Quantum-coherent phase oscillations in synchronization

Talitha Weiss 1, Stefan Walter 1, and Florian Marquardt 1,2

1 Friedrich-Alexander University Erlangen-Nurnberg (FAU),Department of Physics, Staudtstraße 7, 91058 Erlangen, Germany

2 Max Planck Institute for the Science of Light, Gunther-Scharowsky-Straße 1, 91058 Erlangen, Germany(Dated: August 12, 2016)

Recently several studies have investigated synchronization in quantum-mechanical limit-cycle oscillators.However, the quantum nature of these systems remained partially hidden, since the dynamics of the oscillatorphase was overdamped and therefore incoherent. We show that there exists a regime of underdamped phasemotion which would allow for the observation of truly quantum-coherent effects. To this end we study the Vander Pol oscillator, a paradigm for self-oscillating systems, which has recently been used to study synchronizationin the quantum regime. We derive an effective quantum model which fully describes the regime of underdampedphase motion. Furthermore, we find a regime of long-lived quantum coherence which opens up new possibilitiesto study quantum synchronization dynamics. Finally, we identify quantum limit cycles of the phase itself andrelate them to recent experimental observations in the classical regime.

PACS numbers: 05.45.Xt, 03.65.-w, 42.50.-p

Introduction.– Quantum synchronization, i.e. the study ofquantum systems whose classical counterparts synchronize,has recently attracted increasing theoretical attention. In theclassical as well as in the quantum regime a prerequisiteto study synchronization are so-called limit-cycle oscillators,i.e., oscillators subject to negative as well as nonlinear damp-ing. The negative damping causes an instability whereas thenonlinear damping limits the oscillator’s dynamics to a finiteamplitude. Hence, the amplitude of a limit-cycle oscillator isfixed but its phase is free. This allows for synchronization ofthe oscillator to, e.g., an external reference or other limit-cycleoscillators.

Quantum synchronization has theoretically been studied ondifferent platforms including optomechanics [1, 2], atoms andions [3, 4], Van der Pol (VdP) oscillators [5–9], and super-conducting devices [10, 11]. Routes towards a meaningfulmeasure of synchronization in the presence of quantum noisewere discussed in Refs. [4, 12, 13]. There is also a growingnumber of experiments with optomechanical systems whichhave investigated classical synchronization [14–17].

So far, studies of quantum synchronization have only ob-served overdamped phase motion towards some fixed point.This implies that the dynamics, although taking place in quan-tum systems, remains always incoherent and classical-like. Aregime of underdamped phase dynamics would open up newpossibilities to study quantum-coherent phenomena. For in-stance, interesting effects like quantum tunnelling or super-position states of different synchronization phases could beenvisioned. Despite these appealing opportunities, it remainsunclear whether quantum dynamics other than overdampedmotion exists.

In the well-developed field of classical synchronization,overdamped phase motion is indeed the standard ingredientboth of phenomenological equations and microscopically de-rived models. For example, locking to an external force isdescribed by the so-called Adler equation, a first-order dif-ferential equation for the phase. Similarly, synchronized op-

tomechanical systems are described by the first-order phaseequation of the Hopf-Kuramoto model [1, 18, 19]. However,it has been noticed that classical synchronization also allowsfor underdamped phase dynamics. For instance, the classi-cal VdP oscillator features underdamped phase motion andeven (synchronized) phase self-oscillations [1, 20–22]. Bothregimes have recently been observed experimentally using ananoelectromechanical system [24]. A regime of synchro-nized phase self-oscillations, also referred to as phase trap-ping, has been experimentally observed before, e.g. with cou-pled laser modes [25]. Furthermore, synchronized Josephsonjunction arrays can be mapped to the Kuramoto model includ-ing inertia [26, 27], which also allows for richer phase dynam-ics. These signs of classical underdamped phase motion raisehope that a similar quantum regime might exist as well.

In this Letter, we show that such a regime of underdampedquantum phase dynamics can indeed be found. Moreover, wealso identify phase self-oscillations in the quantum regime.For this purpose we study the quantum version of the VdP os-cillator subject to an external drive. The VdP oscillator is aparadigm for limit-cycle oscillators and an excellent model toinvestigate universal synchronization behaviour. We developan effective quantum model that captures the regime of under-damped phase dynamics. It allows us to identify and discussa quality factor for the quantum coherence within this regime.We illustrate the potentially long coherence times by showingthat initial negativities of a Wigner density vanish slowly, ascompared to the phase-oscillation period. Finally, we brieflycomment on possible experimental realizations.

