temperature withγ /ωm = 2 × 10−2. Other parameters are as described in the legend of Fig. 11.

043840-9


Note that although the fluctuations in the displacement arereduced there remain large positive and negative variations inthe momentum p. This may be traced to the micromotion,which, although it has small spatial extent, reflected by thereduced displacement fluctuations in Figs. 11(c) and 11(f),can nonetheless be associated with a large time-oscillatingmomentum due to its high frequency. As for the orientation ofthe phase-space distributions in Figs. 11(b) and 11(e), or 11(c)and 11(f), this depends on the specific choice of dimensionlessinteraction time ωmt since the micromotion is synced with theapplied modulation.

The impact of quantum noise on the behavior of thequantum oscillator is explored in Fig. 12, which is for thesame parameters as used in Fig. 11 but including dampingat T = 0 and for a time ωmt = 300. In all cases the spatialextent of the classical phase-space distributions are muchnarrower than their quantum counterparts, which reflects theabsence of noise in the classical case alluded to earlier. InFigs. 12(a) and 12(d) for the static mechanical potential thephase-space distributions show equal peaks around the minimaof the double-well potential, whereas dynamic stabilizationis clearly evident in Figs. 12(c) and 12(f). Furthermore, theregions indicative of nonclassicality are no longer present inthe presence of damping for these examples.

VII. SUMMARY AND OUTLOOK

We have investigated the concept of dynamic stabilizationof a mechanical oscillator based on an optomechanicalvariation of the Kapitza pendulum problem that involves themodulation of the radiation pressure force. A time-averagedpotential was derived that describes the dynamics of themechanics in situations where the optical field can be adiabat-ically eliminated. Predictions of the time-averaged potentialdescription with numerical simulations of the mechanics werecompared that include the effects of micromotion as well,both in the classical and the quantum regimes. We found thatespecially in those situations where the mechanical dampingcan be ignored, micromotion plays an important role andsignificantly softens the transition from the unstable to dynam-ically stabilized regimes. Mechanical damping significantlyreduces the impact of micromotion, though, especially in theclassical regime. In particular, at zero temperature and inthe absence of thermal noise the dynamics of the classicaloptomechanical oscillator closely follows the predictions ofthe time-averaged potential stability analysis. This is not thecase for a quantum mechanical oscillator, where even at zerotemperature quantum noise significantly softens the thresholdbetween stable and unstable regimes.

Our analysis of the quantum regime relied on the fac-torization of the density operator for the mechanics andthe optical field, eliminating the possibility of bipartiteentanglement between the two systems and the possibilityof considering the potential impact of quantum correlationson dynamic stabilization. We expect that these issues willbe most relevant in situations where the decoherence ofboth subsystems occurs on comparable time scales. Whileit seems unrealistic to realize such a situation in the opticalregime of quantum optomechanics, the situation might provemore favorable with microwave fields, where high Q factors

and long photon lifetimes, of the order of a fraction ofa second, have been previously realized. Future work willexpand on our analysis of dynamic stabilization to focus onthese issues, including the role of quantum noise, includ-ing shot noise and radiation pressure noise, and bipartiteentanglement.

ACKNOWLEDGMENTS

We thank F. Bariani and K. Zhang for stimulating discus-sions. This work is supported by the DARPA QuASAR andORCHID programs through grants from AFOSR and ARO,the US Army Research Office, and the US NSF.

APPENDIX: ADIABATIC ELIMINATION OF THE CAVITYFIELD IN THE QUANTUM REGIME

This appendix presents details of the derivation of thereduced mechanical potential in the quantum regime. Wefollow the approach in Refs. [38–40] to adiabatically elim-inate the cavity field so that the mechanics experiences thereduced potential in the quantum regime. The dynamicsof the optomechanical system is described by the masterequation (33). Introducing the Wigner distribution for themechanics,

W (x,p) =∫∫

dσ

2π

dμ

2πTrm{eiμ(x−x)+iσ (p−p)ρ}, (A1)

the master equation (33), is transformed into

˙W = − i

�[Ho,W ] + LoW +

{− ∂

∂x

p

m+ ∂

∂p

(mω2

mx)}

W

− ig(2)0

(x2 + i�x

∂

∂p− �

2

4

∂2

∂p2

)a†aW

+ ig20

(x2 − i�x

∂

∂p− �

2

4

∂2

∂p2

)W a†a + LmW , (A2)

where Trm{·} denotes partial trace over the mechanics, sothat W (x,p) is a density operator for the cavity field and ac-number quasiprobability distribution for the mechanics, andLmW describes mechanical dissipation.

Taking a partial trace over the Hilbert space for the cavityfield, the time evolution of the Wigner function for themechanics is then given by

Wm(x,p) ={− ∂

∂x

p

m+ ∂

∂p

(mω2

mx)}

Wm

+ ∂

∂p

(2�g

(2)0 xI

) + LmWm, (A3)

where Wm is obtained by taking the partial trace of W over thecavity field, Wm = Tra{W } and I = Tra{a†aW }. Note that thisequation is not closed due to the presence of I on its right-handside.

In order to construct a closed equation of motion for themechanics Wigner function we now adiabatically eliminatethe cavity field. In terms of the normalized time τ = ωmt , di-mensionless mechanical position and momentum, x = x/x0,

043840-10


p = p/p0, the equations of motion for I and α = Tra{aW }are

ε∂I

∂τ= −I + η

κα∗ − η∗

κα + ε

{− ∂

∂xp + ∂

∂px

}I

+ε∂

∂p

(2g

(2)0 x2

0

ωm

xJ

)+ ε

Lm

ωm

I, (A4)

ε∂α

∂τ=

[ic − g

(2)0 x2

0 x2

κ− 1

2

]α + η

κWm

+ ε

{− ∂

∂xp + ∂

∂px

}α + ε

∂

∂p

(2g

(2)0 x2

0

ωm

xK

)

+ εg

(2)0 x2

0

ωm

(∂

∂px + i

4

∂2

∂p2

)α + ε

Lm

ωm

α, (A5)

where ε = ωm/κ is a small parameter in the adiabatic regime,J = Tra{a†aa†aW }, and K = Tra{a†aaW }. Expanding allquantities I,J,K , and α in powers of ε, for exampleI = ∑

m=0 εmIm, one can in principle solve the differentialequations to arbitrary order. However, for κ � ωm and thus ε

approaches to zero, it is sufficient to restrict the description to

order ε0. This zeroth order solution is found to be

α = ηWm

−i(c − g

(2)0 x2

) + κ/2, (A6)

I = ηα∗ − η∗ακ

= |η|2Wm(c − g

(2)0 x2

)2 + κ2/4. (A7)

Substituting Eq. (A7) into Eq. (A3), yields for the mechanicsWigner distribution the closed equation of motion

Wm(x,p) ={− ∂

∂x

p

m+ ∂

∂pU ′

s(x) + Lm

}Wm, (A8)

where

Us(x) = Um(x) − 4P0

ωL

arctan

[c − g

(2)0 x2

κ/2

], (A9)

coincides with the classical reduced mechanical potentialEq. (13). This justifies also using the quantum version of thispotential in the quantum description of Secs. V and VI. Notethat as in the classical case the elimination of the cavity fieldstill holds provided the modulation frequency of the inputpower is much smaller than the cavity decay rate, κ � �.

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