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The Quantum Kuramoto Model

Ignacio Hermoso de Mendoza,1 Leonardo A. Pachon,2 Jesus Gomez-Gardenes,3, 1 and David Zueco1, 4, 5

1Departamento de Fısica de la Materia Condensada,Universidad de Zaragoza, E-50009 Zaragoza, Spain.

2Grupo de Fısica Atomica y Molecular, Instituto de Fısica, Facultad de Ciencias Exactas y Naturales,Universidad de Antioquia UdeA; Calle 70 No. 52-21, Medellın, Colombia

3Instituto de Biocomputacion y Fısica de Sistemas Complejos,Universidad de Zaragoza, E-50018 Zaragoza, Spain.

4Instituto de Ciencia de Materiales de Aragon (ICMA),CSIC-Universidad de Zaragoza, E-50012 Zaragoza, Spain.

5Fundacion ARAID, Paseo Marıa Agustın 36, E-50004 Zaragoza, Spain.(Dated: September 17, 2013)

Synchronization is an ubiquitous phenomenon occurring in social, biological and technological sys-tems when the internal rhythms of a large number of units evolve coupled. This natural tendencytowards dynamical consensus has spurred a large body of theoretical and experimental research dur-ing the last decades. The Kuramoto model constitutes the most studied and paradigmatic frameworkto study synchronization. In particular, it shows how synchronization shows up as a phase transitionfrom a dynamically disordered state at some critical value for the coupling strength between theinteracting units. The critical properties of the synchronization transition of this model have beenwidely studied and many variants of its formulations has been considered to address different physi-cal realizations. However, the Kuramoto model has been only studied within the domain of classicaldynamics, thus neglecting its applications for the study of quantum synchronization phenomena.Here we provide with the quantization of the Kuramoto model. Based on a system-bath approachand within the Feynman path-integral formalism, we derive the equations for the Kuramoto modelby taking into account the first quantum fluctuations. We also analyze its critical properties beingthe main result the derivation of the value for the synchronization onset. This critical coupling turnsup to increase its value as quantumness increases, as a consequence of the possibility of tunnellingthat quantum fluctuations provide.

I. INTRODUCTION

Synchronization is perhaps the most cross-disciplinaryconcept of emergence of collective behavior [1] as it ismanifested across many branches of natural and socialsciences. Ensembles of neurons, fireflies or humans areprone to synchronize their internal rhythms when theybecome coupled enough, producing a macroscopic dy-namically coherent state. In all these seemingly unrelatedsituations, no matter the precise nature of the coupledunits, interaction drives system’s components to behavehomogeneously. Thus, the study about the microscopicrules that drive ensembles towards synchrony has a longand fruitful history since the seminal observations madeby Christiaan Huygens [2–4].The mathematical formulation of the first models

showing synchronization phenomena dates back to the70’s when, after some preliminary works by Peskin andWinfree [11], Kuramoto [5] formalized his celebratedmodel. The Kuramoto model incorporates the minimumdynamical ingredients aimed at capturing a variety ofphysical phenomena related with the onset of synchro-nization. In particular, the Kuramoto model links phys-ical concepts such as self-organization, emergence, or-der in time and phase transitions, thus revealing as themost paradigmatic framework to study synchronization[1, 7, 11].Despite the large body of literature devoted to the Ku-

ramoto model and its variants, its study has always been

restricted to the classical domain. At first sight, given theusual nature (scale) of the systems in which synchroniza-tion is typically observed, it seems superflous thinking ofa quantum theory for the Kuramoto model. However,there is not doubt about the fundamental importanceof studying quantum fluctuations within the emergenceof synchronized states [9–12]. Moreover, the Kuramotomodel has been implemented on circuits and micro andnanomechanical structures [13, 14], systems which havealready met the quantum domain [15, 16]. At the quan-tum level, synchronization, understood as the emergenceof a coherent behaviour from an incoherent situation inthe absence of external fields, is reminiscent of the phe-nomena such as condensation of Bose-Einsten and hasbeen observed in interacting condensates of quasiparti-cles [17, 18]. Thus, moved by its fundamental and appliedimportance, in this work we provide the quantum versionof the Kuramoto model in an attempt for understandingthe influence that quantumness has on the emergence ofsynchronized states.

Our work in this paper consists, as stated by Caldeiraand Leggett [19], on finding consistent equations that inthe classical limit matches the Kuramoto model. Ourderivation relies on the quantization of open systemsin the framework of Feynman’s path-integral formalism.Being the classical limit well defined, i.e. the Kuramotomodel, and once we have derived its quantum counterpartwe analyze its critical properties by deriving the criticalpoint from which synchronization shows up and deter-

2

mine how quantum fluctuations affect this synchroniza-tion transition. As a semiclassical approach the quan-tum Kuramoto model derived takes into account the firstquantum fluctuations, and its results represent the firststep towards the study of synchronization in quantumextended systems.

II. THE CLASSICAL KURAMOTO MODEL.

The original Kuramoto model [5] considers a collectionof N phase-oscillators, i.e., it assumes that the charac-teristic time scale of their amplitudes is much faster thanthat for the phases. Thus, the dynamical state of thei-th unit is described by an angular variable θi ∈ (0, 2π]whose time evolution is given by:

θi = ωi +K

N

N∑

j=1

sin(θi − θj) . (1)

The above equation thus describes a set of weakly cou-pled phase-oscillators whose internal (natural) frequen-cies ωi are, in principle, different as they are assignedfollowing a frequency distribution g(ω) that it is assumedto be uni-modal and even around the mean frequency Ωof the population, g(Ω + ω) = g(Ω− ω).In the uncoupled limit (K = 0) each element i de-

scribes limit-cycle oscillations with characteristic fre-quency ωi. Kuramoto showed that, by increasing the cou-plingK the system experiences a transition towards com-plete synchronization, i.e., a dynamical state in whichθi(t) = θj(t) ∀i, j and ∀t. This transition shows up whenthe coupling strength exceeds a critical value whose exactvalue is:

Kc =2

πg(Ω). (2)

To monitor the transition towards synchronization,Kuramoto introduce a complex order parameter:

r(t)eiΨ(t) =1

N

N∑

j=1

eiθj(t) . (3)

The modulus of the above order parameter, r(t) ∈ [0, 1],measures the coherence of the collective motion, reachingthe value r = 1 when the system is fully synchronized,while r = 0 for the incoherent solution. On the otherhand, the value of Ψ(t) accounts for the average phase ofthe collective dynamics of the system.In Figure 2 we have illustrated the synchronization in

the Kuramoto model. The panels in the top show, fordifferent values of the coupling K, how the oscillatorsconcentrate as K increases. Below we have shown theusual synchronization diagram r(K) for which the exactvalue of r for eachK is the result of a time average of r(t)over a large enough time window. In this diagram we canobserve that Kc = 1 as a result of using the distributiong(ω) shown in the right.

Let us note that the all-to-all coupling considered orig-inally by Kuramoto can be trivially generalized to anyconnectivity structure by introducing the coupling ma-trix Aij inside the sum in (1) so that each term j ac-counting for the interaction between oscillator i and jis assigned a different weight. The latter allows for thestudy of the synchronization properties of a variety ofreal-world systems for which interactions between con-stituents are better described as a complex network [20].The formalism developed in this work is fully general andvalid for any form of Kij thus making possible the exten-sion of the large number of studies about the Kuramotomodel in any topology [21] to the quantum domain. How-ever, the numerical part of our work will deal with theall-to-all coupling for the sake of comparison with theoriginal Kuramoto work.

III. QUANTIZATION OF THE KURAMOTO

MODEL.