Quantum model.– The quantum VdP oscillator subject toan external drive is described by the following master equation(~ = 1)

˙ρ = −i[−∆b†b+ iF (b− b†), ρ

]+ γ1D[b†]ρ+ γ2D[b2]ρ ,

(1)

with D[O]ρ = OρO† − O†O, ρ/2. Here, ∆ = ωd − ω0 isthe detuning of the oscillator’s natural frequency ω0 from the

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FIG. 1. (color online). Synchronization in the quantum regime. Thesteady-state Wigner density Wss(x, p) and phase probability distri-bution P (φ) of (a) an undriven (F/γ1 = 0) and (b) an externallydriven (F/γ1 = 10) VdP oscillator. (a) The Wigner function is ringshaped thus indicating limit-cycle motion. (b) With increasing de-tuning ∆/γ1, the synchronization phase deviates from φ = π andsynchronization becomes weaker, i.e., the state broadens in phasespace. Parameters: γ2/γ1 = 5 × 10−3, (a) ∆/γ1 = 0, (b) “1, 2, 3”correspond to ∆/γ1 = 0, 0.5, and 1.

frequency of the external drive ωd and F is the driving force.The two dissipative terms in Eq. (1) describe gain and loss ofone and two quanta at rates γ1 and γ2, respectively.

In Fig. 1 we show the steady-state Wigner function alongwith the corresponding phase probability distribution P (φ) =∑∞n,m=0

ei(m−n)φ

2π 〈n|ρss|m〉 by numerically solving Eq. (1)for its steady state ρss. Previously, P (φ) has been usedto quantify the amount of synchronization in the quantumregime [4]. In the absence of an applied external force(F = 0), the two competing dissipation rates γ1 and γ2 leadto limit-cycle motion of the VdP oscillator, see Fig. 1(a). Fora finite applied force (F 6= 0) and sufficiently small detun-ing ∆ the VdP oscillator synchronizes to the external forceand a fixed phase-relation between the VdP oscillator and theforce is present. In the rotating frame, this corresponds to alocalized Wigner density and a phase distribution P (φ) witha distinct peak. With increasing detuning the VdP oscillatoris less synchronized to the external force [which is related tothe height and width of P (φ)] and the synchronization phase[peak position of P (φ)] is shifted, see Fig. 1(b).

At this level the steady-state properties do not provide anyinformation on the underlying synchronization dynamics, es-pecially if we are trying to discover possible underdampedphase motion. To test for such a regime of synchroized under-damped quantum phase dynamics, we now derive an effectivequantum model which is obtained by linearizing around thesynchronized steady-state solution.

Effective quantum model.– In the synchronized regime theclassical equation of motion for 〈b〉 = β,

β = i∆β +γ1

2β − γ2|β|2β − F , (2)

(a)

0.2 0.4 0.6 0.8 100

0.5

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FIG. 2. (color online). Classical phase diagram and squeezing. (a)Overview of the classical synchronization regimes with sketches oftypical phase space trajectories. The boundaries are obtained froma linear stability analysis of Eq. (2). (b) Asymmetry of squeezingellipses, max(λcov)/min(λcov), obtained from the effective model asan indicator for the squeezing of the steady state [30]. For parametersat the black crosses we show the squeezing ellipses (not to scale)with their radial direction aligned along ~e1. Two cuts at F/γ1 = 0.5(red line) and F/γ1 = 1.5 (blue line) are shown above the figure.Parameters: γ2/γ1 = 0.1.

has a stable fixed point βss. We linearize the master equa-tion (1) around βss by defining b = βss + δb, where δb de-scribes fluctuations around βss. Neglecting terms of orderO(δb3) and higher, we obtain

˙ρeff = −i[Heff, ρeff

]+ γ1D[δb†]ρeff + 4γ2 |βss|2D[δb]ρeff ,

(3)

with the effective Hamiltonian

Heff = −∆δb†δb− iγ2

2

(β2ssδb

†δb† − β∗2ss δbδb). (4)

A detailed comparison of the full model Eq. (1) and the ef-fective model Eq. (3) can be found in the Supplemental Ma-terial [28]. The effective model is a squeezing Hamilto-nian where the amount of squeezing depends on the classi-cal steady-state solution βss. It can be diagonalized using aBogoliubov transformation which leads to

Hdiag = −Ωeffc†c+ const. . (5)

Here, δbe−iθ/2 = cosh(χ)c+sinh(χ)c†,Aeiθ := −iγ2β2ss/2,

tanh(2χ) = 2A/∆, and Ωeff =√

∆2 − 4A2 is the effec-tive oscillation frequency. The corresponding master equationreads

