Nonlinear Dynamics and Quantum Entanglement in Optomechanical Systems

Guanglei Wang,1 Liang Huang,2,1 Ying-Cheng Lai,1,3,* and Celso Grebogi31School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA2School of Physical Science and Technology and Key Laboratory for Magnetism and Magnetic Materials of MOE,

Lanzhou University, Lanzhou, Gansu 730000, China3Institute for Complex Systems and Mathematical Biology, King’s College, University of Aberdeen,

Aberdeen AB24 3UE, United Kingdom(Received 27 October 2013; revised manuscript received 7 January 2014; published 18 March 2014)

To search for and exploit quantum manifestations of classical nonlinear dynamics is one of the mostfundamental problems in physics. Using optomechanical systems as a paradigm, we address this problemfrom the perspective of quantum entanglement. We uncover strong fingerprints in the quantumentanglement of two common types of classical nonlinear dynamical behaviors: periodic oscillationsand quasiperiodic motion. There is a transition from the former to the latter as an experimentally adjustableparameter is changed through a critical value. Accompanying this process, except for a small region aboutthe critical value, the degree of quantum entanglement shows a trend of continuous increase. The timeevolution of the entanglement measure, e.g., logarithmic negativity, exhibits a strong dependence on thenature of classical nonlinear dynamics, constituting its signature.

DOI: 10.1103/PhysRevLett.112.110406 PACS numbers: 03.65.Ud, 03.67.Lx, 05.45.Mt, 42.65.Sf

Entanglement [1,2], a form of quantum superposition, isa fundamental phenomenon constituting the cornerstone ofquantum computing and quantum information science [3].The phenomenon has been observed in experiments oflarge molecules [4] and even small diamonds [5]. Recently,there is also great interest in entanglement in optomechan-ical systems [6–12].While the principle of superposition is defined with

respect to linear systems, our physical world, when viewedclassically, is nonlinear. A nonlinear dynamical systemcan exhibit all kinds of interesting phenomena such asperiodic oscillations, quasiperiodicity, and chaos [13], andthey can have distinct fingerprints or manifestations whenthe system is treated quantum mechanically. Especially,the studies of quantum manifestations of classical chaosconstitute the field of quantum chaos, and there have beentremendous efforts in the past four decades [14] in thisfield. With respect to entanglement, there is no classicalcorrespondence in the strict sense, as the measurementof a physical state at one place immediately influencesthe measurement at the other. In view of the ubiquity ofnonlinear dynamics in physical systems and of the funda-mental importance of quantum entanglement, curiositydemands that we ask the following question: What is theinterplay between nonlinear dynamics and quantum entan-glement? In this regard, earlier works indicated that thetransition from integrability to quantum chaos is relatedto quantum-phase transition in the Dicke model [15], andthe transition is accompanied by the emergence of anentanglement singularity in the quantum cusp catastrophemodel [16,17]. Signature of nonlinear behavior in entan-glement may have significant practical implications in the

development of devices and systems for quantum comput-ing and quantum information processing.In this Letter, we address the nonlinear-dynamics–

quantum-entanglement issue by using optomechanicalsystems, a field of intense recent investigation [18–31].An optomechanical system consists of an optical cavity anda nanoscale mechanical oscillator, such as a cantilever.When a laser beam is introduced into the cavity, a resonantoptical field is established that exerts a radiation forceon the mechanical cantilever, causing it to oscillate. Themechanical oscillations in turn modulate the length of theoptical cavity, hence, its resonant frequency. There is, thus,coupling between the optical and the mechanical degrees offreedom. This coupling, or interaction, can lead to coolingof the mechanical oscillator toward the quantum groundstate, a topic of great interest [32,33]. The optomechanicalcoupling, thus, provides a straightforward way to entanglethe optical with the mechanical modes, and this can haveprofound implications to optical information science andquantum computing [11,34]. With regard to nonlineardynamics, there were experimental works reporting chaosin optomechanical systems [31,35,36].There are recent theoretical works on quantum entangle-

ment in optomechanical systems [6–12]. For example, aprotocol was proposed for entanglement swapping withapplication to optomechanical systems [9], and recent workrevealed robust photon entanglement in optomechanicalinterfaces [10] and ways to achieve strong steady-stateentanglement in a three-mode optomechanical system[11]. The interplay between synchronization and entangle-ment inoptomechanical systemshas alsobeenexplored [12].However, existing works focused on the situation where the

