Quantum-coherent phase oscillations in synchronization
QQuantum-coherent phase oscillations in synchronization
Talitha Weiss, Stefan Walter, and Florian Marquardt
Friedrich-Alexander University Erlangen-N¨urnberg (FAU),Department of Physics, Staudtstraße 7, 91058 Erlangen, Germanyand Max Planck Institute for the Science of Light, Staudtstraße 2, 91058 Erlangen, Germany (Dated: April 19, 2017)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 oscillator’sphase was overdamped and therefore incoherent. We show that there exist regimes of underdamped and evenquantum-coherent phase motion, opening up new possibilities to study quantum synchronization dynamics. Tothis end, we investigate the Van der Pol oscillator (a paradigm for a self-oscillating system) synchronized to anexternal drive. We derive an effective quantum model which fully describes the regime of underdamped phasemotion and additionally allows us to identify the quality of quantum coherence. Finally, we identify quantumlimit cycles of the phase itself.
PACS numbers: 05.45.Xt, 03.65.-w, 42.50.-p
Introduction.–
Synchronization is commonly studied in so-called limit cycle (LC) oscillators that arise from an interplayof linear and nonlinear effects [1]. For instance, linear am-plification causes an instability, whereas nonlinear dampinglimits the oscillator’s dynamics to a finite amplitude. Notably,the phase remains free, which allows synchronization of theoscillator to an external periodic drive or other LC oscillators.A transition from the intrinsic LC motion towards synchro-nized oscillations occurs depending on the coupling strengthto (and frequency mismatch of) the external reference.Quantum synchronization, i.e., the study of quantum sys-tems whose classical counterparts synchronize, has recentlyattracted increasing theoretical attention. So far, studies ofquantum synchronization have only explored overdampedphase motion. This implies that the dynamics, although tak-ing place in quantum systems, remains always incoherent andclassical-like, ruling out the observation of interesting effectslike quantum tunnelling or superposition states of differentsynchronization phases. In the present article, we discoverquantum-coherent phase dynamics.Theoretical studies of quantum synchronization have beenperformed for different platforms, including optomechan-ics [2, 3], atoms and ions [4, 5], Van der Pol (VdP) oscil-lators [6–10], and superconducting devices [11, 12]. Mea-sures of synchronization in the presence of quantum noisehave been proposed in Refs. [5, 13, 14].On the experimental side, only classical synchronizationhas been studied so far for a wide range of systems [15], in-cluding more recently optomechanical systems [16–19]. Inthe well-developed field of classical synchronization, over-damped phase motion is the standard ingredient both of phe-nomenological equations and microscopically derived mod-els. For example, locking to an external force is described bythe so-called Adler equation, a first-order differential equa-tion for the phase. Similarly, synchronized optomechanicalsystems are described by the first-order phase equation ofthe Hopf-Kuramoto model [2, 20, 21]. However, it has beennoticed that classical synchronization also allows for under- damped phase dynamics. For instance, the classical VdP os-cillator features underdamped phase motion and even (syn-chronized) phase self-oscillations [1, 22–24]. Both regimeshave recently been observed experimentally using a nanoelec-tromechanical system [25]. Classical phase self-oscillations,also called phase trapping, have also been observed with cou-pled laser modes [26]. Furthermore, synchronized Josephsonjunction arrays can be mapped to the Kuramoto model includ-ing inertia [15, 27].Here we will show that a regime of quantum-coherent dy-namics exists and that underdamped phase dynamics is a nec-essary but not sufficient condition to observe this regime.Rather, it is the dynamically generated non-equilibrium de-phasing rate that has to become smaller than the oscillationfrequency. Additionally, we identify phase self-oscillations inthe quantum regime.We obtain these insights for a paradigmatic model, thequantum version of the VdP oscillator subject to an externaldrive. The synchronization of the VdP oscillator to this ex-ternal drive is an excellent test case to investigate universalsynchronization behaviour. We derive an effective quantummodel that captures the regime of underdamped phase dynam-ics. This allows us to identify a quality factor for the quantumcoherence. We illustrate the potentially long coherence timesby showing that initial negativities of a Wigner density van-ish slowly. Finally, we briefly discuss possible experimentalrealizations.