˙ρdiag = −i[Hdiag, ρdiag

]+ Γ↑D[c†]ρdiag + Γ↓D[c]ρdiag , (6)

with Γ↑ = 4γ2 |βss|2 sinh2(χ) + γ1cosh2(χ), Γ↓ =

4γ2 |βss|2 cosh2(χ) + γ1sinh2(χ), and we have neglected fastrotating terms, such as ccρeff. The diagonalized, effectivemodel is a damped harmonic oscillator with frequency Ωeff

3

time

0.3

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FIG. 3. (color online). Quantum coherence. (a) Wigner densities W (x, p, t = 0) of the initial superposition state |Ψ(t = 0)〉 ∼(|βss + 2〉+ |βss − 2〉) and (b) W (x, p, t) at a later time. Due to the underdamped phase dynamics, the state rotates around the classicalsteady-state solution (indicated by a cross). (c) Wigner density W (p, x = 0, t) with negativities that remain visible for many oscillations. (d)Effective temperature in the underdamped regime, indicated by neff. The white area on the left corresponds to the overdamped regime. (e)Quality factor Ωeff/Γdeph in the underdamped regime. (f) and (g) show the effective oscillation frequency Ωeff (blue line), damping Γ (greenline), and dephasing rate Γdeph (red line) as a function of the detuning ∆. (f) At small force F/γ1 = 1.5 the dephasing remains the dominantrate, whereas at larger force F/γ1 = 103 (g) the oscillation frequency Ωeff can significantly exceed both the dephasing and the damping.Parameters: γ2/γ1 = 0.1, and for (a)-(c) F/γ1 = 1.5× 103 and ∆/γ1 = 7× 102.

and damping Γ = Γ↓ − Γ↑. This unambiguously allows usto identify an underdamped phase dynamics regime follow-ing the standard procedure for a harmonic oscillator, i.e., werequire ∆2 > 4A2, which leads to a real-valued effective fre-quency Ωeff. This is consistent with the corresponding classi-cal dynamics derived from the master equation (3), leading tothe following second-order differential equation of the phase,

δφ+ Γδφ+ Ω2δφ = 0 . (7)

Here φ = φss + δφ, φss is the phase correspond-ing to the steady state βss = Rsse

iφss , and Ω =√∆2 + (γ2R2

ss − γ1/2) (3γ2R2ss − γ1/2) is the bare oscil-

lation frequency which is related to the effective frequencyΩeff =

√Ω2 − Γ2/4 =

√∆2 − γ2

2 |βss|4, cf. [24, 28].Before we move on to discuss results from our effective

quantum model, we briefly review the corresponding classical“phase diagram” of synchronization, see Fig. 2(a). We ob-tain the boundaries between the different regimes of the clas-sical phase dynamics from a linear stability analysis of Eq. (2).Notably, we distinguish two qualitatively different transitionsfrom synchronization to no synchronization: At small forcesthe transition between the synchronized (overdamped phasedynamics) and the limit-cycle regime is characterized by asaddle-node bifurcation. At larger forces a regime of un-derdamped phase motion opens up before a Hopf bifurcationmarks the onset of a limit cycle which does not necessarilyencircle the origin.

These regimes also have important consequences for thequantum dynamics. In particular, we find a qualitative changeof behaviour in the squeezing properties of the steady state.Since Heff is quadratic in δb the system is fully character-ized by its covariance matrix σij = Tr[ρeff~vi, ~vj/2] where~v = (X1, X2)T is the vector of the quadratures X1 = (δb +

δb†)/√

2 and X2 = −i(δb − δb†)/√

2. The eigenvalues λcovof the covariance matrix determine the shape of the squeezingellipse [29]. In Fig. 2(b) we show the ratio of these eigen-

values. Notably, at small forces, the squeezing increases withlarger detuning and the states are preferably squeezed alongthe radial direction. In contrast, at larger forces, the ellipsebecomes more circular while increasing the detuning ∆. Thisimplies that we observe less squeezing in the underdampedphase dynamics regime, than in the overdamped one. The ef-fective model becomes unstable if Γ = 0, which correspondsto the classical fixed point losing its stability.

