Quantum many-body dynamics in optomechanical arrays

Max Ludwig1, ∗ and Florian Marquardt1, 2

1Institute for Theoretical Physics, Universitat Erlangen-Nurnberg, Staudtstraße 7, 91058 Erlangen, Germany2Max Planck Institute for the Science of Light, Gunther-Scharowsky-Straße 1/Bau 24, 91058 Erlangen, Germany

We study the nonlinear driven dissipative quantum dynamics of an array of optomechanical sys-tems. At each site of such an array, a localized mechanical mode interacts with a laser-driven cavitymode via radiation pressure, and both photons and phonons can hop between neighboring sites.The competition between coherent interaction and dissipation gives rise to a rich phase diagramcharacterizing the optical and mechanical many-body states. For weak intercellular coupling, themechanical motion at different sites is incoherent due to the influence of quantum noise. When in-creasing the coupling strength, however, we observe a transition towards a regime of phase-coherentmechanical oscillations. We employ a Gutzwiller ansatz as well as semiclassical Langevin equationsgoing beyond the mean-field approach, and we propose a realistic experimental implementation inoptomechanical crystals.

Introduction. - Recent experimental progress hasbrought optomechanical systems into the quantumregime: A single mechanical mode interacting with alaser-driven cavity field has been cooled to the groundstate [1, 2]. Several of these setups, in particular op-tomechanical crystals, offer the potential to be scaled upto form optomechanical arrays. Applications of such ar-rays for quantum information processing [3, 4] have beenproposed. Given these developments, one is led to ex-plore quantum many-body effects in optomechanical ar-rays. In this work, we analyze the nonlinear photon andphonon dynamics in a homogeneous two-dimensional op-tomechanical array. In contrast to earlier works [3–6],here we study the array’s quantum dynamics beyond aquadratic Hamiltonian. To tackle the non-equilibriummany-body problem of this nonlinear dissipative system,we employ a mean-field approach for the collective dy-namics. First, we discuss photon statistics in the array,in particular how the photon blockade effect [7] is al-tered in the presence of intercellular coupling. The mainpart of the article focuses on the transition of the collec-tive mechanical motion from an incoherent state (due toquantum noise) to an ordered state with phase-coherentmechanical oscillations. For these dynamics, the dissi-pative effects induced by the optical modes play a cru-cial role. On the one hand, they allow the mechanicalmodes to settle into self-induced oscillations [8] once theoptomechanical amplification rate exceeds the intrinsicmechanical damping. On the other hand, the fundamen-tal quantum noise (e.g. cavity shot noise) diffuses themechanical phases and prevents the mechanical modesfrom synchronizing. This interplay leads to an elaboratephase diagram characterizing the transition. We developa semiclassical model to describe the effective dynamicsof the mechanical phases and to study the system beyondthe mean-field approximation.

While true long-range order is prohibited for atwo-dimensional system with continuous symmetry, atleast for equilibrium systems, a Beresinskii-Kosterlitz-Thouless transition towards a state with quasi-long range

10

displacement electric !eld

0 1

Figure 1. Example implementation of an optomechanicalarray: A two-dimensional snowflake optomechanical crys-tal [18, 19] supports localized optical and mechanical modesaround defect cavities. Here, we propose arranging them in asuper structure, forming the array. The insets show electricfield ~E and displacement field ~u of an isolated defect cavity(obtained from finite element simulations). Due to the finiteoverlap between modes of neighboring sites [20], photons andphonons can hop through the array, see Eq. (2). A wide laserbeam drives the optical modes of the array continuously andthe reflected light is read out.

order is possible. The ordered mechanical phase thusresembles the superfluid phase in two-dimensional coldatomic gases [9] or Josephson junction arrays [10]. No-tably, optomechanical arrays combine the tunability ofoptical systems with the robustness and durability of anintegrated solid-state device. Other driven dissipativesystems that have been studied with regard to phasetransitions recently include cold atomic gases [11–14],nonlinear cavity arrays [15, 16] and optical fibers [17].In a very recent work and along the lines of [11], thepreparation of long-range order for photonic modes wasproposed using the linear dissipative effects in an optome-chanical array [5]. Our work adds the novel aspect of amechanical transition to the studies of driven dissipativemany-body systems.

Model. - We study the collective quantum dynamics ofa two-dimensional homogeneous array of optomechanicalcells (Fig. 1). Each of these cells consists of a mechanicalmode and a laser driven optical mode that interact via

2

the radiation pressure coupling at a rate g0 (~ = 1):

Hom,j = −∆a†j aj + Ωb†j bj−g0(b†j + bj)a†j aj +αL(a†j + aj).