The most important problem when facing the quan-tization of the Kuramoto model is its non-Hamiltoniancharacter since, as introduced above, equation (1) as-sumes the steady-state for the dynamical state of theamplitude of the oscillators. Thus, a question arises, howdo we introduce quantum fluctuations in the Kuramotomodel? One possible choice is to resort to the originalmicroscopic dynamics of amplitude and phases and thenidentify the underlying Hamiltonian dynamics. However,many different dynamical setups can have the Kuramotomodel as their corresponding limiting case of fast am-plitude dynamics. Thus, in order to keep the flavor ofgenerality of the Kuramoto model, it is desirable not toresort to any specific situation (Hamiltonian) and intro-duce quantum fluctuations directly.A similar problem was faced by Caldeira and Leggett

in the eighties [19] when they studied the influence ofdissipation in quantum tunneling. In their case, the cor-responding classical dynamics dates back to the stud-ies on activation theory by Kramers [22]. Classically, aparticle in a potential experience an energy barrier tosurmount, that is typically acquired from thermal fluctu-ations. On the other hand, a quantum particle finds intunnelling an alternative way to bypass an energy barrier.Caldeira and Leggett were thus interested in quantifyingthe catalytic effect of tunnelling in (effectively) lower-ing the energy barriers. However, as in the Kuramotomodel, Kramers activation theory is based in Langevinequations, i.e. stochastic equations that are not directlyobtained from any Lagrangian. Furthermore, most of re-action rate equations were phenomenological. Therefore,they searched for a consistent way for introducing quan-tum fluctuations regardless of the microscopic origin ofthe effective classical evolution. As a byproduct theirwork opened the field of quantum Brownian motion inthe most general way.We take here the same route followed by Caldeira and

3

FIG. 1. Synchronization in the classical Kuramoto model. Each panel on the top shows the collection of oscillators

situated in the unit circle (when each oscillator j is represented as eiθj (t)). The color of each oscillator represents its naturalfrequency. From left to right we observe how oscillators start to concentrate as the coupling K increases. In the panels belowwe show the synchronization diagram, i.e., the Kuramoto order parameter r as a function of K. It is clear that Kc = 1 asobtained by using the distribution g(ω) shown in the right panel.

Leggett to introduce quantum fluctuations in the Ku-ramoto model. In order to accomodate our dynamicalsystem (1) to the framework provided in [19] we start bywritting its corresponding Langevin equation:

θi = −∂V

∂θi+ ξi , (4)

with

V (θ1, ..., θN ) ≡ −∑

i

ωiθi +K

N

∑

i,j

cos(θi − θj) . (5)

As usual, ξi is a Markovian stochastic fluctuating forcewith 〈ξi(t)〉 = 0 and 〈ξi(t)ξj(t

′)〉 = 2δijDδ(t − t′). Inthe limit D → 0, equation (4) reduces to the Kuramotomodel in equation (1).Equation (4) is nothing but a Langevin equation in the

overdamped limit. It is first rather than second order intime as the inertia term is neglected. Consequently, theKuramoto model can be viewed as a set of phases evolv-ing in the overdamped limit. The absence of fluctuationsin the limit D → 0 means that the system of phases isat zero temperature, D ∼ T . Such identification with aLangevin equation has been already used for generaliza-tions of the original Kuramoto model taking into accountnoise and/or inertial effects [1]. In particular, in [10] itis shown that the critical value Kc reads:

Kc =2

∫∞

−∞D

D2+ω2 g(ω) dω, (6)

which, in the limit D → 0, recovers the Kuramoto criticalcoupling (2).The key point of deriving the Langevin equation (4)

corresponding to the Kuramoto model is that it can beobtained from a fully Hamiltonian framework by cou-pling the system, in our case the coupled phases θi, toa macroscopic bath or reservoir [22]. In this way, boththe damping and fluctuations are seen to be caused bythe coupling of the system of phases to the bath. TheHamiltonian description properly casted in the system-bath approach reads as follows:

Htot = Hsys +Hbath +Hint , (7)

where the bath is an infinite collection of harmonic oscil-lators with frequencies ωα (note that greek subindexeswill denote the oscillators in the bath). In the case weare dealing with the total Hamiltonian reads:

Htot =∑

i

π2i

2+V (θ1, ..., θN )+

1

2

∑

i,α

P 2i,α+ω

2α(Qi,α−λαθi)

2 ,

(8)where (θi, πi) and (Qi,α, Pi,α) denote the system andbath canonical coordinates, respectively, while λα standsfor the coupling constant between bath and system coor-dinates.Under well defined conditions, the equations of motion

for the system coordinates derived from the Hamiltonian(8) lead to the the afore-derived overdamped Langevin

4

equation (4). In particular, one needs to assume: (i)thermalized initial conditions for the bath:

〈Qi,αQi′,α′〉 = δi,i′δα,α′ kBT/ω2α , (9)

〈Pi,αPi′,α′〉 = δi,i′δα,α′ kBT , (10)

(ii) the frequency spectrum of the bath oscillators is flat(this assumption leads to the widely used Ohmic dissipa-tion), and finally (iii) the changes in time of the velocity(acceleration) induced by the energy potentials are farslower than the energy loss induced by the coupling be-tween the system and the bath (this is the situation whenthe system and the bath are strongly coupled) so that wecould neglect the inertial term.Once we have a Hamiltonian description for the Ku-

ramoto equation (1), we are ready to perform its quan-tization. First, we associate the phases and their associ-ated momenta together with the positions and momentafor the bath by providing them with the canonical com-mutation rules. The hardest work is to find an effectivequantum evolution depending only on phase operators,i.e. the so-called quantum Langevin equation. It turnsout that such operators equation is a non-local differen-tial equation, which makes it extremely difficult to ma-nipulate in general. However, the quantum version ofequation (4) in the overdamped limit is a c-number lo-cal differential equation [3, 5, 23, 26–28]. The resultingquantum evolution reads as follows:

θi = −V ′i

Fi+

Λ

Fi

∑

j

(

βV ′j V

′′ij − V ′′′

jji

)

−Λ

2FiV ′′′iii +

√

1

Fi· ξi ,

(11)where we have used the compact notation V ′...′

i,...,k ≡∂θi,...,θkV

Fi = e−ΛD∂2V/∂θ2i (12)

while Λ is a parameter tuning the degree of quantumnessand ξi is an stochastic force with the same statistics as in(4). Let us note that the quantum model is perturbativein Λ, which has a clear physical meaning. It stands forthe quantum corrections to the thermal averages for thevariance of the phases θi. In the Supplementary Material[29] we provide the details of the derivation of the model.

IV. THE TRANSITION TO

SYNCHRONIZATION IN THE QUANTUM

MODEL.

Once we derived the quantum version of the Kuramotoequation, it is natural to unveil the effects that quantumfluctuations induce in the transition to synchronization.As introduced previously, to study the synchronizationtransition one resorts to the order parameter r [intro-duced in equation (F33)] that reveals the synchronizedstate of the system. We solve both the classical Ku-ramoto model (Λ = 0) and the quantum one (Λ > 0)

FIG. 2. Classical vs. Quantum synchronization tran-

sitions. Panel (a) shows the synchronization diagrams r(K)for the classical (Λ = 0) and the quantum (Λ = 0.1) Ku-ramoto models. In both cases the thermal noise is chosensuch that D = 1. The number of oscillators is N = 103

and the distribution of natural frequencies is Lorentzian:g(ω;ω0, α) = π−1α/[(ω − ω0)

2 + α2] centered in ω0 = 0 andα = 0.5. It is clear that the synchronization onset is delayedas soon as quantumness enters into play. In panels (b) and(c) we show the probability P (θ) of finding an oscillator at agiven phase θ as a function of K. Note that for each valueof K, the phases has been equally shifted so that the meanphase is located at θ = π. A thick grey line indicates the crit-ical values Kc and Kq

c for classical and quantum dynamics,respectively.