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classical dynamics are either steady-state [6,8–12] or peri-odic oscillations [7]. Our goal is to search for nonlineardynamical behaviors beyond and to investigate the effectsof such behaviors on quantum entanglement. Our mainfindings are the following. In an experimentally realizableparameter regime, as the power of the driving laser isincreased, there is a transition from periodic to quasiperiodicmotions, where in the latter, the system is strongly nonlinearwith two incommensurate frequencies. Entanglement isenhanced towards the transition, vanishes as the transitionpoint is being reached, but is restored abruptly after thetransitionandcontinues tobeenhancedas the systemevolvesdeeply into the quasiperiodic regime. A surprising result isthat,with respect to timeevolution, there aredirect signaturesof classical nonlinear dynamics in quantum entanglement.In particular, for classically periodic dynamics, the timeevolution of the entanglement measure is also periodic, butwhen the classical system enters into the quasiperiodicregime, the quantum entanglementmeasure exhibits a beats-like behavior with two distinct frequencies. Entanglement,especially when the classical system is quasiperiodic, isrobust with respect to temperature variations.Model.—We consider a generic type of optomechanical

systems, as shown in Fig. 1(a). Such a system is essentiallya Fabry-Perot cavity with a fixed, partially reflecting mirrorat the left side and a movable, perfectly reflecting endmirror on the right side. The cavity has equilibrium lengthl0 and finesse F. The movable mirror is attached to amechanical oscillator of mass m, characteristic frequencyωM, and dissipation rate ΓM. The fixed end of the cavity hasthe optical decay rate κ ¼ πc=ð2Fl0Þ. We assume that asingle optical mode of frequency ωc is excited by anexternal laser field of frequency ω0, where ωc ∼ ω0 so that

the detuning is on the order of the mechanical frequencyωM. The Hamiltonian of the system can be written as[6,37,38] H ¼ ℏωca†aþ ℏωMb

†b − ℏg0a†aðb† þ bÞþiℏðEe−iω0ta† − E�eiω0taÞ þ Hκ þ HΓM

, where aða†Þ andbðb†Þ are the annihilation and creation operators associatedwith the optical and mechanical modes, respectively. Thefirst two terms in the Hamiltonian describe the frequenciesof the uncoupled optical and mechanical components,respectively, and the third term represents the change inthe characteristic frequency of the optical mode as a resultof the mirror movement. The quantity g0 ¼ G

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiℏ=ð2mωMÞ

pis the vacuum optomechanical coupling strength character-izing the interaction between one single photon and onesingle phonon—see Sec. S1 in the Supplemental Material[39]. The quantity

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiℏ=ð2mωMÞ

pis the mechanical zero-

point fluctuation with effective mass m, and G≃ ωc=l0.The fourth term describes the interaction between thecavity field and the driving laser field of complex amplitudeE, where the input laser power is jEj2 ¼ 2κP=ℏω0. The lasttwo terms in the Hamiltonian represent the dissipationresulting from the cavity decay and mechanical friction,respectively.Methods.—To explore and exploit nonlinear behaviors

in the optomechanical system, we use the Heisenbergequations of motion [40]. The dynamics of the optical-cavity field can be described by its corresponding complexamplitude, and the motion of the mechanical mode can becharacterized by a pair of canonical coordinates q and p.By replacing the photon annihilation (creation) operatorsby the complex light amplitude and the position operator ofthe cantilever by its classical counterpart in the Heisenbergequations, we obtain the classical equations of motion [40].This can effectively be regarded as a classical systemdescribed by four independent variables, two associatedwith the optical mode and two with the mechanical mode.In the language of nonlinear dynamics, the optomechanicalsystem in Fig. 1 has a four-dimensional phase space.The degree of quantum entanglement can be quantita-

tively assessed by using the corresponding quantumLangevin equations. Following the standard input-outputtheory [41] and making use of the reference frame rotatingat the laser frequency, we can get the following nonlinearquantum Langevin equations [6,7]:

q: ¼ ωmp; p

: ¼ −ωmq − ΓMpþffiffiffi2

pg0a†aþ ξ;

a: ¼ −ðκ þ iΔ0Þaþ i

ffiffiffi2

pg0aqþ Eþ

ffiffiffiffiffi2κ

pain; (1)

where Δ0 ¼ ωc − ω0 is the cavity detuning. Here, weassume that the optical bath is in thermal equilibrium,i.e., in some incoherent state, so that ain can be interpretedas a noise term [41]. The vacuum noise-input term can bedescribed by a zero-mean Gaussian random process. Theautocorrelation function of the vacuum noise is hainðtÞain†ðt0Þi¼δðt−t0Þ. The mechanical mode is under the influenceof stochastic Brownian noise that satisfies the non-Markovian autocorrelation relation [6,42,43]: hξðtÞξðt0Þi ¼ðΓM=ωMÞ

R ð2πÞ−1e−iωðt−t0Þωfcoth ½ℏω=ð2kBTÞ� þ 1gdω.