Quantum model.–
The quantum VdP oscillator subject toan external drive is described by the master equation ( (cid:126) = 1 ) ˙ˆ ρ = − i (cid:104) − ∆ˆ b † ˆ b + iF (ˆ b − ˆ b † ) , ˆ ρ (cid:105) + γ D [ˆ b † ]ˆ ρ + γ D [ˆ b ]ˆ ρ , (1)with D [ ˆ O ]ˆ ρ = ˆ O ˆ ρ ˆ O † − { ˆ O † ˆ O, ˆ ρ } / . Here, ∆ = ω d − ω isthe detuning of the oscillator’s natural frequency ω from thefrequency of the external drive ω d and F is the driving force.The two dissipative terms in Eq. (1) describe gain and loss ofone and two quanta at rates γ and γ , respectively. a r X i v : . [ qu a n t - ph ] A p r x/x zpf p / p z p f (a) (b) x/x zpf ⇡ ⇡ ⇡/ / ⇡ phase ⇡ ⇡/ / ⇡ phase P ( ) max ⇡ ⇡ W ss ( x , p ) ⇡ ⇡
12 31 2 3
FIG. 1. (color online).
Quantum synchronization.
Steady-stateWigner density W ss ( x, p ) and phase probability distribution P ( φ ) of (a) an undriven ( F/γ = 0 ) and (b) an externally driven ( F/γ =10 ) VdP oscillator. (a) The ring-shaped Wigner function indicatesLC motion. (b) With increasing detuning ∆ /γ , the synchronizationphase changes and synchronization becomes weaker. Parameters: γ /γ = 5 × − , (a) ∆ /γ = 0 , (b) “1, 2, 3” correspond to ∆ /γ = 0 , . , and . In Fig. 1 we show the steady-state Wigner functionalong with the corresponding phase probability distribution P ( φ ) = (cid:80) ∞ n,m =0 e i ( m − n ) φ π (cid:104) n | ˆ ρ ss | m (cid:105) [5] by numerically solv-ing Eq. (1) for its steady state ˆ ρ ss . In the absence of an exter-nal force ( F = 0 ), the two competing dissipation rates γ and γ lead to LC motion of the VdP oscillator, Fig. 1(a). Fora finite applied force ( F (cid:54) = 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 [related to the heightand width of P ( φ ) ] and the synchronization phase [peak po-sition of P ( φ ) ] is shifted, Fig. 1(b).These steady-state properties do not provide any informa-tion on the underlying synchronization dynamics, especially ifwe are trying to discover possible underdamped and quantum-coherent phase dynamics. To test for these regimes, we nowderive an effective quantum model. Effective quantum model.–
In the synchronized regime, theclassical equation of motion for (cid:104) ˆ b (cid:105) = β = Re iφ , ˙ β = i ∆ β + γ β − γ | β | β − F , (2)has a stable fixed point β ss = R ss e iφ ss . We linearize Eq. (1)around β ss by defining ˆ b = β ss + δ ˆ b , where δ ˆ b describesfluctuations around β ss . Neglecting terms of order O ( δ ˆ b ) and higher, we obtain ˙ˆ ρ eff = − i (cid:104) ˆ H eff , ˆ ρ eff (cid:105) + γ D [ δ ˆ b † ]ˆ ρ eff + 4 γ | β ss | D [ δ ˆ b ]ˆ ρ eff , (3) detuning f o r c e F / / ��� ��� ��� ��� ��������������� (a) . . . . . . ������������������������������������������������������������������ sa dd l e - nod e b i f u r ca t i on overdampedlimit-cycle motion H o p f b i f u r c a t i o n under-damped . . . . . . max( cov ) / min( cov ) .