5

FIG. 3. System of two coupled Kuramoto oscillators. The top-left part of the figure shows the analogy between thesystem of two coupled oscillators and an overdamped particle in a washboard potential. The two possible regimes are shown:synchronized state (the particle is at rest ϕ = 0 at a local minimum) and unsynchronized phase (the particle drifts across thepotential). Below we illustrate the possibility that tunneling provides to anticipate the drifting state. On the right part weshow the result of the computation of the velocity ϕ as a function of ∆ω/K for the classical (red line) and quantum (blue line)systems.

numerically, extracting from the dynamics the station-ary value of r. Through this work, the numerical calcu-lations are performed with N = 103 oscillators and thedistribution of natural frequencies is Lorentzian:

g(ω;ω0, α) =1

π

α

(ω − ω0)2 + α2, (13)

with α = 0.5 and centered around ω0 = 0.

Figure 2.a shows the typical synchronization diagram,namely, the value of r as a function of the couplingstrength K. The comparison of the quantum (for Λ =0.1) and classical curves r(K) evinces that quantum fluc-tuations delay the onset of synchronization, i.e., the crit-ical point Kc is seen to move to larger values with Λ. Wehave also considered the evolution for the distribution ofthe phases as a function of K to monitor the microscopicfingerprint of the synchronization transition. These evo-lutions for the classical and quantum Kuramoto modelsare shown in figures 2.b and 2.c, respectively.

To explain the delay in the synchronization onset in-troduced by quantum fluctuations we resort to the sim-plest situation: two coupled Kuramoto oscillators. Inthis case the Kuramoto model (4) consists on just twocoupled equations for the evolution of θ1 and θ2. By tak-ing the difference of those two equations and introducingas a new variable the phase difference, ϕ := θ1 − θ2, we

obtain for its evolution the following equation:

ϕ = ∆ω −K sinϕ+ 2Dξ (14)

The latter equation describes the evolution of an over-damped particle in a washboard potential (see Figure 3).With this image in mind, we map the synchronous move-ment of the two oscillators (defined as a state in which

the frequencies of the oscillators are locked: θ1 = θ2)with the resting state of the overdamped particle insidea local minimum of the potential energy (ϕ = 0). Onthe other hand, when the two oscillators are not syn-chronized the particle drifts across the potential (ϕ 6= 0).Both situations are shown in Fig. 3.The quantum version for the diffusion of an over-

damped particle in a periodic potential has been pre-viously studied in Ref. [27]. The main result is that thescape rate of the particle, and thus its unlocking mech-anism, is enhanced through quantum fluctuations. Thiseffect can be seen as a consequence of the enhancementof the transition probability for energies below the heightof the barrier which is nothing but the well-known tunneleffect [30]. In Fig. 3 we show, for both the classical andquantum (Λ = 0.1) systems of two coupled Kuramotooscillators, the value of ϕ = 0 as a functio of the ratiobetween the difference of the natural frequencies of thetwo oscillators |∆ω| and the coupling K. It is clear that,as stated above, quantum tunnelling facilitates the drift

6

or, equivalently, delays the transition to the synchronousstate.

V. ANALYTICAL EXPRESSION FOR THE

SYNCHRONIZATION ONSET

Coming back to the original model of N interactingoscillators, we now make an analytical estimation of thevalue for critical coupling at which the synchronizationtransition occurs. The procedure is a generalization ofthe one presented in Ref. [10] and takes advantage ofthe mean field description of the Kuramoto model. Thederivation ( presented in the Supplementary Material[29]) yields a rather simple equation for the critical cou-pling:

Kqc = (1 + Λ)Kc , (15)

being Kc the classical critical value shown in Eq. (6).

The above result tells that quantum fluctuations actby effectively decreasing the coupling strength with thedegree of quantumness Λ. Coming back to the physicalimage of a particle in a washboard potential, we can con-sider the effect of the quantum correction by consideringthe first and third terms in the right hand side of equa-tion (F33). In this way, the quantum corrections can becasted in the form of an effective potential:

Veff = V + ΛV ′′, (16)

that in the particular case of the washboard potentialreads:

Veff = −∆ω ϕ− (K − Λ) cosϕ . (17)

The above equation makes clear that tunnelling is for-mally reflected by an effective barrier reduction thatyields the observed shift to higher values for the criti-cal coupling.

Our analytical estimation forKqc (Λ) is plotted in figure

2 (vertical arrows) confirming its validity. To corroboratefurther the correctness of equation (15), we explore thesynchronization transition for different values of Λ in fig-ure 4.a. As expected the onset of synchronization shiftsto higher values as the degree of quantumness increases.Again, the predicted value for Kq

c is plotted (vertical ar-rows) corroborating the validity of equation (15).

To complete our study, we show in 4.b the dependenceof the synchronization diagram with the thermal fluctu-ations, D, both for the classical and quantum (Λ = 0.1)cases. In all the curves explored the coupling K isrescaled by the corresponding critical coupling Kc in theclassical regime. In this way we show both for the clas-sical and quantum cases, the robustness of the criticalvalue (15) against temperature changes.

VI. DISCUSSION

The search for quantum corrections to classical phe-nomena has been pervasive in physics. Some examplesrelated to our work are the generalization to the quan-tum domain of chaos [32], dissipation [33], random walks[34], etc. Each of these examples finds its own difficul-ties when incorporating quantum fluctuations and unveil-ing their role. Some of these obstacles are the quantumlinearity versus the typical non-linearity of classical sys-tems and the quantization of non-Hamiltonian system orphenomenological equations. Overcoming these obsta-cles provides with a consistent quantum description thatopens the quantum door to a variety of classical problemsand their associated physical phenomena.Among the most studied phenomena in (classical) com-

plex systems is synchronization. This emergent phenom-ena is as intriguing as beautiful, since it covers from thedescription of the sympathy of clocks to the neuronalfunctioning in our brain, thus overcoming the disparatelydiversity in the spatial and time scales associated to thebunch of systems in which synchronization is observed.However, the concept of synchronization was usually as-sociated to the classical domain as the typical examples ofclocks, fireflies or humans are too macroscopic to thinkabout the need of introducing quantum fluctuations inthe description of the associated dynamical models.Recently, some experimental works have shown that

synchronization can be observed in the lab withinJosepshon Junction arrays [35], nanomechanical [13] oroptomechanical systems [36]. All of these systems shareone prominent property: they behave quantum mechan-ically at sufficiently low temperatures. Therefore, adapt-ing the concept of synchronization among coupled en-tities within the quantum theory is, apart from an in-teresting theoretical issue, a must imposed by the rapidexperimental advances.A first step consists in taking the most widely used

framework for studying synchronization phenomena, theKuramoto model, and adapting it to the quatum domain.Being a paradigmatic theoretical setup, the quantizationof the Kuramoto model opens the door to the theoreti-cal study of quantum synchronization in the widest pos-sible manner. To this end, and to overcome the non-Hamiltonian character of the Kuramoto equations, wehave mapped the model to an overdamped Langevinequation which has a Hamiltonian description by embed-ding the system in a bath of oscillators. In this way, thequantization of the Kuramoto model is straightforwardand it includes its classical counterpart as a limiting case:the quantum version incorporates quantum fluctuationsfor the phases while the strength of these quantum cor-rections are encoded in a single parameter.The route chosen here must be understood as com-

plementary to the study of particular models of coupledquantum systems. The reason is twofold. First, we aimto be as general as possible. The essence of an emer-gent phenomena is its ability of describe very different

7

FIG. 4. Analysis of the behavior of the critical coupling. In panel (a) we show the synchronization diagrams r(K)for different values of the degree of quantumness Λ. Note that the coupling strength has been normalized to its value in theclassical (Λ = 0) limit. The analytical estimation in equation (15) is shown by the vertical lines confirming its validity. In panel(b) we plot r(K) for two different temperatures, corresponding to D = 1 and D = 0.2, for the classical and quantum (Λ = 0.1)models. Again the analytical estimation is shown by the vertical lines.

situations with different microscopic dynamics. This isthe goal of the Kuramoto model, as it explains the syn-chronization without resorting to the specific dynamics.Second, a force brute study of many body quantum enti-ties is a very difficult task that usually implies the reduc-tion of the system to a few coupled systems. However,the observation of a true synchronization transition de-mands hundreds or thousands of interacting dynamicalsystems.