Quantum coherence.– Within the effective model we havebeen able to identify the quantum regime of underdampedphase motion. Now we demonstrate that within this regime,it is possible to preserve quantum coherence for a significanttime. To this end, we show in Fig. 3 that negativities in theWigner density persist quite long compared to the character-istic timescale of the dynamics Ωeff. As an initial state, wechoose |Ψ(t = 0)〉 ∼ |βss + 2〉 + |βss − 2〉 which possessesnegativities in its Wigner function, see Fig. 3(a). The dynam-ics due to Eq. (1) leads to a rotation of the state around theclassical steady state βss, Fig. 3(b). Notably, this dynamicalevolution has little influence on the coherence of the initialsuperposition. As shown in Fig. 3(c) the negativities of theWigner density survive many oscillations of the system andvanish only slowly.

Our effective model allows us to identify where within theunderdamped regime the observation of quantum coherent dy-namics is possible and quantify its quality. The timescale onwhich the quantum system approaches the steady state is ap-proximately given by the damping Γ. Thus, a necessary con-dition to observe quantum-coherent motion is Ωeff > Γ. Ap-proaching the classical Hopf bifurcation the damping Γ canbecome arbitrarily small. However, a small damping rate Γdoes not imply a small dephasing rate Γdeph = Γ↑ + Γ↓.The dephasing rate ultimately determines the lifetime of neg-ativities, i.e., quantum coherence. With Γ↑ = Γneff andΓ↓ = Γ(neff+1), the dephasing rate Γdeph depends on both thedamping Γ and the effective occupation of the VdP oscillator

4

neff. This effective occupation comes about due to the driven-dissipative character of the quantum oscillator, even at zeroenvironmental temperature, and is also referred to as quantumheating [31]. As shown in Fig. 3(d), neff increases towardsthe boundaries of the underdamped regime, thus counteract-ing the decreasing damping. Physical insight is obtained bydefining Ωeff/Γdeph as the quality factor determining the life-time of negativities in the Wigner density. In Fig. 3(e) weshow that close to the instability and, more importantly, atlarge forcing and detuning, Ωeff/Γdeph increases and can be-come significantly larger than 1. This is the regime where thenegativities of the Wigner density can survive many oscilla-tions of the system, Fig. 3(c). Regarding Fig. 3(e), the only re-maining dimensionless parameter (apart from the normalizedforce and detuning) is the ratio of the damping rates γ2/γ1. Itinfluences the region of stability of the effective model. For in-stance increasing γ2/γ1 shifts the instability (Γ = 0) to largerdetuning. This allows to achieve a comparable quality factorΩeff/Γdeph at smaller forcing but similar detuning. The mainreasons for this is the increase of Ωeff with the detuning. InFig. 3(f) and (g) we show all relevant rates in the underdampedregime at small and large forcing, respectively. In both casesΩeff increases, while Γ and Γdeph decrease with larger detun-ing. At small force, Fig. 3(f), the dephasing rate remains thelargest rate in the entire underdamped regime. Notably, forlarge F we find that Ωeff can become significantly larger thanboth Γ and Γdeph, see Fig. 3(g). This is the key element toobserve long-lived negativities in the Wigner density.

Spectrum.– To shed more light on the possibility to exper-imentally observe the transition from overdamped to under-damped synchronization dynamics we investigate the spec-trum S(ω) =

∫∞−∞ dteiωt〈b†(t)b(0)〉. We obtain S(ω) with

the help of the quantum regression theorem using the steadystate from Eq. (1). Since the effective model Eq. (3) isquadratic, we also analytically calculate the spectrum (onlyvalid in the overdamped and underdamped regime), see Sup-plemental Material [28]. The spectrum carries informationon the frequencies of the driven VdP oscillator. Figure 4(a)shows the spectrum for a fixed external force F/γ1 = 2×104

for various detunings. The curves correspond to the over-damped (black) and underdamped (blue, red) regime. In theoverdamped regime the spectrum shows a single peak whichis close to ω = 0, indicating synchronization to the externalforce. With increasing detuning, the spectrum develops froma single peak to two peaks which now sit at approximately±Ωeff. A small remainder of the central peak at ω = 0 be-comes visible for a larger splitting of the main peaks. Theemerging double peaks clearly indicate the transition from theoverdamped regime to the regime of underdamped phase dy-namics, cf. Fig. 4(b). The increasing asymmetry of S(ω) re-sults from the coupling of amplitude- and phase-dynamics.For even larger detuning, synchronization is lost which ulti-mately leads to a single peak in the spectrum sitting at ω = ∆.In the experiment of Ref. [15] two nanomechanical oscilla-tors have been synchronized by coupling to a common cavitymode. Curiously, the cavity output spectrum showed side-