(1)

The mechanical mode (bj) is characterized by a frequencyΩ. The cavity mode (aj) is transformed into the framerotating at the laser frequency (∆ = ωlaser − ωcav) anddriven at the rate αL. In the most general case, bothphotons and phonons can tunnel between neighboringsites 〈ij〉 at rates J/z and K/z, where z denotes thecoordination number. The full Hamiltonian of the arrayis given by H =

∑j Hom,j + Hint, with

Hint = −Jz

∑〈i,j〉

(a†i aj + aia

†j

)− K

z

∑〈i,j〉

(b†i bj + bib

†j

).(2)

To bring this many-body problem into a treatable form,we apply the Gutzwiller ansatz A†i Aj ≈ 〈A†i 〉Aj +

A†i 〈Aj〉 − 〈A†i 〉〈Aj〉 to Eq. (2). The accuracy of this ap-

proximation improves if the number of neighboring sites zincreases. For identical cells, the index j can be droppedand the Hamiltonian reduces to a sum of independentcontributions, each of which is described by

Hmf = Hom − J(a†〈a〉+ a〈a†〉

)−K

(b†〈b〉+ b〈b†〉

).(3)

Hence, a Lindblad master equation for the single cell den-sity matrix ρ, dρ/dt = −i[Hmf , ρ] + κD[a]ρ+ ΓD[b]ρ canbe employed. The Lindblad terms D[A]ρ = AρA† −12 A†Aρ − 1

2 ρA†A take into account photon decay at a

rate κ and mechanical dissipation (here assumed due toa zero temperature bath) at a rate Γ.

Photon statistics. - Recently, it was shown that theeffect of photon blockade [7] can appear in a single op-tomechanical cell: The interaction with the mechanicalmode induces an optical nonlinearity of strength g2

0/Ω[7, 21] and the presence of a single photon can hinderother photons from entering the cavity. To observe thiseffect, the nonlinearity must be comparable to the cav-ity decay rate, i.e. g2

0/Ω & κ, and the laser drive weak(αL κ) [7, 22].

To study nonclassical effects in the photon statistics,we analyze the steady-state photon correlation functiong(2)(τ) = 〈a†(t)a†(t+ τ)a(t+ τ)a(t)〉/〈a(t)†a(t)〉2 [23] atequal times (τ = 0). Here (Fig. 2), we probe the in-fluence of the collective dynamics by varying the opticalcoupling strength J , while keeping the mechanical cou-pling K zero for clarity. We note that, when increasingJ , the optical resonance effectively shifts: ∆ → ∆ + J .To keep the photon number fixed while increasing J , thedetuning has to be adapted [24]. In this setting, we ob-serve that the interaction between the cells suppressesanti-bunching (Fig. 2 (b)). Photon blockade is lost if theintercellular coupling becomes larger than the effectivenonlinearity, 2J & g2

0/Ω. Above this value, the photon

detuning

optic

al c

oupl

ing

stre

ngth

photon correlations

-1 0-0.5-1.5

0.5

0

1

1

2

4

3

coupling strength0.001 0.01 0.1 1

1

3

phot

on c

orre

latio

ns

a b

Figure 2. Loss of photon blockade for increasing opticalcoupling in an array of optomechanical cavities. (a) Theequal time photon correlation function shows anti-bunching(g(2)(0) < 1) and bunching (g(2)(0) > 1) as a function ofdetuning ∆ and optical coupling strength J . The smallestvalues of g(2)(0) are found for a detuning ∆0 = −g20/Ω. (b)When increasing the coupling J while keeping the intracavityphoton number constant, i.e. along the dashed line in panel(a), photon blockade is lost (black solid line). For a smallerdriving power (blue solid line, αL = 5 · 10−5κ), anti-bunchingis more pronounced and the behavior is comparable to that ofa nonlinear cavity (dashed line). The hatched area in (a) out-lines a region where a transition towards coherent mechanicaloscillations has set in. κ = 0.3 Ω, αL = 0.65κ, g0 = 0.5 Ω,Γ = 0.074 Ω.

statistics shows bunching, and ultimately reaches Pois-sonian statistics for large couplings. Similar physics hasrecently been analyzed for coupled qubit-cavity arrays,[24]. For large coupling strengths, though, Fig. 2(a) re-veals signs of the collective mechanical motion (hatchedarea). There we observe the correlation function to oscil-late (at the mechanical frequency) and to show bunching.We will now investigate this effect.