67 4 . . ~e detuning / limit-cycle motion und e r d a m p e do ve r d a m p e d H o p f b i f u r c a t i o n . F/ = 0 . F/ = 1 . (b) f o r c e F / . . . . detuning / FIG. 2. (color online).
Classical phase diagram and squeezing. (a)Overview of the classical synchronization regimes with sketches oftypical phase-space trajectories. (b) Asymmetry of the steady-statesqueezing ellipses, max ( λ cov ) / min ( λ cov ) , obtained from the effectivemodel [30]. At the black crosses we show the squeezing ellipses (notto scale) with their radial direction aligned along (cid:126)e . Two cuts atdifferent forcing are shown below the figure. Parameters: γ /γ =0 . . with the effective Hamiltonian ˆ H eff = − ∆ δ ˆ b † δ ˆ b − i γ (cid:16) β ss δ ˆ b † δ ˆ b † − β ∗ ss δ ˆ bδ ˆ b (cid:17) . (4)This effective model captures the major features of the fullquantum model and thus allows at least qualitative predic-tions about the behaviour of the system, while quantitativeagreement varies with parameters. A comparison of the out-comes of Eqs. (1) and (3) can be found in the SupplementalMaterial [28]. The effective model is a squeezing Hamilto-nian where the amount of squeezing depends on the classicalsteady-state solution β ss . Diagonalizing Eq. (4) leads to ˆ H diag = − Ω eff ˆ c † ˆ c + const. . (5)Here, δ ˆ be − iθ/ = cosh ( χ )ˆ c + sinh ( χ )ˆ c † , Ae iθ := − iγ β ss / ,tanh (2 χ ) = 2 A/ ∆ , and Ω eff = √ ∆ − A is the effectivefrequency. The corresponding master equation reads ˙ˆ ρ diag = − i (cid:104) ˆ H diag , ˆ ρ diag (cid:105) + Γ ↑ D [ˆ c † ]ˆ ρ diag + Γ ↓ D [ˆ c ]ˆ ρ diag , (6)with Γ ↑ = 4 γ | β ss | sinh ( χ ) + γ cosh ( χ ) , Γ ↓ =4 γ | β ss | cosh ( χ ) + γ sinh ( χ ) , and neglecting fast rotat-ing terms, such as ˆ c ˆ c ˆ ρ eff . The diagonalized, effective model isa damped harmonic oscillator with frequency Ω eff and damp-ing Γ = Γ ↓ − Γ ↑ . This unambiguously allows us to iden-tify an underdamped phase dynamics regime following thestandard procedure for a harmonic oscillator, i.e., we require ∆ > A , which leads to a real-valued effective frequency Ω eff . This is consistent with the corresponding classical dy-namics derived from Eq. (3), leading to a second -order differ-ential equation of the phase, δ ¨ φ + Γ δ ˙ φ + Ω δφ = 0 . (7) time . . t ⌦ e↵ / ⇡ = 0 . t ⌦ e↵ / ⇡ = 0 (a) (b) p / p z p f x/x zpf x/x zpf p / p z p f (c) f o r c e ⌦ e↵ deph W ( x , p , t ) detuning (d) n e↵ (e) detuning eff. thermal occupation quality factor F / / / / / limit cycles limit cycles underdamped underdamped o v e r da m ped li m i t - cyc l e m o t i on (f) (g) under-damped ⌦ e↵ / / deph / underdampedand quantumcoherent t ⌦ e↵ / ⇡ FIG. 3. (color online).
Quantum coherence.