Being general, the results obtained allow to make gen-eral statements about the impact that quantumness hason the synchronization of coupled dynamical units. Themost important one is that quantum fluctuations delaythe appearance of a synchronized state. The explana-tion of this effect relies on the fact that in the quan-tum domain the phases not only have a different natu-ral frequency but also the fluctuations around the classi-cal trajectories are different depending on those internalrhythms. To illustrate this interpretation we recall thesimple case of two coupled Kuramoto oscillators. In thiscase quantum fluctuations are nothing but thermal as-sisted tunneling favoring the phase unlocking. Therefore,the coupling needed to synchronize the two oscillators ishigher in the quantum limit.

Finally, we want to point out that in a recent pub-lication the question about synchronization in quantum

evolutions was also discussed [9]. Under rather generalconditions they find bounds for the degree of synchro-nization based on the Heisenberg uncertainty principle:the phases, derived as averages of non-conmuting opera-tors, cannot take values infinitely close. Instead, in ourcase, focused on the quantum version of the Kuramotomodel, we have discussed, not the maximum degree ofsynchronization but the critical onset for the appearanceof partially synchronized states. In this case quantum-ness also limits the emergence of a synchronous state.Therefore, pretty much like in what happens in quan-tum chaos, synchronization seems to be a quasi-classicalphenomena [11].

ACKNOWLEDGMENTS

This work has been partially supported by the SpanishMINECO under projects FIS2011-14539-E (EXPLORAprogram) FIS2011-25167 and FIS2012-38266-C02-01, bythe Comunidad de Aragon (Grupo FENOL). J.G.G. issupported by MINECO through the Ramon y Cajal pro-gram. LAP was supported by CODI of Universidadde Antioquia under contract number E01651 and un-der the Estrategia de Sostenibilidad 2013-2014 and byCOLCIENCIAS of Colombia under the grant number111556934912.

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[24] Ankerhold, J., Grabert, H. & Pechukas, P. QuantumBrownian motion with large friction. CHAOS 15, 26106(2005).

[25] Maier, S.A. & Ankerhold, J. Quantum Smoluchowskiequation: A systematic study. Physical Review E 81,086802 (2010).

[26] Machura, L., Kostur, M., Hnggi, P., Talkner, P. &Luczka, J. Consistent description of quantum Brownianmotors operating at strong friction. Physical Review E70 031107 (2004).

[27] Luczka, J., Rudnicki, R. & Hanggi, P. The diffusion in thequantum Smoluchowski equation. Physica A 351, 60–68(2005).

[28] Machura, L., Kostur, M., Talkner, P., Luczka, J. &Hanggi, P. Quantum diffusion in biased washboard po-tentials: Strong friction limit. Physical Review E 73

031105 (2006).[29] See Supplementary Material at ...[30] Garcia-Palacios, J.L. and Zueco, D. The Caldeira

Leggett quantummaster equation in Wigner phase space:continued-fraction solution and application to Brownianmotion in periodic potentials. J. Phys. A: Math. Gen.37, 10735 (2004).

[31] Sakaguchi H. Cooperative Phenomena in Coupled Oscil-lator Systems under External Fields. Progress of Theo-retical Physics 79 39–46 (1988).

[32] Ott E, Chaos in Dynamical Systems (Cambridge 2nd1994)

[33] Weiss U, Quantum Dissipative Systems (World Scientific2nd 1999)

[34] Sanchez-Burillo E , Duch J , Gomez-Gardenes J , ZuecoD, Quantum Navigation and Ranking in Complex Net-works. Scientific Reports 2, 605 (2012)

[35] Wiesenfeld, K., Colet, P. & Strogatz, S.H. Frequencylocking in Josephson arrays: Connection with the Ku-ramoto model. Phys. Rev. E 57, 1563 (1998).

[36] Heinrich, G., Ludwig, M., Qian, J., Kubala, B. & Mar-quardt, F. Collective dynamics in optomechanical arrays.Phys. Rev. Lett. 107, 043603 (2011).

Appendix A: The system-Bath Approach

Through the work we make use of the system-bath Hamiltonian. Let us introduce the main notation and quantitiesof interest.

In what follows we are interested in thermal environments. In other words, we are imaging a system at some

9

temperature T . On general grounds, the physics can be described by the system-bath Hamiltonian:

Htot =Hsys +Hbath +Hint (A1)

=N∑

i

p2i2m

+ V (x1, ..., xN ) +∞∑

α

N∑

i

P 2α,i

2M+

1

2MΩ2

α,i

(

xi −cα,iMΩ2

i

Qα,i

)2

,

here we have chosen latin indexes to label the number of particles of the system, while greek ones stand for the bathinfinite degrees of freedom. Notice that each particle is coupled to one bath. A key quantity in this formalism is thespectral density:

Ji(Ω) := π∑

α

c2α,i2mΩα,i

δ(Ω− Ωα,i), (A2)

which condenses the relevant information of the bath modes.

Appendix B: The Kuramoto Model as a Langevin Equation

It is shown in many places that by choosing the so-called Ohmic spectral density:

Ji(Ω) = mγΩ, (B1)

the classical description of the dynamics of a system coupled to a thermal reservoir can be formulated by means ofthe Langevin (L) equation

mxi = −mγxi −∂V

∂xi+ ξi(t). (B2)

This effective stochastic formulation is driven by the Markovian zero-mean-fluctuating force ξi(t), which is character-ized by

〈ξi(t)〉 = 0, (B3)

〈ξi(t)ξj(t′)〉 = 2mγKBTδi,j δ(t− t′). (B4)

By defining the characteristic scales for time, position and energy:

τ0 =mγx20∆V

, (B5)

θi =xix0,

∆V = ~ω0,

together with the dimensionless quantities:

τ =t

τ0, (B6)

V =V

∆V,

β = β∆V ,

Λ =Λ

x20:=

1

D,

the Langevin equation (B2) can be rewritten as:

1

γτ0θi + θi = −

∂V

∂θi+ ξ. (B7)

The stochastic noise is now dimensionless with statistics:

〈ξi(τ)ξj(τ′)〉 = 2Dδijδ(τ − τ ′). (B8)

10

In the overdamped limit γτ0 ≫ 1, which means that γ−1 is the fastest time scale, the dimensionless Langevin equationthe above can be approximated by:

θi = −∂V

∂θi+ ξ. (B9)

Finally, by defining the Kuramoto potential [1],

V (θ1, ..., θN ) ≡ −∑

i

ωiθi +∑

i,j

Kij cos(θi − θj), (B10)

it turns out that the corresponding overdamped Langevin dynamics is nothing than the Kuramoto model with drivingnoise. At zero temperature, D → 0 the equation (B9) reduces to the original Kuramoto model.

Appendix C: The Fokker-Planck Equation

In Hamiltonian dynamics, the expectation value of a given observable A can be computed by means of a normalizedprobability density P ,

∫

dxdxP (x, x, t) = 1, as

〈A〉 =

∫

dxdxP (x, x, t)A(x, x, t). (C1)

The dynamics of P are given by the Liouville equation,

∂P

∂t= H,P. (C2)

In the presence of a thermal bath the dynamics is not longer Hamiltonian and the Liouville equation needs to becomplemented. This type of complementation is what takes the Newtonian equation of motion into the Langevinequation [Cf. equation (B2)]. For the case of the Liouville equation this refinement of the dynamics is incorporated inthe so-called Fokker-Planck (FP) equation [2], which in the overdamped regime leads to the Smoluchowski equation(S). In the notation of phases appearing in K -model, the S equation can be written as

∂P (x, t)

∂t=∑

i

∂2

∂x2i[D + V (x)]P (x, t). (C3)

For the case on non-interacting particles, V (x) = 0, the S equation leads to the Diffusion equation. For futureconvenience, we re-write equation (C3) as

∂tP =∑

i

∂xiLclP, (C4)

with

Lcl = D1(x) +∑

j

∂xjD2,j(x), (C5)

where D1 =∑

j δi,j∂xiV and D2,j = δi,jD.