Collective mechanical quantum effects. - To describethe collective mechanical motion of the array, we focuson the case of purely mechanical intercellular coupling(K > 0, J = 0) for simplicity. Note, though, that theeffect is also observable for optically coupled arrays, asdiscussed above.

As our main result, Figs. 3(a) and (d) show thesharp transition between incoherent self-oscillations anda phase-coherent collective mechanical state as a functionof both laser detuning ∆ and coupling strength K: In theregime of self-induced oscillations, the phonon number〈b†b〉 reaches a finite value. Yet, the expectation value

〈b〉 remains small and constant in time. When increas-

ing the intercellular coupling, though, 〈b〉 suddenly startsoscillating and reaches a steady state

〈b〉(t) = b+ re−iΩeff t. (4)

Here, we introduced the mechanical coherence r and theoscillation frequency Ωeff , which is shifted by the opticalfields and the intercellular coupling, cf. Eq. (S.11).

Our more detailed analysis (see below) indicates thatthis transition results from the competition between thefundamental quantum noise of the system and the ten-

3

0 0.25 0.50

0.5

1

0

15

15 0

0

15

15 0

5

0

-5-5 0 5

5

0

-5-5 0 5

mec

hani

cal c

oupl

ing

detuning

mechanical coherence

incoherent

coherent

onse

t of s

elf-o

scill

atio

ns0

0.5

1.5

1

-1 0-0.5-1.5

0.25

0

0.5

10.5

bc

b b

c c

mec

hani

cal

coupling

opticalreadout

cohe

renc

e

a

b

c

d

Figure 3. Transition from the incoherent to the synchronized(coherent) phase: (a) Mechanical coherence r (Eq. (4)) as afunction of laser detuning ∆ and mechanical coupling K. Atweak coupling, the self-oscillations are incoherent, r = 0, dueto quantum noise. When increasing the coupling strength, thesystems shows a sharp transition towards the ordered regime,where the mechanical oscillations are phase-coherent, r > 0.(b,c) Modulus of the density matrix elements (in Fock space)and Wigner density of the collective mechanical state in theincoherent (b) and the coherent regime (c), as marked in (a).(d) Mechanical coherence r as a function of coupling strengthK along the dashed line in (a). The dotted line shows theoptical readout of coherence, i.e. the oscillating componentof the photon number 〈a†a〉, proportional to the intensity ofthe reflected beam and thus directly accessible in experiment.g0 = κ = 0.3 Ω, αL = 1.1κ, Γ = 0.074 Ω

dency of phase locking between the coupled nonlinearoscillators. Below threshold, the quantum noise from thephonon bath and the optical fields diffuses the mechan-ical phases at different sites and drives the mechanicalmotion into an incoherent mixed state. The reduceddensity matrix ρ(m) is predominantly occupied on thediagonal, see Fig. 3(b), and the Wigner distribution,W (x, p) = 1

π~∫∞−∞〈x−y|ρ

(m)|x+y〉e2ipy/~dy, has a ring-like shape, reflecting the fact that the mechanical phaseis undetermined [25, 26]. Above threshold, the mechan-ical motion at different sites becomes phase locked, andthe coherence parameter r reaches a finite value. Theemergence of coherence also becomes apparent from theoff-diagonal elements of ρ(m) (Fig. 3(c)). The corre-sponding Wigner function assumes the shape of a co-herent state with a definite phase oscillating in phasespace. Thus, this transition spontaneously breaks thetime-translation symmetry. In a two-dimensional imple-mentation, true long range order is excluded, but thecoherence between different sites is expected to decay asa power law with distance. We also note that this tran-

sition is the quantum mechanical analogon of classicalsynchronization, which was studied for optomechanicalsystems in [27–29]. An important difference is, though,that the classical nonlinear dynamics was analyzed foran inhomogeneous (with disordered mechanical frequen-cies) system in the absence of noise [27–29], while in ourcase disorder is only introduced via fundamental quan-tum noise. Quantum synchronization has also been dis-cussed in the context of linear oscillators [30] and non-linear cavities [31] recently.

The laser detuning determines both the strength of theself-oscillations and the influence of the cavity shot noiseon the mechanical motion. It turns out that the diffusionof the mechanical phases is pronounced close to the onsetof self-oscillations and at the mechanical sideband [20].As we will show below, even the coherent coupling be-tween the mechanical phases (ultimately leading to syn-chronization) is tunable via the laser frequency. As a re-sult, the synchronization threshold depends non-triviallyon the detuning parameter ∆, see Fig. 3(a).