Wigner densities W ( x, p, t ) of (a) the initial state | Ψ( t = 0) (cid:105) ∼ ( | β ss + 2 (cid:105) + | β ss − (cid:105) ) and (b)at a later time. The underdamped phase dynamics rotates the state around the classical steady-state solution (yellow 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 n eff . The left white area corresponds to the overdamped regime. (e) Quality factor Ω eff / Γ deph in the underdamped regime. (f) and (g) show theeffective oscillation frequency Ω eff (dashed blue), damping Γ (dash-dotted green), and dephasing rate Γ deph (red) as a function of the detuning ∆ . (f) At small force F/γ = 1 . the dephasing remains the dominant rate. (g) In contrast, at larger force F/γ = 10 the frequency Ω eff cansignificantly exceed both the dephasing and the damping (quantum-coherent regime). Parameters: γ /γ = 0 . , and (a)-(c) F/γ = 1 . × and ∆ /γ = 7 × . (a)-(c) show numerical solutions to the full model Eq. (1), while (d)-(g) show the rates obtained from our effectivequantum model which characterize the behaviour of the full system. Here, δφ = φ − φ ss describes phase devia-tions from the steady-state phase φ ss and Ω = (cid:112) ∆ + ( γ R ss − γ /
2) (3 γ R ss − γ / is the barefrequency which is related to the effective frequency Ω eff = (cid:112) Ω − Γ / (cid:112) ∆ − γ | β ss | ; cf. [25, 28].Before we discuss results from our effective quantummodel, we briefly review the relevant features of the cor-responding classical “phase diagram” of synchronization,Fig. 2(a). This phase diagram and its quantum analoguewill help us to identify the parameter regime of underdampedphase motion, where we then can check for quantum coher-ence. We obtain the boundaries between the regimes of theclassical phase dynamics from a linear stability analysis ofEq. (2); cf. [1, 28]. Notably, we distinguish two qualitativelydifferent transitions from synchronization to no synchroniza-tion: At small forces, a saddle-node bifurcation characterizesthe transition from synchronized (overdamped) dynamics di-rectly to the LC regime. At larger forces, a regime of un-derdamped phase motion opens up before a Hopf bifurcationmarks the onset of a LC which does not necessarily encirclethe origin.Since we are actually interested in a quantum regime, it isworthwhile to see that these two qualitatively different transi-tions have also important consequences for the quantum dy-namics. In particular, we find a qualitative change of be-haviour in the squeezing properties of the steady state. Since ˆ H eff is quadratic in δ ˆ b the system is fully characterized by itscovariance matrix σ ij = Tr [ˆ ρ eff { ˆ X i , ˆ X j } / with the quadra-tures ˆ X = ( δ ˆ b + δ ˆ b † ) / √ and ˆ X = − i ( δ ˆ b − δ ˆ b † ) / √ . Theeigenvalues λ cov of the covariance matrix determine the shapeof the squeezing ellipse [29]. Their ratio (the asymmetry ofthe ellipses) is shown in Fig. 2(b). Notably, at small forces,it increases with larger detuning. In contrast, at larger forceswhere we predict underdamped phase dynamics, the ellipsesbecome more circular while increasing the detuning ∆ . Thus, the squeezing behaviour can be used as an indicator for theexistence of a quantum regime of underdamped phase mo-tion. The effective model becomes unstable if Γ = 0 , whichcorresponds to the classical fixed point losing its stability. Ad-ditional details on squeezing can be found in the SupplementalMaterial [28].