The FP equation and the way we have rewritten it in equation (C5) will be crucial in making a connection with thequantum case. Moreover, it may be useful for analytical investigations. In particular, if one is interested in stationarysolutions: ∂tP = 0, this implies that LclP = 0, which in some cases can analytically evaluated [2].

Appendix D: Quantum Master Equations

In the next sections we detail the technical steps for obtaining the quantum generalization to the classical Langevin-like equation (B9). In the same way as for the classical Kuramoto model, we assume to be in the overdamped limit. Ithas been shown that the overdamped regime corresponds to a kind of semi-classics [3]. This means that the equilibriumdensity matrix is essentially diagonal in the position basis: 〈x1, ..., xN |β |x

′1, ..., x

′N 〉 ∼

∏

δ(xi−x′i). Following Ref. [3]

our task consists on i) finding the reduced equilibrium density matrix in the system-bath model, β and ii) proposea Fokker-Planck equation like equation (C4) such that the stationary state is that β.

11

1. The reduced density matrix

We are interested in the dynamics of the phases. Therefore we must integrate out the bath degrees of freedom. Ina more quantum language, we are interested in observables acting on the phases’ Hilbert space. Therefore, the objectof interest will be:

≡ Trbath(W ). (D1)

Here W is the total density matrix and Trbath means partial trace over the Hilbert space for the bath. With thisdefinition, the averages over any observable acting on the phases (the system in this case) is:

〈A〉 = Trtot [Asys ⊗ IbathW ] = Trsys [Asys] , (D2)

i.e. the averages can be done by considering the reduced density operator defined in equation (D1).

Appendix E: Equilibrium Density Matrix: Path Integral Formalism

As we anticipated, we need first to compute the equilibrium density matrix. In particular we are interested in thereduced density matrix (at equilibrium):

β = TrbathWβ, (E1)

where Wβ is the total equilibrium density operator, Wβ ∼ e−β(Hsys+Hbath+Hint). The equilibrium reduced densitymatrix can be expressed as [4]

β(x,x′) =

1

Z

∫ x′

1

x1

Dx1 · · ·

∫ x′

N

xN

DxN e−1~SEeff [x], (E2)

with the effective action

SEeff [x] =

∫ ~β

0

dτ(

∑

j

1

2mx2j + V (x1, ..., xN )

)

+1

2

∑

j

∫ ~β

0

dτ

∫ ~β

0

dσK(τ − σ)xj(τ)xj(σ), (E3)

which contains the kernel

K(τ) =m

~β

∑

n

|νn|γ(|νn|)eiνnτ , (E4)

being νn the Matsubara frequencies,

νn =2πn

~β. (E5)

and the Laplace transform of the damping kernel is given by:

γ(z) =2

m

∫ ∞

0

dω

π

J(ω)

ω

z

z2 + ω2. (E6)

1. Overdamped Equilibrium

Based on previous works [3, 5] for the single particle case, we compute the equilibrium distribution in the overdampedlimit. As we already mention in this supplementary material, the overdamped dynamics refers to a regime in theparameter space where the damping is sufficiently strong to suppress the non-diagonal elements, coherences, of thereduced density matrix, i.e., a regime where 〈x1, ..., xN |β |x

′1, ..., x

′N 〉 ∼

∏

δ(xi − x′i). These semiclassical diagonalcontributions can be computed perturbatively on the quantum fluctuations, to be defined below. The versatility andpower of the of the path-integral approach in the semiclassical regime is what justifies its use here. In fact, the zerothorder calculation corresponds to compute the minimal path action for the diagonal contributions.

12

a. Minimal path

Let us denote the minimal action (ma) path as xmai ≡ xi. Besides, since we are interested in the diagonal contribu-

tions in the imaginary-time path integral in equation (E2), this means for us to take the trajectories with

xi(0) = xi(~β) ≡ xi (E7)

i.e. periodic trajectories with frequencies νn. The minimal action path satisfies the generalized Lagrange equations[6]

m¨xi −∂V

∂xi−

∫ ~β

0

dσk(τ − σ)xi(σ) = 0. (E8)

The periodic condition in equation (E7) suggests to Fourier expand xi(τ), such that

xi(τ) =∑

n

xn,ieiνnτ , (E9)

where the Fourier components satisfy

− ν2nxn,i + γ(νn)xn,i + vn,i = bi, (E10)

with

vn,i =

∫

~β

0

dτ∂V

∂xie−iνnτ (E11)

and the inhomogenous term

bi = ˙xi(~β) − ˙xi(0), (E12)

comes from the jumps and cups singularities arising from fact that the Fourier series expansion for xi(τ) periodicallycontinues the path outside the interval 0 ≤ τ ≤ ~β [6]. Note that terms like ai = xi(~β) − xi(0) are, in general,expected; however, since we are interested in the diagonal contributions, they do not contribute to the present case.At this point, we first notice that by making n = 0 for bi we obtain

bi =~β

m

∂V

∂xi. (E13)

Besides, the components xn,i with n 6= 0,

xn,i =−bi

ν2n + γ(νn), (E14)

are suppressed by dissipation. Hence

x0,i ∼= xi(0) +bi~Λ, (E15)

where Λ measures the quantumness:

Λ =2

mβ

∑

n

1

ν2n + γνn(E16)

=~

mπγ

(

Ψ

[

~βγ

2π

]

− C+2π

~βγ

)

,

being C = 0.577... the Euler-Mascheroni constant. Note that in the limit ~ → 0, Λ → 0, as it must be. Thus,recovering the classical result.The contribution of the minimal action can be further simplified by considering that

1

2

∫

˙x2idτ =1

2

(

xi( ˙xi(~β)− ˙xi(0))−

∫

xi ¨xidτ

)

(E17)

13

together with (E7) and replacing equation (E8) in the second term at the r.h.s of equation (E17), such that

S =1

2

∑

i

xibi +

∫

~β

0

dτ(

V −1

2

∑

i

xi∂xiV)

. (E18)

By using the relation (E13) and by noticing that xi ∼= x0,i [xn,i are suppressed, see equation (E14)], we have thatxi − xi = biΛ/~ [Cf. equation (E15)]. Hence,

Sma = ~βV −1

2

∑

i

~β2Λ(∂xiV )2. (E19)

b. Fluctuations around the minimal action path

We study now the fluctuation around the minimal path

xi = xi + yi, (E20)

subjected to the boundary conditions:

yi(0) = yi(~β) = 0. (E21)

Consequently, the correction to the path integral reads,

F (q) =

∫

Dy1 · · ·

∫

DyNe−1/~∫

~β

0dτ〈y|L|y〉, (E22)

where we have used an economical notation, a la Dirac, for the quadratic form 〈y|L|y〉 =∑

Lijyiyj , being L = Lijdefined as

L = −I

(

md2

dτ2+

∫ ~β

0

k(τ − σ)dσ

)

+ V′′, (E23)

where I is the identity matrix and the second-derivative-potential-matrix V′′ = V ′′

ij is defined as,

V ′′ij :=

∂V

∂xi∂xj. (E24)