FIG. 1 (color online). (a) Schematic illustration of a typicaloptomechanical system driven by a single-mode laser. The laserbeam enters the optical cavity through the fixed mirror, butphotons inside the cavity can also decay through the mirror. Themovable mirror is totally reflecting and linked to a frictionalmechanical oscillator. [(b),(c)] Time series of the dimensionlesscanonical coordinate ~q for the situations where the classicaldynamics are periodic and quasiperiodic, respectively.

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For the quantum effect of the mechanical mode to beimportant, the quality factor of themechanical oscillator mustbe sufficiently high: Q ¼ ωM=ΓM ≫ 1. In this regime, wehave the following Markovian delta-correlated relation:hξðtÞξðt0Þþξðt0ÞξðtÞi=2¼ΓMð2nþ1Þδðt− t0Þ, where n ¼1=ðexp ½ℏωM=ðkBTÞ� − 1Þ is the mean mechanical phononnumber and T is the temperature of the mechanical bath.Equation (1) is a set of quantum stochastic equations that canbe simulated numerically (cf., Sec. S3 in the SupplementalMaterial [39]), and the standard ensemblemethod can beusedto calculate the degree of quantum entanglement. Anothermethod was introduced in Ref. [7], where a time-dependentcovariance matrix is calculated to fully characterize theevolution of the Gaussian state. We use both methods andset the system in the weakly coupling regime so that themagnitudes of noise are small compared to the zeroth-ordereffects (cf., Sec. S2 in the Supplemental Material [39]).In the Gaussian-evolution method, we assume small

fluctuations so that the relevant quantum operators can beexpanded about their respective mean values: OðtÞ ¼hOiðtÞ þ δOðtÞ, where O≡ ðq; p; a; a†Þ, yielding a set ofinhomogeneous equations governing the time evolutionof the fluctuations u

: ðtÞ ¼ AðtÞuðtÞ þ nðtÞ, where AðtÞ is a4 × 4 matrix and nðtÞ characterizes the input noise (seeSec. S4 in the Supplemental Material [39]). We can alsoobtain a set of equations for the mean values of the operators,but they have the same forms as the corresponding classicalequations of motion. Whereas Eq. (1) is nonlinear, u

: ðtÞ is aset of stochastic linear equations, meaning that the fluctua-tions will evolve asymptotically into zero-mean Gaussianstates but only when none of the Lyapunov exponents in thecorresponding classical system is positive. The covariance-matrix approach is, thus, not applicable to situations wherethe classical dynamics are chaotic [44]. For a nonchaoticdynamical process, the properties of the quantum fluctua-tions can be determined from the 4 × 4 covariance matrixthat obeys [7,42] V

: ðtÞ ¼ AðtÞVðtÞ þ VðtÞATðtÞ þD,where the element of the covariance matrix is defined asVij ¼ huiuj þ ujuii=2, and D ¼ diagð0; γMð2nþ 1Þ; κ; κÞ.This comes from hniðtÞnjðt0Þþnjðt0ÞniðtÞi=2¼δðt−t0ÞDij,characterizing the magnitudes of the noisy terms. Forconvenience, we can express V as

V ¼�VA VC

VTC VB

�;

where VA, VB, andVC are 2 × 2matrices associated with themechanical mode, the optical mode, and the optomechanicalcorrelation,respectively.Thedegreeofquantumentanglementbetween the mechanical and optical modes can, thus, beassessed by calculating the so-called logarithmic negativity,defined as [45] EN ≡max½0;− ln ð2η−Þ�, where η− ≡ð1= ffiffiffi

2p Þ½ΣðVÞ −

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΣðVÞ2 − 4 detV

p�1=2 and ΣðVÞ ¼

detðVAÞ þ detðVBÞ − 2 detðVCÞ (cf., Sec. S5 in theSupplemental Material [39]). We define Ep ≡ − ln ð2η−Þ asthe pseudoentanglement measure so that EN ¼ maxð0; EpÞ.