Quantum coherence.–
Studying the effective model, wehave identified the quantum regime of underdamped phasemotion. Now we demonstrate that within this regime, it ispossible to preserve quantum coherence for a significant time.To this end, we choose an initial state which possesses nega-tivities in its Wigner function, Fig. 3(a), and show that thesenegativities persist for a long time compared to the charac-teristic timescale of the dynamics Ω − eff . The dynamics dueto Eq. (1) leads to a rotation of the state around the classicalsteady state β ss , Fig. 3(b). Notably, this dynamical evolutionhas little influence on the coherence and the negativities ofthe Wigner density survive many oscillations of the system,Fig. 3(c). After the loss of coherence, the state remains ina classical mixture of two displaced states and settles into thesteady state only on an even longer timescale; see Ref. [28] fora complete overview. All Wigner densities in Figs. 3(a)-3(c)are obtained by numerically solving the full master equation(1).This behaviour is successfully predicted by our effectivemodel, which allows us to quantify quantum coherence withinthe underdamped regime and eventually identify a quantum-coherent regime . The time scale on which the quantum sys-tem approaches the steady state is approximately given by thedamping Γ . Thus, a necessary condition to observe quantum-coherent motion is Ω eff > Γ . Approaching the classical Hopfbifurcation, the damping Γ becomes arbitrarily small. How-ever, a small damping does not imply a small dephasing rate Γ deph = Γ ↑ + Γ ↓ . The dephasing rate ultimately determinesthe lifetime of negativities, i.e., quantum coherence. With Γ ↑ = Γ n eff and Γ ↓ = Γ( n eff + 1) , the dephasing rate Γ deph depends on both the damping Γ and the effective occupationof the VdP oscillator n eff . This effective occupation comesabout due to the driven-dissipative character of the quantumoscillator even at zero environmental temperature, also calledquantum heating [31]. It increases towards the boundaries ofthe underdamped regime, Fig. 3(d), counteracting the decreas-ing damping. Additional insight is obtained by identifying aquality factor for quantum coherence, Ω eff / Γ deph , which deter-mines the lifetime of negativities in the Wigner density. Closeto the instability and, more importantly, at large forcing anddetuning, Ω eff / Γ deph increases and can become significantlylarger than , Fig. 3(e). This is the quantum-coherent regimewhere negativities of the Wigner density can survive manyoscillations of the system, Fig. 3(c). Regarding Fig. 3(e), theonly remaining dimensionless parameter (apart from the nor-malized force and detuning) is the ratio of the damping rates γ /γ . It influences the region of stability of the effectivemodel. For instance, increasing γ /γ shifts the instability( Γ = 0 ) to larger detuning. This allows to achieve a compa-rable quality factor Ω eff / Γ deph at smaller forcing but similardetuning - mainly because Ω eff increases with ∆ . In Figs. 3(f)and (g) we show all relevant rates in the underdamped regimeat small and large forcings, respectively. In both cases Ω eff increases, while Γ and Γ deph decrease with larger detuning. Atsmall force, Fig. 3(f), the dephasing rate remains the largestrate in the entire underdamped regime. Notably, for large F ,Fig. 3(g), we find that Ω eff can become significantly largerthan both Γ and Γ deph , thus entering the quantum-coherentregime. This is the key element to observing long-lived quan-tum coherence. 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 ) = (cid:82) ∞−∞ dte iωt (cid:104) ˆ b † ( t )ˆ b (0) (cid:105) . We obtain S ( ω ) fromthe steady state of Eq. (1) by applying the quantum regressiontheorem or analytically from the effective model; see Supple-mental Material [28]. The spectrum carries information on thefrequencies of the driven VdP oscillator. Figure 4(a) shows S ( ω ) for a fixed external force and various detunings, corre-sponding to the overdamped (upper black spectrum) and un-derdamped (middle blue, lower red spectra) regime. In theoverdamped regime the spectrum shows a single peak close to ω = 0 , indicating synchronization to the external force. Withincreasing detuning, the spectrum develops from a single peakto two peaks which now sit at approximately ± Ω eff . A smallremainder of the central peak at ω = 0 becomes visible fora larger splitting of the main peaks. The emerging doublepeaks clearly indicate the transition from overdamped to un-derdamped phase dynamics, Fig. 4(b). The increasing asym-metry of S ( ω ) results from the coupling of amplitude andphase dynamics. For even larger detuning, synchronization islost which ultimately leads to a single peak in the spectrum at ω = ∆ . A recent experiment synchronized two nanomechani-cal oscillators by coupling to a common cavity mode [17]. Cu-riously, the cavity output spectrum showed sidebands next to (c) x/x zpf ⇡ ⇡ ⇡/ / ⇡ phase max . . (d)(a)(b) p / p z p f P ( ) detuning S ( ! ) frequency S ( ! ) W ss ( x , p ) over-damped underdamped S ( ! ) !/ ! / !/ / ⇥ FIG. 4. (color online).