We proceed as above and Fourier expand the fluctuations around the minimal path [Cf. equation (E21)],

yi =1

~β

∑

n

yn,ieiνnτ , (E25)

which allows us to effectively replace the boundary condition yi(0) = 0 in terms of a product of Dirac’s delta functions,∏

i δ[yi(0)] =∏

i δ[1/~β∑

n yn,i], in the integral expressions above, i.e., by changing

∏

i

δ[yn,i] ∼

∫

∏

i

dµiei/~β〈µ|yn〉 (E26)

where 〈µ|yn〉 =∑

i µiyn,i. Therefore,

F (q) ∼

∫

∏

i

dµi

∫

∏

n

∏

j

dyn,jei/~β〈µ|yn〉e−1/~β〈yn|An|yn〉, (E27)

with,

An = Iλn + V′′ and λn = ν2n + |νn|γ. (E28)

14

This a Gaussian integral that can be performed by resorting twice to the formula

∫

∏

j

dxje−〈x|A|x〉+〈b|x〉 =

√

πN

detAe−〈b|A−1|b〉. (E29)

So that

F (q) ∼

√

∏

n detA−1n

∑

n detA−1n

. (E30)

Up to first order in 1/γ, we get [Cf. equation (E28]:

A−1n

∼=1

λnI−

1

λ2nV′′. (E31)

To be consistent, we also need to compute the determinants at first order in 1/γ [7]

detA−1n

∼=1

λNne−Tr[V′′]/λn . (E32)

Based on all the consideration above, in the next appendix we explicitly present the thermal equilibrium statewith first order corrections in the fluctuations along the semiclassical minimal path results and derive the associatedSmoluchowski equation.

Appendix F: The Quantum Master Equation for the Kuramoto Model: q-K

We proceed here as Ankerhold et al. in Refs. [3, 5]: (i) We compute the equilibrium density matrix β in theoverdamped limit. (ii) We assume that in the quantum regime the qme will have the form equation (C4) andequation (C5). (iii) Finally, we find the the coefficients in equation (C4) and equation (C5) by forcing the equilibriumdensity matrix to be stationary, i.e., LPβ(q) = 0– note that this a natural condition. This is a consistent equilibrium-based derivation for the QME. Since we are going to be interested in the long time dynamics (stationary), by imposingthe correct equilibrium in the dynamics we consider ourselves in a safe place.

1. Thermal Equilibrium Density Matrix β

Based on the result obtained in section E, the equilibrium density matrix in the overdamped limit reads:

β(q, q) ≡ Pβ(q) =1

Ze−βΛ

∑i

∂2V

∂x2i e−βV+ 1

2β2Λ

∑i(∂xi

V )2 , (F1)

where we have introduced the notation Pβ(q) since in the overdamped limit only the diagonal elements β(q, q) matter.

2. One-Particle Master Equation

Let us consider first the one-particle model. This will be a useful analogy later. Besides, since the one particle casehas been studied in Refs. [3, 5], we can use it for the purposes of comparison.In the classical case, the Fokker-Planck equation can be expressed as

∂tP = ∂xLP (F2)

where

L = D1(x) + ∂xD2 (F3)

with,

D1 = V ′ = ∂xV (F4)

15

D2 =D

γ2=kβT

mγ=

1

mγβ=

1

Γβ, (F5)

where Γ denotes mγ.To derive the corresponding quantum Fokker-Planck equation, let us start by considering the quantum equilibrium

for the single particle case, which after equation (F1) reads:

Pβ =1

ZF (x)eS =

1

Ze−βΛV

′′

e(−βV+ β2Λ2V ′2), (F6)

where Z is the partition function and

S = −βV +β2Λ

2V ′2. (F7)

Up to leading order in Λ,

Pβ =1

Z(1− βΛV ′′) e−βV

(

1 +β2Λ

2V ′2

)

. (F8)

We choose for D2:

D2 =D

γ2(1 + βΛV ′′) =

D

γ2F, (F9)

where F = 1−βΛV ′′ is called the quantum correction factor [3, 5] and is the same factor that we found in equation (F6).Imposing Lβ = 0 together with this election for D2, we end up with a D1 coefficient given by

D1 = −D

γ2F∂xS. (F10)

It is easy to calculate the term ∂xS:

∂xS = (−βV ′)(1− βΛV ′′). (F11)

So the term D1 is:

D1 =D

γ2βV ′ =

1

mγV ′. (F12)

This yields the QME for the single case in the overdamped limit:

∂tP = ∂x

[

1

mγV ′ + ∂x(

D

γ2(1 + βΛV ′′))

]

P. (F13)

3. N-Particles Master Equation

Let us obtain the equations in the multivariable case. With N particles the problem is quite more difficult. Thefirst and second derivatives of the potential are not scalars, but a vector V = Vi and a matrix V = Vij, whosecomponents are given by

V ′i =

∂V

∂xiand V ′′

ij =∂V

∂xi∂xj, (F14)

where i, j run from 1 to N , the number of particles. Again, we have to find the Fokker-Planck coefficients which leadsto a stationary solution of the Fokker-Planck equation. In the multivariable case, the FP equation is the same thatequation (F2), but instead of equation (F3) L reads now:

L = D1,i(x) + ∂x,iD2,i, (F15)

16

whereas the stationary solution Pβ in equation (F1) can be rewritten as,

Pβ =1

Ze−βΛTr(V′′)e−βV+β2Λ

2V

′·V′

, (F16)

where Tr(V′′) denotes trace of the matrix V′′. The stationary solution has the general form 1

ZF (x)eS . We can choose

freely what part of Pβ is included in F and what part in eS . The form of the D1,i and D2,i terms will be determinedby our election of F .There are three immediate choices: F = 1 is the simplest one, but it does not lead to the quantum harmonic oscillator

in the one-particle limit. The second choice is an F , like the one-particle, e−βΛTr(V′′). This has one inconvenient: inthe uncoupled case, we will have, for each particle, an equation that depends on the factor F ; moreover, with thatchoice, F depends on all the oscillators in the system. This problem can be solved with the third choice: The factorF can be seen as a product of factors Fi, which are uncoupled from each other. This third option is the one we followbelow.Before giving a particular functional form to F , we will find the general form that the master equation will have

depending on our choice. Let us do something similar to what we did in the single-oscillator case. We consider aquantum white noise term and then impose the stationary solution in order to find the appropriate D1.Since we want to simplify the term D2Pβ in the equation LPβ, we write D2,i =

Dγ2Fi

. By Imposing the stationary

solution Pβ together with our election for D2,i, we find that

D1,iPβ + ∂xiD2,iPβ =

D1,i

∏

j

FjeS +

D

γ2∂xi

(∏

j 6=i

Fj)eS =

eS

D1,i

∏

j

Fj +D

γ2(∏

j 6=i

Fj)∂xiS +

D

γ2

∑

j 6=i

(F ′j,i

∏

k 6=j 6=i

Fk)

= 0

which yields,

D1,i

∏

j

Fj +D(∏

j 6=i

Fj)∂xiS +D

∑

j 6=i

(F ′j,i

∏

k 6=j 6=i

Fk) = 0 . (F17)

We finally obtain:

D1,i = −D

γ2Fi∂xi

S −∑

j 6=i

D

γ2Fi

F ′j,i

Fj. (F18)

Now we have the general form of the terms D1,i and D2,i. If we now apply our particular choice for Fi, we will obtainthe master equation of the system. If Pβ has the form 1

ZF (x)eS , we easily identify the quantum correction factor F :

F = e−βΛTr(V′′). (F19)

This F can be separated in N factors Fi as follows:

F =∏

i

Fi =∏

i

e−βΛV′′

ii . (F20)

After substituting these factors in the expression for D2,i, we obtain:

D2,i =D

γ2Fi=D

γ2e−βΛV

′′

ii . (F21)

By doing so for D1,i and by calculating the term ∂xiS:

∂xiS = −βV ′

i + β2Λ∑

j

∂V

∂xj

∂2V

∂xi∂xj, (F22)

as well as the the partial derivate F ′j,i:

F ′j,i =

∂Fj∂xi

= −βΛV ′′′jjiFj , (F23)

17

we finally get that

D1,i =V ′i

ΓFi−

β

ΓFiΛ∑

j

∂V

∂xj

∂2V

∂xi∂xj+

Λ

ΓFi

∑

j 6=i

V ′′′jji. (F24)

The expressions (F21) and (F24) allows us for explicitly writing the Fokker-Planck equation that describes a systemof many quantum particles in the Smoluchowski regime:

∂tP =∑

i

∂

∂xi

V ′i

ΓFi−

β

ΓFiΛ∑

j

∂V

∂xj

∂2V

∂xi∂xj+

Λ

ΓFi

∑

j 6=i

V ′′′jji

+∂

∂xi

[

D

γ2Fi

]

P. (F25)

This expression constitutes the main core of our quantization of the Kuramoto model.