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FIG. 4. (color online). Spectrum. (a) and (b) show the spectrumS(ω) of a synchronized VdP oscillator at different detuning ∆/γ1 =0 (black), ∆/γ1 = 2 × 104 (blue), and ∆/γ1 = 5 × 104 (red). Inthe overdamped regime (black line) the spectrum features a singlepeak at ω = 0, while in the underdamped regime (blue and red line)double peaks at±Ωeff appear. With increasing detuning the spectrumbecomes more asymmetric. Parameters: γ2/γ1 = 2× 103, F/γ1 =2 × 104. (c) The steady-state Wigner density Wss(x, p) and phaseprobability distribution P (φ) of a VdP oscillator showing phase self-oscillations, i.e. a ring-like Wigner density not encircling the origin(as indicated by the dashed black line). The corresponding spectrum(d) features multiple peaks at higher harmonics. Parameters for (c)and (d): γ2/γ1 = 5× 10−3, F/γ1 = 10,∆/γ1 = 1.55.

bands next to the common frequency of the locked oscillators.These sidebands were suggested to arise from (classical) un-derdamped phase motion of the oscillators which is also con-sistent with the classical limit of our theory.

Interestingly, we find that the phase can even undergoself-oscillations. In the quantum regime, these phase self-oscillations appear (in analogy to the classical scenario) at theboundary of underdamped phase motion just before the lossof synchronization occurs. A circular limit cycle opens uparound the former stable fixed point. In the quantum regimethis is smeared by quantum fluctuations and becomes visibleonly once the limit cycle is large enough. If that limit cy-cle expands even further, it will eventually come to resemblethe original unsynchronized state: The limit cycle encirclesthe origin of phase-space and the corresponding phase dis-tribution is flat, cf. Fig. 1(a). However, in Fig. 4(d), this isnot yet the case, i.e., the limit cycle does not encircle the ori-gin. The oscillator has still a tendency to be locked to thephase of the external force. This is also reflected in the corre-sponding phase distribution P (φ) which becomes asymmet-ric and shows the onset of a double peak structure. Notably,phase self-oscillations are accompanied by the appearance ofa series of peaks in the spectrum, see Fig. 4(d), representinghigher harmonics of the main phase-oscillation frequency.

Experimental realization.– The regime of quantum un-derdamped phase motion and even quantum phase self-oscillations could be experimentally studied in a variety ofsystems. For instance, trapped ions are promising candi-date systems for studying synchronization in the quantumregime [5, 7]. The possibility to prepare nonclassical statesexperimentally [32] allows for probing the quantum-coherentnature of the underdamped phase dynamics. Based on theparameters for trapped 171Yb+ ions from Refs. [5, 33, 34],

5

we estimate that it should be possible to observe significantquantum coherence in the regime of underdamped phase mo-tion. In this scenario, the negative and nonlinear damping areboth of the order of kHz, with a ratio γ2/γ1 ∼ 1. In order toobserve quantum-coherent underdamepd phase dynamics thedetuning ∆ and the external force F should be a few hundredkHz each. This is realistic, with frequencies of the motionalstate in the MHz regime.

Furthermore, mechanical self-oscillations in cavity op-tomechanics have been discussed theoretically [35] and ob-served experimentally [36, 37]. Thus, optomechanical sys-tems are also well-suited to study synchronization, and clas-sical synchronization phenomena have already been demon-strated experimentally [14–17]. Yet another possible plat-form to observe quantum-coherent phase motion would besuperconducting microwave circuits. These are exceptionaland highly tuneable platforms for experimentally investigat-ing quantum systems. In principle arbitrary quantum statescan be realized [38–40]. Even, the faithful engineering of two-photon losses in such systems has been demonstrated [41].This makes them very interesting for studying underdampedphase motion and phase self-oscillations of a quantum VdPoscillator.

Conclusion.– We have shown that the quantum VdP os-cillator synchronized to an external drive shows a regime ofunderdamped phase motion. In order to explore this novelregime of underdamped quantum phase dynamics, we havedeveloped an effective quantum model and identified wherethe dephasing rate becomes sufficiently small to observequantum-coherent phase motion. As a direct consequence wehave shown that this preserves a nonclassical quantum statefor many phase oscillations. We estimate that this could read-ily be observed in state of the art experiments. While we haveanalyzed the simplest synchronization phenomenon, to an ex-ternal drive, the regime identified here will also show up in thequantum phase dynamics of two coupled oscillators or evenlattices [1]. In the latter case, phenomena such as quantummotion of phase vortices may potentially become observable.

Acknowledgements: We acknowledge financial support bythe Marie Curie ITN cQOM and the ERC OPTOMECH.