Spectrum. (a) and (b) show the spec-trum S ( ω ) of a synchronized VdP oscillator for different detunings ∆ /γ = 0 (upper black spectrum), ∆ /γ = 2 × (middleblue spectrum), and ∆ /γ = 5 × (lower red spectrum). Inthe overdamped regime (upper black) the spectrum shows a singlepeak at ω = 0 , while in the underdamped regime (middle blue andlower red) double peaks at ± Ω eff emerge. Parameters: γ /γ =2 × , F/γ = 2 × . (c) Steady-state Wigner density W ss ( x, p ) and phase probability distribution P ( φ ) of a VdP oscillator showingphase self-oscillations, i.e., a ring-like Wigner density not encirclingthe origin (indicated by the dashed black line). The correspondingspectrum (d) shows multiple peaks at higher harmonics. Parametersfor (c) and (d): γ /γ = 5 × − , F/γ = 10 , ∆ /γ = 1 . . Allsubfigures (a)-(d) show numerical solutions of the full model Eq. (1). the common frequency of the locked oscillators. These side-bands were suggested to arise from (classical) underdampedphase motion of the oscillators, which is also consistent withthe 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 LC opens up aroundthe former stable fixed point. In the quantum regime this issmeared by quantum fluctuations and becomes visible onlyonce the LC is large enough. If that LC expands even further,it will eventually come to resemble the original unsynchro-nized state: The LC encircles the origin of phase space andthe corresponding phase distribution is flat, Fig. 1(a). How-ever, in Fig. 4(c), this is not yet the case, i.e., the LC doesnot encircle the origin. The oscillator has still a tendency tobe locked to the phase of the external force. This is also re-flected in the corresponding phase distribution P ( φ ) whichbecomes asymmetric and shows the onset of a double peakstructure. Notably, phase self-oscillations are accompanied bythe appearance of a series of peaks in the spectrum, Fig. 4(d),representing higher harmonics of the main phase-oscillationfrequency. Experimental realization.–
The regime of underdampedquantum 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 [6, 8]. The possibility to prepare nonclassical statesexperimentally [32] allows for probing the quantum-coherentnature of the underdamped phase dynamics. Based on param-eters for trapped Yb + ions [6, 33, 34], we estimate that itshould be possible to observe significant quantum coherence.In this scenario, the negative and nonlinear damping are bothof the order of kHz, with γ /γ ∼ . To observe quantum-coherent underdamepd phase dynamics the detuning ∆ andthe external force F should be a few hundred kHz each. Thisis realistic, with frequencies of the motional state in the MHzregime. Furthermore, mechanical self-oscillations in cavityoptomechanics have been discussed theoretically [35] and ob-served experimentally [36, 37]. Thus, they are well-suited tostudy synchronization, and classical synchronization phenom-ena have already been demonstrated experimentally [16–19].Yet another possible platform to observe quantum-coherentphase motion are superconducting microwave circuits. Theseare exceptional and highly tuneable platforms for experimen-tally investigating quantum systems. In principle, arbitraryquantum states can be realized [38–40]. Even, the faith-ful engineering of two-photon losses in such systems hasbeen demonstrated [41]. This makes them very interestingfor studying quantum-coherent phase motion and phase self-oscillations of a quantum VdP oscillator. Conclusion.–
We have shown that the phase of a syn-chronized quantum Van der Pol oscillator exhibits intrigu-ing underdamped and even quantum-coherent phase dynam-ics around the synchronized steady state. In order to explorethis interesting regime, we have developed an effective quan-tum model and identified where the dephasing rate becomessufficiently small to observe quantum-coherent phase motion.As a direct consequence, we have shown that this preserves anonclassical quantum state for many phase oscillations. Weestimate that this could readily be observed in state-of-the-artexperiments. While we have analyzed the simplest synchro-nization phenomenon, to an external drive, the regime identi-fied here will also show up in the quantum phase dynamics oftwo coupled oscillators or even lattices [2]. In the latter case,phenomena such as quantum motion of phase vortices maypotentially become observable.