4. The Langevin Equation

Once we have derived the master equation, we can easily find the associated Langevin equation, in the form

∂xi∂t

= Ai(x, t) +∑

k

Bik(x, t)ξk(t), (F26)

following the guidelines explained in Ref. [9]. Here ξk is Gaussian δ-correlated white noise with zero mean and variance2D [Cf. Eqs. (B3) an (B4)]. Let us write the master equation in terms of the Langevin coefficients

∂tP =−∑

i

∂

∂xi

Ai +D∑

jk

Bjk∂Bik∂xj

P

(F27)

+D∑

ij

∂2

∂xi∂xj

[

∑

k

BikBjk

]

P

.

So that the Langevin coefficients reads:

Ai = −V ′i

ΓFi+βΛ

ΓFi

∑

j

∂V

∂xj

∂2V

∂xi∂xj−

Λ

ΓFi

∑

j 6=i

∂3V

∂x2j∂xi−

Λ

2ΓFi

∂3V

∂x3i, (F28)

Bik =1

γ

(

1

Fi

)12

δik =1

γe

βΛ2

V′′

iiδik. (F29)

The full Langevin equation reads

∂xi∂t

=−V ′i

mγFi(F30)

+βΛ

mγFi

∑

j

∂V

∂xj

∂2V

∂xi∂xj

−Λ

mγFi

∑

j 6=i

∂3V

∂x2j∂xi

−Λ

2mγFi

∂3V

∂x3i

+1

γ

(

1

Fi

)12

ξi(t).

18

Let us discuss the third and the fourth terms in detail. If we calculate the third partial partial derivate of theKuramoto potential (note that this happens only to this potential as a particular case), we realize that third and thefourth terms are:

∑

j 6=i

(

V ′′′ijj

)

=1

N

∑

j

Kij sin(xj − xi), (F31)

V ′′′iii =

1

N

∑

j

Kij sin(xj − xi). (F32)

Note that when i = j, the sin function vanishes, so the fourth term is exactly one half of the third term. Thus, theequation is simpler than it looks like.We can rewrite the equation in a Euler scheme, for convenience, just multiplying the equation by mγ. This reads:

mγ∂xi∂t

= −V ′i

Fi+βΛ

Fi

∑

j

∂V

∂xj

∂2V

∂xi∂xj(F33)

−Λ

Fi

∑

j 6=i

∂3V

∂x2j∂xi−

Λ

2Fi

∂3V

∂x3i+m

(

1

Fi

)12

ξi(t).

5. Classical and One-Particle Limits

The quantum Langevin equation (F33) must reduces the classical Langevin in the limit when Λ → 0 and to theone-particle one when N = 1.It is easy to note that the equation leads to the classic one: when Λ → 0, the quantum correction factor F tends to

1. Most authors take also m = 1 and γ = 1, so Γ = mγ = 1 just for simplify. So the master equation (F25) triviallyleads to:

∂tP =∑

i

∂

∂xi

[V ′i ] +

∂

∂xi[D]

P, (F34)

which is the classical master equation for N particles.

Recall now the quantum master equation (F25) and consider N = 1:

∂tP =∂

∂x

[

V ′

ΓF−

β

ΓFΛ∂V

∂x

∂2V

∂x2

]

+∂

∂x

[

D

γ2F

]

P. (F35)

As we already did in the one-particle case [see equation (F8)], we approximate the exponential functions up to leadingorder in Λ, so that for F we have

F = e−βΛV′′

≈ 1− βΛV ′′. (F36)

The D1 term can now be read as:

V ′

ΓF−

β

ΓFΛV ′V ′′ =

V ′

ΓF(1− βΛV ′′) =

V ′

Γ, (F37)

whereas the term Ai in the Langevin equation reads now

A = −V ′

ΓF+βΛ

ΓF

∂V

∂x

∂2V

∂x2= −

V ′

ΓF+βΛ

ΓFV ′V ′′ (F38)

Taking a common factor −V ′ and simplifying, this is just:

A = −V ′

Γ(F39)

19

6. Nondimensionalization of the Langevin Equation and the Quantum Kuramoto model

We proceed here as in the classical Langevin equation (B2) where some dimensionless quantities (B6) were in-troduced for finding the Kuramoto model. We start by writing the quantum langevin equation in the overdampedlimit

mγ∂xi∂t

= −1

Fi

∂U

∂xi+βΛ

Fi

∑

j

∂U

∂xj

∂2U

∂xi∂xj(F40)

−Λ

Fi

∑

j 6=i

∂3U

∂x2j∂xi−

Λ

2Fi

∂3U

∂θ3i+m

(

1

Fi

)12

ξi(t),

Using now the definitions (B5) and (B6) together with the scaling for the quantum parameter:

Λ =Λ

x20(F41)

we can arrive to:

∂θi∂τ

= −1

Fi

∂V

∂θi+ β

Λ

Fi

∑

j

∂V

∂θj

∂2V

∂θi∂θj(F42)

−Λ

Fi

∑

j 6=i

∂3V

∂θ2j∂θi−

Λ

2Fi

∂3V

∂θ3i+

(

1

Fi

)12

ξi(τ).

7. Physical meaning for Λ

Finally let us give a physical interpretation for the quantum parameter Λ. For a harmonic oscillator (equation 6.62in Ref. [6])

〈q2〉qm =~

m

1

~β

∞∑

n=−∞

1

ω20 + |νn|2 + ζn

(F43)

with ζn = |νn|γ(|νn|), where γ(z) is the Laplace transform of the damping kernel γ(t). Since ζ−n = ζn, we have

〈q2〉 =1

m

1

β

1

ω20 + |ν0|2 + ζ0

+ 21

m

1

β

∞∑

n=1

1

ω20 + |νn|2 + ζn

(F44)

= 〈q2〉cl +2

mβ

∞∑

n=1

1

ω20 + |νn|2 + |νn|γ(|νn|)

, (F45)

where we used ν0 = 0, ζ0 = 0 and the fact that 〈q2〉cl = (mω20β)

−1. In the Ohmic limit, γ(|νn|) = γ, for overdapmedcoupling, Λ = 2

mβ

∑∞n=1

1ω2

0+|νn|2+|νn|γ(|νn|)reduces to Λ = 2

mβ

∑∞n=1

1|νn|2+|νn|γ

.

The quantum corrections to the classical 〈q2〉 are accounted for by

〈q2〉qm

〈q2〉cl= 1 +

Λ

〈q2〉cl. (F46)

Therefore, Λ measures the quantum corrections to the phase fluctuations at equilibrium.

Appendix G: Critical coupling value

We generalize here the work presented in Ref. [10] to the quantum domain.