Beyond mean-field approximation. - In order to gainfurther insight into the coupling and decoherence mech-anisms as well as effects of geometry and dimensionality,we analyze the semi-classical Langevin equations of thefull optomechanical array:

βi =(− iΩ− Γ

2

)βi + ig0|αi|2 + i

K

z

∑〈ij〉

βj +

√Γ

2ξβ

αi =(i∆ + 2ig0(βi + β∗i )− κ

2

)αi − iαL +

√κ

2ξα. (5)

The fluctuating noise forces ξσ=α,β(t) mimic the effectsof the zero temperature phonon bath and the cavity shotnoise, respectively. They are independent at each siteand obey 〈ξσ〉 = 0 and 〈ξσ(t)ξ∗σ(t′)〉 = δ(t− t′), where, inthis context, 〈...〉 denotes the average over different real-izations of the stochastic terms. For a single optomechan-ical cell, such a description has proven good qualitativeagreement with the full quantum dynamics [25, 32].

First, we study the onset of quasi long-range order ina finite system. To this end we evaluate the correla-tions C(d = |i − j|) = 〈eiϕie−iϕj 〉, where eiϕi = βi/|βi|.Numerical calculations on a 30 × 30 square lattice (seeFig. 4(a)) indicate that for weak intercellular couplingthe mechanical phases at different sites are uncorrelatedeven for small distances d. When increasing the couplingstrength, however, the mechanical motion becomes cor-related over the whole array with only a slow decreasewith distance. The coupling threshold, here defined bysetting a lower bound of C(14) > 0.01, varies with coor-dination number, see Fig. 4(b). Within the mean-fieldapproximation, i.e. for a lattice with global coupling ofall sites, fluctuations between neighboring sites and hencethe threshold value are underestimated. The couplingthreshold grows with the quantum parameter [25], i.e.the ratio of optomechanical coupling and cavity decay

4

0 0.5 10

0.10.2

0.1 0.110

0.5

1

0 4 8 120

0.5

1

corr

elat

ions

corr

elat

ions

coupling

thre

shol

d

thresholddistance on lattice

quantum parameter

mean-!eld

mean-!eld

square

hexag.

squaremean-!eld,quantum

a b

c

Figure 4. Langevin dynamics beyond mean-field approxi-mation: (a) Correlations C(d = |i − j|) = |〈eiϕie−iϕj 〉|in a 30 × 30 optomechanical array. Quasi-long range or-der sets in for sufficiently large coupling strengths. K =0.09, 0.105, 0.107, 0.12, 0.15Ω (b) Correlations over a dis-tance of d = 14 as a function of mechanical coupling strengthK for a square lattice (z = 4, squares), a hexagonal lattice(z = 6, triangles) slightly below the mean-field result (circles).(c) Coupling threshold as a function of quantum parameterg0/κ (squares: square lattice, empty (filled) circles: semi-classical (quantum) mean-field approach). ∆ + g20/Ω = 0.34,g0 = 0.1κ in (a),(b), g0αL = 0.33κ in (c), other parametersas in Fig. 3.

rate, g0/κ, see Fig. 4(c): For g0 ≈ κ, single photonsand phonons interact strongly and quantum fluctuationshamper synchronization.

Synchronization threshold. - For an analytical ap-proach, the complexity of the Langevin equations canbe reduced by integrating out the dynamics of the opti-cal modes and the mechanical amplitudes and by goingback to the mean-field approximation [20]. The resultingequation describes the coupling of the mechanical phaseon a single site, ϕ, to a mean field Ψ:

ϕ = −Ω(A) +KR cos(Ψ− ϕ) +K1R sin(Ψ− ϕ)

+√

2Dϕξϕ +O(R2). (6)

Here, the order parameter is defined as 〈eiϕj 〉 ≡ ReiΨ.The rate K1 = (dΩ − K/2)K/γ determines the cou-pling of phases mediated by slow amplitude modulationsbetween neighboring sites. These beat modes coupleback to the phase dynamics via the amplitude depen-dent optical spring effect, Ω(A) + dΩ · (A− A)/A, wheredΩ = A dΩ

dA |A=A, and the bare mechanical coupling K,leading to two opposing terms in K1. Here, A denotesthe steady state mechanical amplitude and γ the ampli-tude decay rate set by the optical field. The fluctuatingnoise force ∼ ξϕ comprises the effects of mechanical fluc-tuations and radiation pressure noise and is characterizedby a diffusion constant Dϕ [20, 32].