Acknowledgements:
We acknowledge financial support bythe Marie Curie ITN cQOM and the ERC OPTOMECH. [1] J. Kurths, A. Pikovsky, and M. Rosenblum,
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Talitha Weiss, Stefan Walter, and Florian Marquardt
Friedrich-Alexander University Erlangen-N¨urnberg (FAU), Department of Physics, Staudtstraße 7, 91058 Erlangen, Germanyand Max Planck Institute for the Science of Light, Staudtstraße 2, 91058 Erlangen, Germany
DETAILS ON THE CLASSICAL SYNCHRONIZATION PHASE DIAGRAM
Here, we discuss some additional details of the classical phase diagram of a Van der Pol (VdP) oscillator synchronizing toan external force, cf. Fig. 2(a) of the main text. The boundaries for the regimes of overdamped, underdamped, and limit-cyclemotion are obtained from a linear stability analysis of Eq. (2) of the main text (see for instance also Ref. [1] for more details).Using β = β ss + δβ and keeping only first order terms of δβ , the linearized equation of motion is δ ˙ β = i ∆ δβ − γ β ss δβ ∗ + γ δβ − γ | β ss | δβ. (S1)The eigenvalues λ i 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 β ss . In Fig. S1, we show the real and imaginary partof the eigenvalues λ / = − | β ss | γ + γ / ± (cid:112) | β ss | γ − ∆ . Note that, depending on parameters, Eq. (S1) features eitherone or three fixed points. However, if there are three fixed points, see Fig. S1(a), only one of them is stable. We show the realand imaginary parts of the eigenvalues of a stable fixed point as solid lines and the eigenvalues corresponding to unstable fixedpoints as dashed lines. Note that in the synchronized regimes (both overdamped and underdamped) synchronization towards thestable fixed point occurs.The phase diagram, cf. Fig. 2(a) of the main text, shows that the regime of limit-cycle motion can be entered via two differenttypes of bifurcations which we show here in detail: In Fig. S1(a) the regime of limit-cycle motion is entered via a saddle-nodebifurcation, while in Fig. S1(b) a Hopf bifurcation marks the onset of limit-cycle oscillations. In both cases a single, unstablefixed point exists in the limit-cycle regime, i.e., the corresponding real parts of the eigenvalues, associated to damping, arepositive. This indicates amplification and the stabilizing, nonlinear effects that lead to a limit cycle are not included in thelinear stability analysis. At smaller detuning, before the limit-cycle regime, synchronization towards a stable fixed point occurs.The important difference between the two possible transitions is determined by the imaginary part of the eigenvalues of thisstable fixed point. The imaginary part is related to the oscillation frequency and approaching the saddle-node bifurcation, the (b) overdamped und e r d a m p e d f r e q u e n c y damping f r e qu e n cy damping ⌦ e ↵ ⌦ e ↵ overdamped limit-cycle motion limit-cyclemotion detuning detuning . . . . ⇥ ⇥ I m [ i ] R e [ i ] /! ⇥ /! ⇥ (a) . . . ������� ������� ������� ������� ������� - ������ - ������ - ������ - ������������������������ ������ ������ ������ ������ ������ ������ - ������ - ������ - ������ - ������������������������ FIG. S1. (color online).