For the ensemble method, we simulate Eq. (1) a largenumber of times to obtain time series of the fluctuations.We can then obtain the time series of the covariance matrixfrom its definition, i.e., Vij ¼ huiuj þ ujuii=2. The entan-glement degree can be calculated from the definition of EN .Results.—We find that there are wide parameter regimes

in which the optomechanical system exhibits periodicand quasiperiodic motions, the representative time seriesof which are shown in Figs. 1(b) and 1(c), respectively.Whereas quasiperiodic motions and even chaos have beendiscovered recently in a coupled BEC (Bose-Einsteincondensation) type of hybrid optomechanical system[46], to our knowledge there were no previous reports ofquasiperiodic motions in the generic optomechanical sys-tems as in Fig. 1, which typically occur for low values ofthe cavity-decay rate, e.g., κ < 0.2ωM, and for moderatevalues of the detuning, e.g., for Δ0 ≈ −0.8ωM. The dis-sipation rate of the mechanical oscillator can take on valuesfrom a large range, e.g., 10−5ωM ∼ 10−3ωM. For computa-tional convenience, we rescale the dynamical variables:~q ¼ ffiffiffi

2p

g0q=ωM and ~P ¼ 8g20E2=ω4

M. Using the classicalHeisenberg equations, we calculate the bifurcation diagramof the system, i.e., the asymptotic extreme values of thedynamical variables as a function of the power of thedriving laser, as exemplified in Fig. 2(a), for the exper-imentally reasonable parameter setting of ωM ¼ 2πMHz,Q ¼ 25000, and m ¼ 10.67 ng. For this diagram, thecavity is assumed to be driven by a blue detuned laserwith value of the detuning Δ0 ¼ −0.81ωM and wavelengthλ ¼ 1064 nm. The damping of the mechanical mode isrelated to the quality factor Q by ΓM ¼ ωM=Q. To ensurethe validity of the photon-pressure Hamiltonian, we esti-mate x=l0 ¼ ~qωM=ωc ∼ 10−8, so the small-displacementassumption is well justified [37,40]. As the normalizedlaser power is increased through a threshold value ~Pc, atransition from periodic oscillation to quasiperiodic motionoccurs. Figure 2(b) shows the Lyapunov spectrum versus ~P.We observe that for ~P < ~Pc, the maximum Lyapunovexponent is zero but there is only one zero exponent,signifying the existence of a periodic attractor. For ~P > ~Pc,the maximum Lyapunov exponent remains to be zero butthere are two such exponents. In particular, the secondlargest Lyapunov exponent is negative for ~P < ~Pc but itbecomes zero for ~P > ~Pc, a feature characteristic of thetransition from periodicity to quasiperiodicity. In the rangeof ~P values shown, the third and the fourth Lyapunovexponents are all negative. The feature that there is nopositive Lyapunov exponent renders applicable the use ofthe quantum Langevin equation to calculate the degree ofquantum entanglement, because zero-mean Gaussian ran-dom inputs lead to fluctuation patterns that remainGaussian.This preservation of the Gaussian distribution meansthat the fluctuations can be fully determined by the covari-ance matrix. Figure 2(c) shows the pseudo-entanglementmaximum Ep;m as a function of ~P. Note that there isentanglement when the value of Ep is greater than 0, so

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the shaded region in which the values of Ep;m fall belowzero indicates lack of entanglement. The surprising phe-nomenon is that, as the system evolves towards the transitionpoint, the value of Ep;m continues to increase but drops tozero rather abruptly as the transition is reached. In thevicinity of the transition, quantum entanglement disappears.As the classical dynamics becomes quasiperiodic, entangle-ment is restored immediately because the value of Ep;m isrecovered and continues to increase as the laser power isfurther increased. The basic observation is that stronglynonlinear behavior can lead to the enhancement of quantumentanglement between the optical and mechanical modes.To probe further into the dynamics of quantum entangle-

ment through the classical transition point, we calculate thetime evolution of the pseudoentanglementmeasureEpðtÞ forrepresentative values of the driving laser power, as shownin Fig. 3 for four cases. Results from the ensemble approachby simulating the quantum Langevin equations with 3000realizations are also included. We observe a strong correla-tion between EpðtÞ and the evolution of the classicaldynamics, in that EpðtÞ, after a short transient period,exhibits periodic (quasiperiodic) behavior if the classicaldynamics is periodic (quasiperiodic). In all cases of Ep > 0

so that there is entanglement, the phenomenon of death andrebirth of entanglement [47,48] with time occurs, in whichEpðtÞ becomes negative and then restores to some positivevalue after a time interval. From the perspective of applica-tions, the entanglement duration time is of the order of amicrosecond so that the effects of entanglement oscillationsare negligiblewhen the desirable quantum operations can beachieved at sufficiently high speed. The remarkable phe-nomenon is that when quasiperiodicity sets in so that theclassical dynamics possesses two incommensurate frequen-cies, the corresponding quantum pseudoentanglementdynamics, after its “rebirth,” exhibits a surprising “beats”or temporalmodulation phenomenon. There is, thus, a directconsequence of classical nonlinear characteristics in quan-tum entanglement.We remark that there are proposals to quantify the