Equation (S.11) reveals the close connection to the Ku-ramoto model [33] and the two-dimensional xy-model. In

the incoherent regime, the order parameter R is zero andthe phase fluctuates freely. In the coherent regime, therestoring force ∼ K1R leads the phase ϕ towards a fixedrelation with Ψ. The cosine term only renormalizes theoscillation frequency. This statement can be clarified bya linear stability analysis, see [20, 34]. It turns out thatthe incoherent phase becomes unstable for

K1 = 2Dϕ, (7)

defining the threshold of the transition. Moreover, if K1

becomes negative, no stable phase synchronization is pos-sible. This situation arises if dΩ < 0, or for large inter-cellular coupling rates K > 2dΩ, see Fig. 3(d).

Experimental prospects. - We note that observation ofthe mechanical phase transition does not require singlephoton strong coupling (g0 & κ): The quantum fluctua-tions of the light field will dominate over thermal fluctu-ations as long as 4g2

0 |α|2/κ > kBT/Q. This is essentiallythe condition for ground-state cooling, which has beenachieved using high-Q mechanical resonators and cryo-genic cooling [1, 2], see Table I. In contrast, the photon-blockade effect (Fig. 2) requires low temperatures T andg2

0 & Ωκ, or at least, in a slightly modified setup [35, 36],g0 & κ. While still challenging, optomechanical systemsare approaching this regime [37].

Microfabricated optomechanical systems such as mi-croresonators (e.g. [38]), optomechanical crystals (e.g.[2]) or microwave-based setups (e.g. [1]) lend themselvesto extensions to optomechanical arrays. Here, we focuson optomechanical crystals, which are well suited due totheir extremely small mode volumes. The properties oftwo-dimensional optomechanical crystals have been ana-lyzed in [18]. The finite overlap of the evanescent tailsof adjacent localized modes [20] results in a coupling ofthe form of Eq. (2), in analogy to the tight-binding de-scription of electronic states in solids. Sufficiently strongoptical and mechanical hopping rates are feasible, see [27]for one-dimensional and [20] for two-dimensional struc-tures. The simultaneous optical driving of many cellsmay be realized by a single broad laser beam irradiat-ing the slab, see Fig. 1. Alternatively, similar physicsmay be observed for many mechanical modes coupling toone extended in-plane optical mode [6, 27, 28] (therebyeffectively realizing global coupling).

setup T [K] Γnth/Ω Γopt/Ω g0/κ L [µm]

microwave-based 25 mK 10−4 7 · 10−3 10−3 ∼ 100

optomech. crystal 20 K 10−3 4 · 10−3 2 · 10−3 ∼ 4

microtoroid 650 mK 7 · 10−2 6 · 10−2 5 · 10−4 ∼ 30

Table I. Parameters of optomechanical systems [1, 2, 38]:Temperature of phonon bath T , strength of mechanical fluc-tuations Γnth ≈ kBT/Q, strength of cavity shot noise Γopt ≈4g20 |α|2/κ, quantum parameter g0/κ and approximate size L.

5

The transition towards the synchronized phase can bedetected by probing the light reflected from the optome-chanical array and measuring the component oscillatingat the mechanical frequency, see Fig. 3(d). To read outcorrelations between individual sites, the intensities ofindividual defect cavities may be analyzed [20], for ex-ample by evanescently coupling them to tapered fibersor waveguides.

We expect the transition to be robust against disorder[27]. One may also study the formation of vortices andother topological defects induced by engineered irregular-ities and periodic variations, and explore various differentlattice structures or the possibility of other order phases(e.g. anti-ferromagnetic order). Thus, optomechanicalarrays provide a novel, integrated and tunable platformfor studies of quantum many body effects.

The authors would like to thank Oskar Painter andBjorn Kubala for valuable discussions. This work wassupported by the DFG Emmy-Noether program, an ERCstarting grant, the DARPA/MTO ORCHID program andthe ITN network cQOM.

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Supplemental Material for ”Quantum many-body dynamics in optomechanical arrays”

Max Ludwig1,∗ and Florian Marquardt1,2

1Institute for Theoretical Physics, Universitat Erlangen-Nurnberg, Staudtstraße 7, 91058 Erlangen, Germany2Max-Planck-Institute for the Science of Light, Gunther-Scharowsky-Straße 1/Bau 24, 91058 Erlangen, Germany

MEAN-FIELD PHASE EQUATION

In this section, we provide details for the derivation ofthe mean-field phase equation, Eq. (S.11), starting fromthe equations of motion of the optomechanical array, Eqs.(5). Introducing phases ϕj and amplitudes Aj (in unitsof the mechanical ground state width) as new coordinatesof mechanical motion and omitting fast oscillating terms,the so-called Hopf equations are derived directly from (5).We follow [1], but add noise terms:

ϕi = −Ω(A) +K

zAi

∑〈ij〉

Aj cos(ϕj − ϕi) +ξϕAi

Ai = −γ(Ai − A)− K

z

∑〈ij〉

Aj sin(ϕj − ϕi) + ξA.(S.8)

The steady amplitude A and the amplitude decay rateγ are determined by the optical field: γ(A − A) =(Γ+Γopt(A)

)A/2, where Γopt = −4g0〈|α|2 sinϕ〉T , where

〈...〉T denotes the average over one mechanical period.The mechanical oscillation frequency is modified via anamplitude dependent optical spring effect: Ω(A) = Ω −2g0〈|α|2 cosϕ〉T /A. The fluctuating noise forces ξϕ and

ξA comprise the effects of the phonon bath and the cavityshot noise, see [32] and below.

For weak coupling K/z Ω the fluctuations of themechanical amplitudes around the steady state value Aare given by [1]

δAi(t) ≈ −KA

zγ

∑〈ij〉

sin(ϕj(t)− ϕi(t)). (S.9)

These beat modes introduce an effective second ordercoupling between phases at different sites, as can be seenafter plugging Eq. (S.9) into the Hopf equation for ϕi(S.8) and performing the time averages 〈...〉T :

ϕi = −Ω(A) +K

z

∑〈ij〉

cos(ϕj − ϕi) +K dΩ

z γ

∑〈ij〉

sin(ϕj − ϕi)

+K2

2z2γ

∑〈ij〉

∑〈jk〉

(sin(2ϕj − ϕk − ϕi)− sin(ϕk − ϕi)

)+K2

2z2γ

∑〈ij〉

∑〈ik〉

sin(ϕj + ϕk − 2ϕi) + ξϕ, (S.10)

where we introduced dΩ = A dΩdA |A=A. This equation is

equivalent to the Kuramoto model and the xy-model with

additional terms that mainly shift the frequency (the cos-term) and indicate higher-order coupling (the contribu-tions of the double sums).

Ultimately, we apply a mean-field approximation: Wereplace eiϕj for neighboring cells by 〈eiϕj 〉 ≡ ReiΨ andei2ϕj by 〈ei2ϕj 〉 ≡ R2e

iΨ2 , where 〈...〉 denotes the aver-age over all sites [3], and arrive at the effective phaseequation, Eq. (S.11), with additional second order con-tributions:

ϕ = −Ω(A)+KR cos(Ψ−ϕ)+(KdΩ/γ−K2)R sin(Ψ−ϕ)

+K2R2 sin(2Ψ− 2ϕ) +K2RR2 sin(Ψ2 −Ψ− ϕ) + ξϕ.

(S.11)

PHASE DIFFUSION

Here, we list some more details of the phase diffusionin the system based on the analysis given by Rodriguesand Armour [2] for a single optomechanical cell. Thediffusion constant associated with ξϕ (see Eq. S.11) isgiven by

Dϕ =1

A2

(Dϕ +

δΩ2

γ2DA

), (S.12)

where Dϕ and DA correspond to the noise acting onphase and amplitude in Eqs. (S.8), and where the diffu-sion rates are defined as

2Dϕ,A =

∫dτ〈ξϕ,A(t+ τ)ξϕ,A(t)〉T (S.13)

and likewise for Dϕ. For the explicit expressions we referto [2]. In the limit of g0A Ω and for Ω κ and∆ = Ω, the maximum diffusion rates are given by

2Dϕ,A ≈ Γ + Γopt, (S.14)

i.e. the sum of the intrinsic mechanical damping Γ andthe optomechanical damping rate at the mechanical side-band

Γopt ≈ 4g20 |α|2/κ. (S.15)

The diffusion of the mechanical phase can also be stud-ied using the full quantum simulations and evaluating thelinewidth of the correlator 〈b(t)b†(0)〉 ∼ e−(iΩeff+Dϕ)t, seeFig. S5(b). Close to the onset, for small amplitudes A

7

-1 0 10

0.1

0.2-1 0 1

123

a

bphase di!usion

detuning

phonon number

mechanical coupling0.2 0.4

1

0

2

3

4

optic

al c

oupl

ing

0

0.5

1.0

0

c mechanical coherence

Figure S5. Additional details for the quantum dynamics of theoptomechanical array within the mean-field approximation:(a) The phonon number 〈b†b〉 shows, as a function of detuning,maxima at the resonance and at the sideband (∆ ≈ Ω −g20/Ω). (b) The diffusion constant for the mechanical phase,Dϕ, for an uncoupled (K = 0, solid line) and a coupled array(K = 0.1 Ω, dash-dotted line). Other parameters as in Fig. 3.(c) Interplay of mechanical and optical coupling: Mechanicalcoherence is observed as a function of both mechanical andoptical coupling (here for fixed laser detuning ∆ + J = Ω/2).

and weak amplitude damping γ, the mechanical phasesare very susceptible to quantum noise preventing syn-chronization. For finite coupling strengths, the diffusionis also enhanced, most strikingly at the mechanical side-band. As a result, the synchronization threshold showsa minimum between the onset of self-oscillations and thesideband, as observed in Fig. 3(a). From extended simu-lations, we find that this behavior is generic for systemsin the resolved sideband regime (Ω > κ).

