Collective dynamics in optomechanical arrays

Georg Heinrich,1 Max Ludwig,1 Jiang Qian,2 Bjorn Kubala,1 and Florian Marquardt1, 3

1Institute for Theoretical Physics, Universitat Erlangen-Nurnberg, Staudtstr. 7, 91058 Erlangen, Germany2Department of Physics, Center for NanoScience,

and Arnold Sommerfeld Center for Theoretical Physics,Ludwig-Maximilians-Universitat Munchen, Theresienstr. 37, D-80333 Munchen, Germany

3Max Planck Institute for the Science of Light, Gunter-Scharowsky-Straße 1/Bau 24, 91058 Erlangen, Germany

The emerging field of optomechanics seeks toexplore the interaction between nanomechanicsand light (see [1] for a recent review). Rapidprogress in laser cooling of nanomechanical os-cillators [2, 3] promises new fundamental tests ofquantum mechanics [4], while applications bene-fit from ultrasensitive detection of displacements,masses and forces [5–7]. Recently, the excitingconcept of optomechanical crystals has been in-troduced [8–10], where defects in photonic crys-tal structures are used to generate both local-ized optical and mechanical modes that interactwith each other. For instance, this opens theprospect of integrated optomechanical circuitscombining several functions on a single chip (seealso [11, 12]). Here we start exploring the collec-tive dynamics of arrays consisting of many cou-pled optomechanical cells (Fig. 1a,b). We showthat such “optomechanical arrays” can displaysynchronization and that they can be describedby a modified Kuramoto model that allows to ex-plain and predict most of the features that willbe observable in future experiments.

The crucial ingredient of any optomechanical systemis an optical mode (OM) whose frequency shifts in re-sponse to a mechanical displacement: δωopt = −Gx.This coupling, vice versa, gives rise to a radiation force,F = ~G|α|2, where |α|2 is the number of photons circu-lating inside the laser-driven OM. For a laser red-detunedfrom the OM (∆ = ωLaser − ωopt < 0), dynamical back-action effects induced by the finite photon decay time κ−1

lead to cooling of the mechanical motion. In contrast,for blue detuning (∆ > 0), anti-damping results. Oncethis overcomes the internal mechanical friction, a Hopfbifurcation towards a regime of self-induced mechanicaloscillations takes place (Fig. 1c) [13–18]. While the me-chanical amplitude A is fixed, the oscillation phase ϕ isundetermined. Therefore, it is susceptible to externalperturbations. In particular, as we will see, it may lockto external forces or to other optomechanical oscillators.

Synchronization has first been discovered by Huygensand is now recognized as an important feature of col-lective nonequilibrium behavior in fields ranging fromphysics over chemistry to biology and neuroscience [19],with applications in signal processing and stabilization

laser drivecell 1

cell 2

...

optical modemechanical mode

bifurcation

amplitude

phase

a laserpower

b

c

FIG. 1. Optomechanical crystals may be used to build ar-rays with several localized optical and mechanical modes. (a)Potential setup fabricated as a periodically patterned, free-standing dielectric beam on a chip with laser drive via taperedfibre as in [8, 9]. (b) Schematic array of mechanically coupledoptomechanical cells. (c) For a single cell, at sufficient laserdrive power, there is a Hopf bifurcation towards self-inducedoscillations with an undetermined phase ϕ.

of oscillations. A paradigmatic, widely studied modelfor synchronization was introduced by Kuramoto [20].For two oscillators, his phase evolution equation readsϕ1 = Ω1 + K sin(ϕ2 − ϕ1), and likewise ϕ2. Onefinds synchronization (ϕ1 = ϕ2) if the coupling K ex-ceeds the threshold Kc = |Ω2 − Ω1|/2, and the phaselag δϕ = ϕ2 − ϕ1 vanishes for large K according tosin(δϕ) = (Ω2−Ω1)/2K. For the globally coupled, mean-field type version of infinitely many oscillators, there isa phase transition towards synchronization beyond somethreshold Kc that depends on the frequency distribution[21]. In many examples the Kuramoto model is foundas a generic, reduced description of the phase dynamics.Nevertheless, for any specific system, it remains to beseen whether this model (or possibly a structurally sim-ilar variant thereof) applies at all, and how the couplingK is connected to microscopic parameters [22–24]. Wenow turn to this question in the case of optomechanicaloscillators.