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old for synchronization [1, 6]. Below this threshold, the effectivemodel is not applicable since classically a stable fixed point ex-ists, but the quantum system settles on a limit cycle. In order toimprove the readability of (a) we nevertheless also plot the re-gion where the effective model fails which here is only the casefor very small force and detuning.

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7

Supplemental Material for “Quantum-coherent phase oscillations in synchronization”

Talitha Weiss 1, Stefan Walter 1, and Florian Marquardt 1,2

1 Friedrich-Alexander University Erlangen-Nurnberg (FAU), Department of Physics, Staudtstraße 7, 91058 Erlangen, Germany2 Max Planck Institute for the Science of Light, Gunther-Scharowsky-Straße 1, 91058 Erlangen, Germany

DETAILS ON THE CLASSICAL SYNCHRONIZATION PHASE DIAGRAM

Here, we give some more details on the phase diagram of a classical Van der Pol (VdP) oscillator synchronizing to an externalforce, cf. Fig. 2(a) of the main text. The boundaries for the regimes of overdamped, underdamped, and limit-cycle motion areobtained from a linear stability analysis of Eq. (1) of the main text (see for instance also Ref. [1] for more details). Usingβ = βss + δβ and keeping only first order terms of δβ, the linearized equation of motion is

δβ = i∆δβ − γ2β2ssδβ

∗ +γ1

2δβ − 2γ2 |βss|2 δβ. (S1)

The eigenevalues λi of the corresponding Jacobi matrix are related to damping and effective frequency of the VdP oscillator andcontain information about the properties of the corresponding fixed point. Note that Eq. (S1) features up to three fixed points, butnot all of them are physical and at most one of them is stable. In Fig. S3, we show the real and imaginary part of the eigenvaluesλ1/2 = −2|βss|2γ2 + γ1/2±

√|βss|4γ2

2 −∆2 of the relevant fixed point.Limit-cycle motion starts when the real part of the eigenvalue, related to the damping, becomes positive, indicating amplifi-

cation. The corresponding fixed point is no longer stable. However, depending on the force, the regime of limit-cycle motion isentered via a saddle-node bifurcation, see Fig. S3(a), or a Hopf bifurcation, see Fig. S3(b). The difference is determined by theimaginary part of the eigenvalues, related to the oscillation frequency: At small driving force the imaginary part remains zeroup to the bifurcation, where both real and imaginary part have a zero. This implies that, for detunings below the bifurcation,no characteristic oscillation frequency exists. The system approaches the steady state in an overdamped manner. However, atlarger force another transition occurs first, where the real parts of the eigenvalues are still negative (i.e. there exists a stable fixedpoint), but the imaginary parts already becomes nonzero. This implies that in this case the steady state is approached in anoscillatory fashion and determines the regime of underdamped phase motion. Only at even larger detuning the real part becomespositive as well and a limit cycle is created. The boundaries between these regimes are characterized by bifurcations which are

explicitly: (i) a saddle-node bifurcation given by F 2 =

(∓2γ1+

√γ21−12∆2

)(−γ1√γ21−12∆2±(12∆2+γ2

1))

108γ22

, (ii) a transition from

a stable node to a stable focus which is defined by F 2 =(−γ12 + ∆

)2 ∆γ2

+ ∆3

γ2with |∆| > γ1/4, and (iii) a Hopf bifurcation

described by F 2 = 14γ1γ2

∆2 + 164γ31

γ2with |∆| > γ1/4. Note that this linear analysis does not allow us to distinguish between

stable self-oscillations of the phase (limit cycles not evolving around the origin) and ordinary limit cycles where the phase ismonotonously increasing.

(a) (b)

overdamped

unde

rdam

ped

frequency

damping

frequency

damping

e↵

e↵

overdamped

limit-cycle motionlimit-cyclemotion

detuning detuning0.2 0.4 0.6 0.8 10

103

1

0

1

20

103

1

0

1

2

Im[

i]R

e[

i]

-

-

-

-

-

-

-

-

/1

0.2/1

0.1 0.3

FIG. S1. (color online). Classical synchronization transitions. (a) and (b) show the real and imaginary parts of the eigenvalues λi from thelinear stability analysis that are related to the damping and the frequency respectively. The dashed line indicates the frequency of the freeVdP oscillator in the rotating frame, ∆. (a) At small forcing, F/γ1 = 0.2, the VdP oscillator starts out in a regime of synchronization andoverdamped phase dynamics and transitions to limit-cycle motion via a saddle-node bifurcation when increasing the detuning ∆. (b) At largerexternal force, F/γ1 = 1, the transition from synchronization to no synchronization occurs via a Hopf bifurcation, thus first crossing a regionof underdamped phase motion. Parameters as in Fig. 2 of the main text.