entanglement experimentally [6,49,50]. In particular, withthe aid of an ancillary cavity, one can construct thecovariance matrix through homodyne detection and thencalculate the logarithmic negativity.To summarize, using optomechanical systems as a

paradigm, we have addressed the manifestations ofclassical nonlinear dynamics in quantum entanglement,with a focus on two common types of classical dynamics:periodic and quasiperiodic motions. Our result is thatstrong signatures of the classical dynamics exist in therespective quantum entanglement dynamics. For example,when the classical dynamics is quasiperiodic, the corre-sponding quantum entanglement exhibits a surprising“beating” behavior in its time evolution. Not only is thedegree of entanglement enhanced as the classical dynamicstransits from periodic to quasiperiodic motions but theentanglement corresponding to the latter is also more

0 1000 2000 3000−6

−5

−4

−3

−2

−1

0

1(a)

2,500 2,550 2,600

−0.20

0.20.4

0 1000 2000 3000−6

−5

−4

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1(b)

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00.2

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0

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2,450 2,650 2,850

−3−2−1

0

FIG. 3 (color online). (a)–(d) Time evolution of Ep for fourvalues of the laser driving power ~P as specified by thevertical red dotted lines in Fig. 2(a). Insets show the respectivezoom-in regions indicated by the black dashed boxes, and thethin red dashed lines represent the results from the ensembleapproach. The shaded regions correspond to nonphysical resultsand EN ¼ max½0; Ep�.

0.02 0.04 0.06 0.08 0.1 0.12

3

3.2

3.4

3.6

3.8 (a)

0.02 0.04 0.06 0.08 0.1 0.12

Lyap

unov

exp

onen

t

(b)

0.02 0.04 0.06 0.08 0.1 0.12

−0.5

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0.5

1(c)

−0.1

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FIG. 2 (color online). (a) Typical bifurcation diagram of thegeneric optomechanical system in Fig. 1(a), where the asymptoticextreme values of the dimensionless canonical coordinate ~qare plotted versus the normalized external laser power ~P. Thethick purple dashed line indicates the threshold driving power~Pc ¼ 0.05842 at which a transition from periodic to quasiperi-odic motions occur. The cavity length is l0 ¼ 25 mm with finesseF ¼ 6 × 104. (b) The entire spectrum of the four Lyapunovexponents versus ~P and (c) the maximum of the pseudoentan-glement measure Ep versus ~P. The thin red vertical dashed linesat ~P ¼ 0.0400, 0.0567, 0.0589, and 0.0800 in panel (a) indicatethe specific values of ~P at which the time evolution of Ep is to beexamined (see Fig. 3).

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temperature robust (cf., Sec. S6 in the SupplementalMaterial [39]). Pushing the classical system into a highlynonlinear regime so that it exhibits more complicatedmotion than periodic oscillation can, thus, be beneficialfor achieving quantum entanglement [44].

We thank L. Ying for discussions. This work wassupported by AFOSR under Grant No. FA9550-12-1-0095 and by ONR under Grant No. N00014-08-1-0627.L. H. is also supported by the NSF of China under GrantsNo. 11135001 and No. 11375074.

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(2012).[44] A natural question is, thus, whether stronger and more robust

entanglement can be achieved when the system is driven intochaos. While we have indeed found wide parameter regimes inwhich the optomechanical system is chaotic, e.g., κ=ωM ¼ 0.1,ΓM=ωM ¼ 0.0013,Δ0=ωM ¼ 0.66,g0=ωM ¼ 10−5,E=ωM ¼1.7 × 104, at the present no method is available to quantify thequantum entanglement in this case. Especially, chaos is asso-ciatedwith at least one positiveLyapunovexponent, so an initialGaussian distribution of the stochastic fluctuations is quicklydestroyed. For non-Gaussian quantum fluctuations, the method

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to calculate the logarithmic negativity based on the evolution ofthecovariancematrix isnotapplicable.Newmethodologyneedsto be developed to characterize quantum entanglement forclassically chaotic systems. In this regard, the relation betweenentanglement and phase transition in the Dicke model has beeninvestigated [16], where the role of phase transitions in theconnection between entanglement and underlying integrable toquantum chaotic transitions was speculated. The quantum-chaotic properties of the Dicke Hamiltonian were also explored[15], and entanglement in cusp quantum catastrophes wasexamined [17].

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