A single optomechanical cell consists of a mechani-cal mode (displacement x) coupled to a laser-driven OM(light amplitude α):

mx = −mΩ2x−mΓx+ ~G|α|2 (1)

α =[i(∆ +Gx)− κ

2

]α+

κ

2αmax (2)

Here Ω is the mechanical frequency, Γ the intrinsic damp-ing, G the optomechanical frequency pull per displace-

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ment, and αmax is the maximum light-field amplitudeachieved at resonance (set by the laser drive).

Near the Hopf bifurcation (Fig. 1c), we can capturethe essential dynamics by eliminating the light field [16]and switching to the phase- and amplitude-dynamics ofthe resulting Hopf oscillator:

ϕ = −Ω +F (t)

mΩAcos(ϕ) (3)

A = −γ(A− A) +F (t)

mΩsin(ϕ). (4)

Here A is the steady-state amplitude, and γ is the rate atwhich perturbations will relax back to A. Both dependon the microscopic optomechanical parameters, such aslaser detuning and laser drive power, and both vanish atthe bifurcation threshold (see methods section). More-over, we have introduced an external force F (t) (as addedto Eq. (2))

We start our discussion by considering phase-lockingto an external force F (t) = F0 sin(ωF t). To this end wetime-average Eq. (3), keeping only the slow dynamics,under the assumption ωF ≈ Ω. This results in

δϕ = −δΩ +KF sin(δϕ) , (5)

where δϕ = ϕ(t) + ωF t, δΩ = Ω − ωF , and KF =F0/2mΩA. Eq. (5) is a special case of the Kuramotoequation. Direct numerical simulation confirms the goodagreement between the microscopic optomechanical dy-namics and the simplified descriptions via Eqs. (3,4) and(5). In Fig. 2 we show sin δϕ(t) and its time-average〈sin δϕ(t)〉, with the phase ϕ being extracted from thecomplex amplitude of motion, β ≡ x + ix/Ω. Phase-locking sets in when δϕ = 0 has a solution, i.e. for|δΩ| ≤ KF , resulting in an “Arnold tongue” (see Fig. 2d).

We now turn to the dynamics of coupled cells, each ofwhich is described by Eqs. (1, 2). To these equations, weadd a mechanical coupling, set by a spring constant k:mx1 = . . . + k(x2 − x1). In the Hopf model, this yieldsa force F1 = kA2 cos(ϕ2) on the first oscillator (and viceversa). The case of optical coupling will be mentionedfurther below.

In order to arrive at the time-averaged dynamics forthe phase difference, δϕ = ϕ2 − ϕ1, it is necessary togo further than before, keeping A(t) = A + δA(t) in thephase equation, and eliminating the amplitude dynamicsto lowest order (see methods for the derivation; and [25,26] for further examples where the amplitude dynamics iscrucial). Then, we arrive at an effective Kuramoto-typemodel for coupled optomechanical Hopf oscillators:

δϕ = −δΩ− C cos(δϕ)−K sin(2δϕ). (6)

In contrast to the standard Kuramoto model, 2δϕ ap-pears, which will lead to both in-phase and anti-phase

forc

e st

reng

th

-0.02 0.00 0.02

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86420 86420time

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1.0

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(a)(b)

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0

sin

(δϕ)

0

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1

0

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ba0.03

0.02

-0.02 0.00 0.02-0.01 0.01

c d

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sin(δϕ)

0.01

FIG. 2. Phase-locking of an optomechanical cell to an exter-nal force. (a,b) Phase lag δϕ between oscillations and force,displayed via sin δϕ, outside (a) and inside (b) the phase-locked regime (colored in (c)). Time is plotted in units of2π/δΩ. (c) Plot of 〈sin δϕ〉 as a function of frequency dif-ference δΩ, at the force strength indicated by the blue tri-angle in (d), comparing optomechanics (blue, solid) againstthe Hopf model (Eqs. (3,4); magenta, dash-dot) and the Ku-ramoto model (Eq. (5); black, dash). (d) 〈sin δϕ〉 in the forcevs. frequency difference plane, showing the Arnold tongue.Deviations between the different models set in at high forces.(Lines indicate the transition towards phase-locking, styles asin (c)). (Microscopic cell parameters: ∆/Ω = 1, κ/Ω = 1,Γ/Ω = 0.01, ~G2α2

max/mΩ3 = 0.36).

synchronization. This corresponds to two distinct min-ima in the effective potential that can be used to rewriteEq. (6): δϕ = −U ′(δϕ). The coupling K = k2/2m2Ω2γdiverges near the bifurcation, where γ → 0. In the fol-lowing we focus on the case of nearly identical cells wherethe coupling C can be neglected; C/δΩ = k/2mΩ2 1.