8

CLASSICAL DYNAMICS OF THE EFFECTIVE QUANTUM MODEL

The effective quantum model, Eq. (3) of the main text, allows us to discuss the corresponding classical dynamics which isgiven by δβ = Tr[δbρeff] = i∆δβ − γ2β

2ssδβ

∗ + γ12 δβ − 2γ2 |βss|2 δβ. This is equivalent to the linearized equation (S1)

confirming that we have indeed derived the correct linearized quantum model. It is instructive to obtain the correspondingequations for the amplitude δR and the phase δφ where δR = R−Rss and δφ = φ−φss and Rss (φss) is the steady-state valueof the amplitude (phase). Since δR and δφ are small, δR is approximately the change in direction ofRss and δφ is approximatelythe change perpendicular to this. For δβ = reiϕ we then obtain δφ ≈ r sin(ϕ− φss) and δR ≈ r cos(ϕ− φss) and with this

δφ = ∆δR−(γ2R

2ss −

γ1

2

)δφ , (S2)

δR = −(

3γ2R2ss −

γ1

2

)δR−∆δφ , (S3)

which can be combined to a second-order differential equation for the phase,

δφ+ Γδφ+ Ω2δφ = 0 . (S4)

Here we have defined Γ =(4γ2R

2ss − γ1

)and Ω =

√∆2 +

(γ2R2

ss − γ12

) (3γ2R2

ss − γ12

). Notably Eq. (S4) is describes a

common harmonic oscillator which allows for overdamped as well as underdamped motion. The transition from overdampedto underdamped solutions is characterized by Ω2 = Γ2/4, i.e. where the effective oscillation frequency of the system Ωeff =√

Ω2 − Γ2/4 becomes real-valued. The solution to Eq. (S4) becomes unstable if Γ < 0, revealing the onset of limit-cyclemotion.

The parameters Γ and Ωeff obtained from this classical analysis are equal to the damping and effective frequency appearing inthe effective quantum model.

COMPARISON OF THE FULL AND THE EFFECTIVE QUANTUM MODEL

Here we compare results from the full quantum model, Eq. (1) of the main text, to results from the effective model, Eq. (3) ofthe main text, and the outcome of the classical equations (S2) and (S3). In Fig. S2(a) and (b) we show the steady-state Wigner

full model

effective model

x/xzpf

p/p

zpf

0 0.055 0.11

phas

eva

rianc

e

time

detuning

0

0

Hop

f bifu

rcat

ion

(a)

(b)

(c)

(d)

(c)0

5

4

3

2

1

3

2

13

8

11

8

classical modelfull quantum modeleff. quantum model

Var(

r ?)

Wss(x, p)

t1

/1

9

410 0 10

p/p

zpf

10

0

1010

0

10

r

r?

sshr

?i(

t)/R

ss

FIG. S2. (color online). Full and effective quantum model. Steady-state Wigner densities of (a) the full and (b) of the effective quantum model.The corresponding “phase trajectories” in (c) show similar oscillating behaviour, although relaxing to a different steady state. The black,dashed line gives the classical trajectory for comparison. In (d) we show the variance V ar(r⊥) as a function of the detuning. The deviationsof the effective from the full model increase towards the Hopf bifurcation, where the effective model breaks down. Parameters: γ2/γ1 = 0.1,F/γ1 = 4, and ∆/γ1 = 1.8.

9

density obtained from the full quantum model and the effective model respectively. The result of the effective model needs tobe displaced to the classical steady state βss, indicated by the white cross. The Wigner densities obtained from the full andthe effective quantum model match reasonably well. The parameters were chosen such that first deviations become visible: (i)The Wigner density of the full model is no longer centered exactly around the classical solution βss, while the effective modeldoes so by construction. (ii) The effective quantum model is described by a squeezing Hamiltonian, cf. main text. Thus thecorresponding Wigner densities are ellipses, while the full model can lead to additional curvature in the Wigner density (morebanana-shaped).