STABILITY ANALYSIS

Here, we briefly recall details of the stability analysisleading to Eq. (7), which, for the case of the Kuramotomodel, has been given in [4, 5]. We consider the density of

the mechanical phases, %(ϕ). It is normalized,∫ 2π

0%(ϕ) =

1, and the order parameterR and the mean-field Ψ can be

computed from ReiΨ =∫ 2π

0eiϕ%(ϕ)dϕ (and likewise for

R2 and Ψ2). The Fokker-Planck equation correspondingto Eq. (S.11) is given by

∂t%+ ∂ϕ(%v)

= Dϕ∂2ϕ% (S.16)

with a velocity

v = −Ω(A) +K cos(Ψ− ϕ) +K1R sin(Ψ− ϕ) +O(R2),(S.17)

where second order contributions can be neglected forthis linear analysis. In the unsynchronized regime, themechanical phases are equally distributed over the inter-val [0, 2π], and % = (2π)−1. To study the time evolu-tion of a small fluctuation on top of the incoherent back-ground, we employ the ansatz [5]:

%(ϕ) =1

2π+ c(t)eiϕ + c∗(t)e−iϕ, (S.18)

leading to

c = −(i(Ω− K

2

)+(Dϕ −

K1

2

))c. (S.19)

This equation reveals that the incoherent solution % =(2π)−1 becomes unstable for

K1=2Dϕ. (S.20)

and thus defines the coupling threshold for the synchro-nization transition [5].

EXPERIMENTAL IMPLEMENTATION

(a) Competing setups - Here, we provide some detailson possible experimental implementations suitable forobserving the mechanical transition:

• Microdisks [6–8] and microtoroids [9–11] fabricatedon microchips: Strong optical coupling betweenresonators is feasible via evanescent fields [8]. Forboth types of setups, scalability still has to beshown.

• Micro- and nanomechanical beams [12, 13] or mem-branes [14] coupling to superconducting microwavecavities: Mechanical interaction may be achievedvia a common support or, capacitively, by applyinga voltage bias between the mechanical resonators.Two-dimensional arrays of coupled microwave cavi-ties are starting to be developed [15]. Related elec-tromechanical systems (see, e.g., [16] for a setupcomprising a nanobeam coupling to a supercon-ducting single electron transistor) may also be em-ployed.

• Optomechanical crystals [17–19] feature smallmode volumes and are thus very suitable for ex-tensions to optomechanical arrays. Some details ofthe proposed implementation are given below (Fig.S6).

(b) Required parameters: According to our semiclassicalanalysis, the essential requirement is

KA2 & Γopt > Γnth. (S.21)

In this case, quantum noise (Γopt) dominates over ther-mal fluctuations (Γnth), which enter the model by replac-ing Γ → Γnth ≈ kBT/Q in Eqs. (5). The mechanicaltransition can then be studied by varying Γopt via thelaser detuning ∆.

Recent experiments have demonstrated Γopt > Γnth,see Table I. We note that two-dimensional optomechan-ical crystal devices are expected to show very good op-tomechanical properties [18], even exceeding those of ex-isting one-dimensional setups [17].

8

1 2 3 4

9.10

9.15m

echa

nica

l fre

quen

cya

b

0

-1

1op

tical

freq

uenc

y

0

-1

1

1 2 3 4 5 6196

200

204 defect spacing

defect spacing

Figure S6. Variation of coupling strength with separationbetween defect cavities: (a) mechanical and (b) optical fre-quencies of symmetric (cross) and antisymmetric (diamond)normal modes for two defects on a snowflake optomechani-cal crystal (as discussed in the main text). The insets showthe displacement field (electrical field) components of the an-tisymmetric mode for a defect separation of 2. The barefrequencies of the localized eigenmodes are 9.12 GHz and200.5 THz, respectively. We note that the optical splittings(and likewise J) change sign when varying the defect separa-tion. Results obtained from finite element simulations.