To test whether these features are observed in the fulloptomechanical system, we directly simulate the motionand increase the coupling k for a fixed frequency differ-ence δΩ = Ω2−Ω1. The results are displayed in Fig. 3(a-c). Beyond a threshold kc, the frequencies and the phaseslock, indicated by a kink in 〈sin δϕ〉. As the coupling in-creases further, the phases are pulled towards each othereven more, so |δϕ| decreases. Thus, coupled optome-chanical systems do indeed exhibit synchronization. Aspredicted, there is both synchronization towards δϕ→ 0and δϕ→ π.

The dependence of the threshold kc on the frequencydifference δΩ is shown in Fig. 3d. The observed behav-ior kc ∼

√δΩ at small δΩ is correctly reproduced by the

generalized Kuramoto model, Eq. (6). For δΩ > γ devi-ations occur via terms of higher order in δΩ/γ, startingwith −(δΩ/γ)K cos(2δϕ) in Eq. (6). These produce alinear slope kc ∝ δΩ, see Fig. 3d.

In terms of experimental realization, optomechanicalcrystals [8, 9] offer a novel promising way to build ar-rays of optomechanical cells. They are fabricated asfree-standing photonic crystal beams (Fig. 1a). Varia-tions of the µm-scale lattice spacing produce both local-ized optical and vibrational modes. The strong confine-

3

sin

(δϕ)

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time time

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0.0 0.61

→ π

→ 0

b c dmechanical coupling k/mΩ21 frequency difference δΩ/Ω1

k/mΩ

2 1

time tδΩ/2πtime tδΩ/2π

δϕ→0δϕ→π

FIG. 3. Phase-locking for two coupled optomechanical cells. (a) The phase particle in the effective Kuramoto potential U(δϕ), inthe de-synchronized and phase-locked regime (left/right). (b) Time-evolution of sin(δϕ(t)). (c) When the mechanical couplingk exceeds a threshold, the phase difference δϕ between the oscillations locks in spite of different bare mechanical frequencies,here δΩ = 0.003 Ω1. Both in-phase and anti-phase regimes are observed. (d) Phase lag, expressed via 〈sin(δϕ)〉, in the planecoupling k vs. frequency difference δΩ, including a comparison of the critical coupling kc (white, solid) with the one from aHopf model with one fit parameter (green, dash) and the generalized Kuramoto model (blue, dash-dot). (Cell parameters asin Fig. 2).

ment leads to extremely large optomechanical couplings,on the order of G ∼ 100 GHz/nm. Typical parameters,that we use in our simulations for experimentally realis-tic results, are G = 100 GHz/nm, mechanical frequencyΩ = 1 GHz, mass m = 100 fg, mechanical quality fac-tor QM = Ω/Γ = 100, cavity decay rate κ = 1 GHzand laser input powers such that the circulating photonnumber |αmax|2 & 100.

To consider optomechanical arrays like in Fig. 1a, weuse finite element methods (FEM) to simulate two iden-tical cells arranged on the same beam (Fig. 4a,b). Theoptical and vibrational couplings mediated by the beamcan be deduced from the splitting between the resultingsymmetric/antisymmetric modes. The results shown inFig. 4c,d validate the parameters considered above andindicate mechanical couplings k/mΩ2 . 0.01. Due tothe relatively strong optical coupling (∼ THz), distinctOMs in the individual cells can only be achieved by pat-terning them to have frequencies sufficiently different toprevent hybridization (Fig, 4e). This requires differentlaser colors to address each cell.

In experiments, a convenient observable would be theRF frequency spectrum of the light intensity emanatingfrom the cells, |α|2 (ω). We first show the spectrum as afunction of frequency difference δΩ for two mechanicallycoupled cells, driven independently (Fig. 5a). Frequencylocking is observed in an interval around δΩ = 0. Exper-imentally, the mechanical frequencies can be tuned viathe “optical spring effect” [8].

The most easily tunable parameter is the laser drivepower (∝ α2

max). Synchronization sets in right at theHopf bifurcation. For two cells (Fig. 5c), we recoverthe regimes of in-phase and anti-phase synchronization.They differ in the synchronization frequency, Ω(π) −

mechanical “pinch” modea

b optical mode

16 18 20 22 24 26 14 16 18 20 22 24 26

mechanicalcoupling [MHz]

opticalcoupling [THz]

c d e

108

6

4

20

2.52.01.51.00.5

distance / [lattice spacing]

FIG. 4. FEM simulation for two coupled cells localized on thesame beam in an optomechanical crystal setup (cf. Fig. 1a).(a) Horizontal displacement of vibrational “pinch” modes. (b)Transverse electric fields of optical modes. (c,d) The inter-cell couplings decay exponentially as a function of distance.(e) Blue-detuned lasers could illuminate two distinct opticalmodes (top) or a single hybridized “molecular” mode. (Geo-metrical parameters see methods section).