20

1. Periodicity and self-consistency of the master equation

Proceeding as in [10] for the classical case, we use a mean field approach for computing the critical coupling strength,Kq

c . In doing so we recall that the order parameter r is given by:

rei(ω0t+φ0) =1

N

N∑

j=1

eiφj . (G1)

In a mean-field approximation, we have to rewrite our equations in terms the parameter r. Since we have expressedour equations in terms of V and its derivatives, we only need to write them in terms of r

V = −ωψ −Kr cosψ, V ′ = −ω +Kr sinψ,

V ′′ = Kr cosψ, V ′′′ = −Kr sinψ,(G2)

where ω = ωi − ω0 and ψ = φi − ω0t − φ0. Note that if ψ → ψ + 2π, then all the derivatives of V do not change;however, V→V − 2π. Hence, because of the presence of V , our stationary solution in equation (F16) does not satisfythe periodic boundary condition P (ψ;ω) = P (ψ + 2π;ω). Therefore, we have to find a periodic stationary solution.Following a similar procedure as the one performed by Risken (see pgs. 98 and 287-288 in Ref. [2]), we derive thefollowing periodic stationary solution

P (ψ;ω) =e−βVeffP (0;ω)

[

1 +(e−2βπω − 1)

∫ ψ

0eβVeffdφ

∫ 2π

0 eβVeffdφ

]

, (G3)

where for the sake of brevity, we have introduced the appealing shorthand notation Veff = V − 12βΛV

′2 + ΛV ′′. Assuch, in the classical limit Λ → 0, Veff → V , which coincides with the classical periodic stationary solution derived bySakaguchi [10].As we have already said, the first contribution to P (ψ;ω), e−βVeffP (0;ω), namely the original stationary solution,

does not satisfy the periodic boundary condition. Under the change ψ → ψ + 2π, it becomes

e−β(V− βΛ2V ′2+ΛV ′′)→e2πβωe−β(V− βΛ

2V ′2+ΛV ′′).

Hence, we have chosen the second term in such way that it yields a factor e−2πβω that cancels the contribution fromthe original stationary solution.Proving that P (ψ;ω) in equation (G3), in deed, satisfy the periodic boundary conditions, let us define f(φ) = eβVeff

and consider

∫ ψ

0

f(φ)dφ→

∫ ψ+2π

0

f(φ)dφ =

∫ ψ

0

f(φ)dφ+

∫ ψ+2π

ψ

f(φ)dφ.

Next we introduce the change of variable φ = τ + 2π, so that the last two terms read now

e−2πβω

[

∫ ψ

0

f(τ)dτ +

∫ 0

−2π

f(τ)dτ

]

.

As a consequence, a factor e−2πβω has appeared from the V term in the exponentials. Let us introduce a secondchange of variables. In particular, for the first term consider τ → φ whereas for the second consider τ → φ− 2π (suchthat we undo the first change for the second term). This leads to

e−2πβω

[

∫ ψ

0

f(φ)dφ + e2πβω∫ 2π

0

f(φ)dφ

]

= e−2πβω

∫ ψ

0

f(φ)dφ +

∫ 2π

0

f(φ)dφ, (G4)

and the second contribution in equation (G3) reads:

1 +(e−2βπω − 1)

∫ ψ

0 eβVeffdφ∫ 2π

0eβVeffdφ

e−2πβω + (e−2πβω − 1) = e−2πβω

[

1 +(e−2βπω − 1)

∫ ψ

0 eβVeffdφ∫ 2π

0eβVeffdφ

]

. (G5)

Hence, this solution does satisfy the stationary boundary condition.

21

Since we have changed our original stationary solution, we need to check that the new periodic solution is stationaryas well. It is quite easy to prove this. Call P0 to the original stationary solution and ξ to the multiplicative factorthat makes it periodic, so P = P0ξ is the new solution we want to test. We have chosen ξ to fulfil ∂θξ = P−1

0 . Bydoing this,

∂θ(P0ξ) = (∂θP0)ξ + P0(∂θξ) = (∂θP0)ξ + 1, (G6)

∂2θ (P0ξ) = (∂2θP0)ξ + (∂θP0)(∂θξ) = (∂2θP0)ξ + P−10 (∂θP0). (G7)

If we then apply the master equation

∂θ

[

V ′eff +

1

β∂θ

1

F

]

(P0ξ) = V ′′effP0ξ + V ′

eff(∂θP0)ξ + V ′eff + ∂2θ (

1

F)P0ξ +

1

β(∂2θ )ξ +

1

βP−10 (∂θP0),

and consider that (∂θP0) = −βV ′effP0, then we have that the third and sixth terms cancel each other. In the other

terms, after taking ξ as a common factor, we end up with the original solution, which we know is 0. Thus,

∂θ

[

V ′eff +

1

β∂θ

1

F

]

(P0ξ) = 0 (G8)

explicitly shows that the periodic solution is stationary.

2. Critical value

Once we have this solution, we can proceed as Sakaguchi [10] in order to find the critical coupling strength Kqc .

The order parameter r can be expressed in terms of ψ as:

r =

∫ ∞

−∞

g(ω)dω

∫ 2π

0

n(ψ;ω)exp(iψ)dψ. (G9)

Replacing (G5) above we have a self-consistent equation for r. In the right hand of (G9), the imaginary part is alwayszero, because g(ω) is symmetric about ω = 0. The real part can be expanded in powers of Kr/D as:

r = Kr

[∫ ∞

−∞

g(ω)πω/D[1 + Λ(ω2/D2 − 1)][1 + coth(πω/D)]

(ω2/D2 + 1)dω

]

+O[

(Kr/D)2]

. (G10)

Assuming a peaked, around 0, g(ω)-distribution we expand around ω = 0, obtaining:

r = Kr

[∫ ∞

−∞

g(ω)(1− Λ)(1 + πω/D)

(ω2/D2 + 1)dω

]

+O[

(Kr/D)2]

. (G11)

Noticing again that g(ω) is an even function the linear term πω/D do not contribute to the integral. Finally, thecritical coupling strength, as a function of the temperature, is obtained from (G11), and it is:

Kqc (β) =

2

(1− Λ)∫∞

−∞ g(ω) D2

(ω2+D2)dω. (G12)

As K increases, a non-trivial solution branches off the trivial solution r = 0 at K = Kc. This solution reduces to theclassical one [10, 11] when Λ = 0 at the classical critical coupling strength Kc

c . In fact, from above a simple relationbetween the classical and the quantum critical values can be obtained

Kqc (K

cc ; Λ) =

Kcc

(1− Λ). (G13)

Then, we have a simple relation between the quantum and classic cases.

[1] Bonilla, L.L., Vicente, C.J.P. & Ritort, F. The Kuramoto model: A simple paradigm for synchronization phenomena. Rev.Mod. Phys. 77, 137 (2005).

22

[2] Risken, H. The Fokker-Planck Equation (Springer Berlin Heidelberg, 1989).[3] Ankerhold, J., Grabert, H. & Pechukas, P. Quantum Brownian motion with large friction. CHAOS 15, 26106 (2005).[4] Ingold,G.-L. Path integrals and their application to dissipative quantum systems, in Coherent Evolution in Noisy Environ-

ments, Lecture Notes in Physics, Vol. 611, edited by A. Buchleitner and K. Hornberger (Springer Berlin Heidelberg, 2002)pp. 153.

[5] Maier, S.A. & Ankerhold, J. Quantum Smoluchowski equation: A systematic study. Physical Review E 81, 021107 (2010).[6] Grabert, H., Schramm, P. and Ingold, G.-L. Quantum Brownian motion: The functional integral approach, Phys. Rep.

168, 115 (1988).[7] Here we have made use of the results derived in Ref. [8]. In particular, let us define M = D + O, which can be re-

expressed as M = D(I + A) with A = D−1

O. The determinant of M reads detM = det D eTr[log(I+A)], with log(I + A) =A−

12A

2 + 13A

3− · · · if ρ(A) < 1.

[8] Ipsen, I. C.F. and Lee, D. J. Determinant Approximations, arXiv:1105.0437.[9] Garcia-Palacios, J.L. Introduction to the theory of stochastic processes and Brownian motion problems. arXiv:cond-

mat/0701242[10] Sakaguchi H. Cooperative Phenomena in Coupled Oscillators Systems user External Fields. Prog. Theor. Phys. 79, 39

(1988).[11] Strogatz, S.H. From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators.

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