(c) Photon and phonon hopping amplitudes: Using fi-nite element simulations, we studied the hybridizationof the photon and phonon modes of two defect cavitiesinside a two-dimensional snowflake silicon optomechan-ical crystal, to obtain the hopping constants as a func-tion of defect separation. Parameters: relative permit-tivity εr = 11.68, Poisson ratio 0.17, density 2329 kg/m3,Young’s modulus 170 GPa, snowflake design [20] withlattice constant 500 nm, snowflake radius 168 nm andsnowflake width 60 nm. The mechanical and optical split-tings (i.e. couplings 2K/z and 2J/z, respectively) reachvalues of up to 8 % and 4 % of the mechanical and opticaleigenfrequencies, respectively (Fig. S6). These values arecompatible with the requirements given by Eq. (S.21),even for very small oscillation amplitudes of the order ofthe mechanical zero-point width, A ≈ 1.

Other experimental approaches may realize differentcoupling terms, e.g. (xi−xj)2 [21]. Additional fast rotat-

ing terms like bibj are, however, negligible for K/z Ω.

We note that the transition from incoherent to syn-chronized dynamics is also observable for extended opti-cal modes (J Ω, κ), see Fig. S5(c).

(d) Optical drive: In principle, the methods used fordriving single defect modes via tapered and dimpledfibers [22] and optical waveguides [23, 24] can be extendedto larger scales. To limit experimental efforts, however,one may drive the array using a freestanding broad laserbeam, see the schematic picture in Fig. 1. The coupling

between laser mode and the in-plane cavity modes willbe relatively weak, which, however, can be compensatedvia the laser power. Preliminary experimental results onfree space coupling to optomechanical crystals have beenreported in [25].

(e) Detection: To detect the mechanical transition,measurements of the optical field emitted from the arrayare sufficient, see Fig. 3(d). To determine correlationsbetween two separate lattice sites, the intensity emanat-ing from these defect cavities may be probed by two ta-pered fibers in the near-field of the selected cells [22] or byespecially designed waveguides [23, 24]. The correlationsbetween the mechanical sideband components of the op-

tical intensities Ii(t) =∫ t+2π/Ωeff

teiΩeff t

′ |αi|2(t′)dt′, arethen proportional to the mechanical correlations,

〈IiI?j 〉 ∝ 〈eiϕie−iϕj 〉. (S.22)

NUMERICAL METHODS

To study the quantum dynamics of the system, we nu-merically integrate the Lindblad master equation (seemain text) to the steady state (independent of initialconditions) using a fourth-order Runge-Kutta algorithm.The size of the Hilbert space is optimized to enable an ad-equate representation of the physical state while obtain-ing reasonable simulation times. Typically, 10− 20 pho-ton and phonon levels, respectively, were taken into ac-count. When slowly sweeping through parameter space,bistable behavior is revealed. This can be seen most

0.1 0.110

0.5

1

0.09 0.1 0.110

0.5

1

mechanical coupling

a

corr

elat

ions

mechanical coupling

corr

elat

ions

b

open boundary

periodic

1

0

0.5

Figure S7. Langevin dynamics on finite square lattices, con-firming results shown in Fig. 4: (a) Correlations C(d) as afunction of coupling strength for different lattice sizes N ×Nwith N = 5, 10, 20, 30 (d = 2, 4, 9, 14, periodic boundaryconditions). (b) Correlations C(d = 14) on a 30 × 30 squarelattice with periodic boundary conditions (empty squares)and open boundary conditions (filled squares). The insetshows the correlations |〈eiϕie−iϕj 〉| for i = (8, 15) (square)and K = 0.115 Ω and open boundary conditions. The crossmarks the lattice site j = (22, 15) for which the correlationsare shown in the main plot. Other parameters as in Fig. 4.

9

prominently in Fig. S5(c) from the cut line in the rightpart of the plot.

The numerical integration of the Langevin equations(5) was obtained using a fourth order Runge-Kuttamethod for stochastic differential equations [26]. Whenevaluating the dependence of the threshold value on thequantum parameter g0/κ (Fig. 4), the value of g0αL washeld constant. In this case, only the strength of quan-tum fluctuations changes (keeping the classcial solutionconstant). For very large values of g0/κ, fluctuations areoverestimated by the Langevin equations and one has torely on the exact quantum simulations.

The finite element simulations of Figs.1 and S6 wereperformed using COMSOL Multiphysics. We studied ahexagonal silicon slab with sides of length 5.6µm andrestricted our simulations to two space dimensions, i.e.in-plane elastic deformations and electromagnetic waves,see also [20]. Extended simulations in three dimensionshave to be employed to analyze the coupling of the cavitymodes to the out-of plane-modes.

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