Ω(0) = k/mΩ. At higher drive, we find a transitiontowards de-synchronization. This remarkable behaviorcan be explained from our analytical results. We knowthat γ increases away from the Hopf bifurcation (i.e. forhigher drive), leading to a concomitant decrease in theeffective Kuramoto coupling K ∝ 1/γ, and finally a lossof synchronization. Near the transition, the frequenciesfan out (as δΩeff ∝

√αmax − αc). The multitude of peaks

is produced due to nonlinear mixing. In some regimes,

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Hopf bifurcation

FIG. 5. Mechanical frequency spectra of optomechanical arrays. (a,b) Frequency locking upon changing the detuning δΩ = Ω2−Ω1 between the mechanical frequencies (magenta, dashed) of two independently driven, mechanically coupled optomechanicalcells (k/mΩ2

1 = 0.015). (a) Spectrum I(ω) of intensity fluctuations, I(t) = |α1(t)|2 + |α2(t)|2, from an optomechanical model.(b) Spectrum of x1 + x2 in the Hopf model. (c) Spectrum vs. laser input power for two independently driven, mechanicallycoupled cells (k/mΩ2

1 = 0.01, δΩ = 0.005Ω1). (d) Example of trajectories xi(t), in units of κ/G, displaying strong amplitudemodulation, not described by the Kuramoto model (at a power indicated by the blue triangle in (c)). (e) Spectrum I(ω) vs.laser input power for an array of five mechanical modes (k = 0), coupled to a common optical mode. At large drive, a regime ofchaotic motion is entered. Due to the presence of multiple attractors, the regimes observed in such a diagram may depend onhow the parameters are swept. (f) Example of a trajectory in the chaotic regime (blue triangle in (e)). (color scale in all plotsindicates |I(ω)| in units of the peak height at ω = 0 for a system at rest; δ peaks are broadened for clarity; cell parameters are∆i = κi = 100Γi = Ω1, as in Figs. 2,3; yellow triangles indicate the bare mechanical frequencies Ωi)

the Kuramoto model fails (Fig. 5d).For large arrays, it will be most practical to have iden-

tical OMs that combine into extended ’molecular’ modes,one of which is then driven by a single laser via an evanes-cently coupled tapered fibre (Fig. 1a, Fig. 4e). For effi-cient excitation of self-induced oscillations, one has toensure that the detuning ∆ is equal to all the mechani-cal frequencies Ωj in the array, to within |∆− Ωj | < κ.For arrays of reasonable size, the splittings between ad-jacent optical molecular modes will be more than 102

times larger than κ, such that we can ignore all but oneOM. This setup then leads to a global coupling of manynanoresonators to a single OM, such that

α =

i(∆ +∑j

Gjxj)−κ

2

α+κ

2αmax , (7)

and the force on each resonator is given by −~Gj |α|2.This setup comes close to realizing the all-to-all couplingmost often investigated in the literature on the Kuramotomodel.

For illustration, we chose N = 5 cells (Fig. 5e). Asbefore, we find synchronization regimes. In addition, at

higher drive, a transition into chaos takes place. Analyz-ing the time-evolution in more detail, we observe tran-sient fluctuations in amplitude and phase, with a strongsensitivity on changes in initial conditions (Fig. 5f). Notethat in a single optomechanical cell one may also findchaotic behavior [14], but for far larger driving strengths.

Optomechanical arrays open up a new domain to studycollective oscillator dynamics, with room-temperatureoperation in integrated nanofabricated circuits and withnovel possibilities for readout and control, complement-ing existing research on Josephson arrays [22], laser ar-rays [23] and other nanomechanical structures [24, 27].Recent experiments on 2D optomechanical crystals [28]could form the basis for investigating collective dynam-ics in 2D settings with various coupling schemes. Ap-plications in signal processing may benefit from phasenoise suppression via synchronization [29]. Variations ofthe optomechanical arrays investigated here may also berealized in other designs based on existing setups, likemultiple membranes in an optical cavity [30] or arrays oftoroidal microcavities [2, 5].

5

Acknowledgements

We thank O. Painter, H. Tang and J. Parpia for fruitfuldiscussions, and GIF and DFG (Emmy-Noether program,NIM) for funding.