Within the effective model synchronization attracts the system’s dynamics towards the stable fixed point βss. We can capturethe dynamics using small deviations around βss. A natural choice are deviations in radial direction, δR, and in phase direction,δφ. We can define corresponding operators r = cos(φss)x/xzpf + sin(φss)p/pzpf in radial direction and perpendicular to it,r⊥ = sin(φss)x/xzpf − cos(φss)p/pzpf. With this, deviations of the phase can be approximated via δφ ≈ −〈r⊥〉/Rss suchthat the full phase is given by φ(t) ≈ φss − 〈r⊥〉(t)/Rss. We show the phase as a function of time in Fig. S2(c). The systemshows underdamped phase motion, i.e. a few damped oscillations can be observed in the full and effective quantum model, aswell as in the classical simulation. It is consistent with the corresponding Wigner densities, that the trajectories of the full andeffective quantum model are damped towards a different steady state. Only the steady state of the effective quantum modeland the classical equations are equal by construction. Note that the relation of 〈r⊥〉 to the phase deviations is only accurate ifthe deviations are small. In Fig. S2(d) we show the variance V ar(r⊥) as a function of detuning. For small ∆ synchronizationworks best, i.e. the Wigner density is more confined in phase space and thus the resulting variance is small. Deviations betweenthe full and effective quantum model appear with increasing detuning. Then, synchronization becomes weaker and the fullmodel can develop a less ellipse-like Wigner density. Approaching the Hopf bifurcation the variance within the effective modelblows up, signalling the break-down of the model. The full model shows an increasing variance, which is consistent with thesynchronization becoming weaker and the Wigner density becoming more smeared out.

ANALYTICAL SPECTRUM

In the over- and underdamped regime we can also obtain the spectrum from the analytical solution to our effective model. Westart from Eq. (4) of the main text and write down the quantum Langevine equations,

δ˙b = i∆δb− Γ

2δb− γ2

2β2ssδb

† +√

Γξ , (S5)

δ˙b† = −i∆δb† − Γ

2δb† − γ2

2β∗2ss δb+

√Γξ†. (S6)

Here the noise operators ξ and ξ† represent white noise, fulfilling 〈ξ†(t)ξ(t′)〉 = nδ(t− t′) and 〈ξ(t)ξ†(t′)〉 = (n+1)δ(t− t′)and n = 1/(4γ2|βss|2/γ1 − 1) is obtained from the dissipation rates, i.e. we identified γ1 ≡ nΓ and 4γ2|βss|2 ≡ (n + 1)Γ.Eqs. (S5) and (S6) are easily solved in Fourier space where the problem simplifies to finding the inverse of a 2 × 2-matrix.Choosing the convention δb(ω) =

∫ +∞−∞ dteiωtδb(t) and δb†(ω) =

∫ +∞−∞ dte−iωtδb†(t) we find

0

S(!

)

106

frequency !/1

101

105105

S(!)

Se↵(!)

full model

effective model

FIG. S3. (color online). Full and effective spectrum. The spectrum S(ω) of a synchronized VdP oscillator obtained from the full masterequation (1) of the main text, and the corresponding Seff(ω) calculated analytically from the effective quantum model (dashed green lines).The three curves show the spectrum for different detuning ∆/γ1 = 0 (black), ∆/γ1 = 2 × 104 (blue), and ∆/γ1 = 5 × 104 (red). Theeffective spectrum features two (possibly degenerate) peaks close to ±Ωeff, and cannot capture the third peak appearing at larger detuning.Parameters as in Fig. 4(a) of the main text.

10

δb(ω) =[−i(ω −∆) + Γ/2]

√Γ

∆2 − (ω + iΓ/2)2 − γ22 |βss|4

ξ(ω)− −γ2β2ss

√Γ

∆2 − (ω + iΓ/2)2 − γ22 |βss|4

ξ†(−ω) ,

δb†(ω) = − −γ2β∗2ss

√Γ

∆2 − (ω + iΓ/2)2 − γ22 |βss|4

ξ(ω) +[−i(ω + ∆) + Γ/2]

√Γ

∆2 − (ω + iΓ/2)2 − γ22 |βss|4

ξ†(−ω).

Within the effective model the fluctuation spectrum Seff(ω) =∫ +∞−∞ dteiωt〈δb†(t)δb(0)〉 =

∫ +∞−∞

dω′

2π 〈δb†(−ω)δb(ω′)〉 can beobtained from this solution by evaluating the relevant noise correlators. We find

Seff(ω) =Γγ2

2 |βss|4 + nΓ[(Γ/2)2 + γ2

2 |βss|4 + (ω + ∆)2][

(ω −√

∆2 − γ2|βss|4)2 + (Γ/2)2] [

(ω +√

∆2 − γ2|βss|4)2 + (Γ/2)2] . (S7)

This spectrum features peaks close to ω = ±√

∆2 − γ22 |βss|4 ≡ Ωeff.

[1] J. Kurths, A. Pikovsky, and M. Rosenblum, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press,2001).