Methods

The dynamics of a single optomechanical cell,Eqs. (1, 2), in the self-induced oscillation regime canbe mapped to a Hopf model close to the Hopf bifurca-tion. This model is described by the steady state ampli-tude A and amplitude decay rate γ (see Eqs. (3, 4)).The dependence of γ, A on the microscopic parame-ters can be deduced by expanding the average mechan-ical power input provided by the radiation pressureforce, ~G〈|α|2 x〉 (see Eq. (1)), in terms of a = GA;

~G〈|α|2 x〉 = (~α2maxπ2)a2 + (~α2

maxπ4/Ω2)a4 + O(a6).

This yields

γ = 2PΩπ2 − Γ

and (GA/κ

)2= γΩ/(−2π4Pκ2),

where the dimensionless coefficients π2(∆/Ω, κ/Ω) andπ4(∆/Ω, κ/Ω) only depend on the rescaled detuning andcavity decay rate. P = ~G2α2

max/mΩ3 is the rescaledlaser input power. Numerically, the Hopf parametersmay be found (even away from threshold) by simulatingthe exponential relaxation of the cell’s oscillation ampli-tude towards A, after a small instantaneous perturbationof the steady-state dynamics. In general, to compare op-tomechanics to results from Hopf (e.g. Fig. 2), the opticalspring effect also needs to be considered.

Two mechanically coupled optomechanical cells aremodeled as distinct Hopf oscillators with phase dynam-ics ϕ1(t), ϕ2(t) and amplitude dynamics A1(t), A2(t)according to Eqs. (3) and (4). The coupling forcesread F1 = kA2 cos(ϕ2), F2 = kA1 cos(ϕ1). WithAi(t) = Ai + δAi(t), the solution for the amplitude dy-

namics is δAi(t) =´ t−∞ e−γ(t−t′)fi(t

′) dt′ where f1(t) =k(A2+δA2(t))

m1Ω1cosϕ2(t) sinϕ1(t) and likewise for f2(t). In

the following, we consider small couplings, where δAi Ai. Then δA1 (and likewise δA2) is found to be

δA1(t)

A1=

k

2m1Ω1×

Im

(ei(ϕ2(t)+ϕ1(t))

γ − i(Ω2 + Ω1)− ei(ϕ2(t)−ϕ1(t))

γ − i(Ω2 − Ω1)

).

Thus we can eliminate the amplitude dynamics tolowest order from the phase equations, by expanding

A2(t)/A1(t) ≈ A2/A1 + δA2/A1 − A2δA1/A1 in the fol-lowing equation (likewise for ϕ2):

ϕ1 = −Ω1 +k

m1Ω1

(A2

A1

)cos(ϕ2) cos(ϕ1).

We now perform a time average, keeping only theslow dynamics near frequencies 0 and ±|Ω2 − Ω1|.This leads to the stated result for the effective Ku-ramoto model, Eq. (6), after setting δϕ = ϕ2 −ϕ1. The coupling constants are given by C =(k/2)

(A2/m1Ω1A1 − A1/m2Ω2A2

)and K = (1 +

(ξ1/ξ2 + ξ2/ξ1)/2)k2/4m1Ω1m2Ω2, where ξj = mjΩjA2j .

The optomechanical simulation in Fig. 3 shows resultsfor experimentally realistic microscopic parameters us-ing an input power well above the bifurcation threshold.This allows to observe the essential features predictedfrom Hopf and the effective Kuramoto-type model in anappropriate range of frequency detuning δΩ. However,to achieve quantitative agreement of the Hopf model inFig. 3, its parameter γ has to be treated as an adjustableparameter (here γ = 0.02 Ω). Each simulation initiallystarts with a system at rest and considers an instanta-neous switch-on of the laser input power. Whether thesystem synchronizes towards δϕ → 0 or δϕ → π alsodepends on the initial conditions.

In principle, an amplitude dependence of the mechan-ical frequency, Ω(A) ' Ω(A) + (∂Ω/∂A)δA can yield ad-ditional terms and an alternative synchronization mech-anism for two Hopf oscillators. However, numerical sim-ulations verified that for optomechanical cells this aspectcan be neglected.

For the finite-element simulation in Fig. 4 the unit cellin the periodic part is a 1,396nm-wide, 362nm-long rect-angle with a co-centric rectangular hole of 992nm widthand 190nm length. The thickness of the beam is 220nm.The isotropic Young’s modulus of 168.5 GPa and the re-fractive index is 3.493. Each defect is 15 units in length,symmetric across the 8th(central) cell. The lattice con-stants vary linearly from 362nm at the edge to 307.7nmat the center. The holes in the defects stay co-centricwith the unit cell and remain constant in size. For a sin-gle cell these parameters have been reported in Ref. [9].

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