Classical synchronization transitions. (a) and (b) show the real (blue) and imaginary (red) parts of the eigenvalues λ i from the linear stability analysis that are related to the damping and the frequency respectively. The dashed blue and red lines correspondto unstable fixed points, while the solid lines correspond to a stable fixed point. Each fixed point has two associated eigenvalues that might bedegenerate.The black dashed line indicates the frequency of the free VdP oscillator in the rotating frame, ∆ . (a) At small forcing, F/γ = 0 . ,the VdP oscillator starts out in a regime of synchronization and overdamped phase dynamics and transitions to limit-cycle motion via asaddle-node bifurcation when increasing the detuning ∆ . (b) At larger external force, F/γ = 1 , the transition from synchronization to nosynchronization occurs via a Hopf bifurcation, thus first crossing a region of underdamped phase motion. Parameters as in Fig. 2 of the maintext. imaginary parts remain zero. This implies that, for detunings below the bifurcation, no characteristic oscillation frequencyexists. The non-zero imaginary parts visible in Fig. S1(a) belong to unstable fixed points, which are not of importance in thecontext of synchronization. In contrast, at larger force, Fig. S1(b), a Belyakov-Devaney transition [2] occurs even before theHopf bifurcation. There the real parts of the eigenvalues become equal but remain negative (i.e. there exists a stable fixed point),yet the imaginary parts become nonzero. This implies that in this case the steady state is approached in an oscillatory fashionand determines the regime of underdamped phase motion. Only at even larger detuning a Hopf bifurcation occurs where the realparts become positive as well and a limit cycle is created.The boundaries between the different regimes of classical synchronization phase dynamics are determined by the followingexplicit expressions: ( i ) a saddle-node bifurcation given by F = (cid:16) ∓ γ + √ γ − (cid:17)(cid:16) − γ √ γ − ± ( + γ ) (cid:17) γ , ( ii ) a transi-tion from a stable node to a stable focus (Belyakov-Devaney transition) which is defined by F = (cid:0) − γ + ∆ (cid:1) γ + ∆ γ with | ∆ | > γ / , and ( iii ) a Hopf bifurcation described by F = γ γ ∆ + γ γ with | ∆ | > γ / . Note that this linear analysisdoes not allow us to distinguish between stable self-oscillations of the phase (limit cycles not evolving around the origin) andordinary limit cycles where the phase is monotonously increasing. 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 δ ˙ β = Tr [ δ ˆ b ˙ ρ eff ] = i ∆ δβ − γ β ss δβ ∗ + γ δβ − γ | β ss | δβ . This is equivalent to the linearized equation (S1)confirming that we have indeed derived the correct linearized quantum model. It is instructive to consider to first split thecomplex amplitude β into amplitude R and phase φ such that β = Re iφ , and then obtain the corresponding equations for theamplitude and phase deviations δR and δφ . These deviations are simply defined as the difference between the actual amplitude R (phase φ ) from the steady-state amplitude R ss (steady-state phase φ ss ), i.e. δR = R − R ss and δφ = φ − φ ss . Since δR and δφ are small, δR is approximately the change in direction of R ss and δφ is approximately the change perpendicular to this. For δβ = re iϕ we then obtain δφ ≈ r sin( ϕ − φ ss ) and δR ≈ r cos( ϕ − φ ss ) and with this δ ˙ φ = ∆ δR − (cid:16) γ R ss − γ (cid:17) δφ , (S2) δ ˙ R = − (cid:16) γ R ss − γ (cid:17) δR − ∆ δφ , (S3)which can be combined to a second-order differential equation for the phase, δ ¨ φ + Γ δ ˙ φ + Ω δφ = 0 . (S4)Here we have defined Γ = (cid:0) γ R ss − γ (cid:1) and Ω = (cid:113) ∆ + (cid:0) γ R ss − γ (cid:1) (cid:0) γ R ss − γ (cid:1) . Notably Eq. (S4) is describes acommon harmonic oscillator which allows for overdamped as well as underdamped motion. The transition from overdampedto underdamped solutions is characterized by Ω = Γ / , i.e. where the effective oscillation frequency of the system Ω eff = (cid:112) Ω − Γ / becomes real-valued. The solution to Eq. (S4) becomes unstable if Γ < , revealing the onset of limit-cyclemotion. The limit-cycle motion itself depends on nonlinear effects to stabilize and thus cannot be described with the linearizedequations.The parameters Γ and Ω eff 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 Wignerdensity 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 β ss , 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 β ss , 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). full modeleffective model x/x zpf p / p z p f .
055 0 . pha s e v a r i an c e timedetuning H op f b i f u r c a t i on (a)(b) (c)(d) (c) ⇡ ⇡ ⇡ classical modelfull quantum modeleff. quantum model V a r ( ˆ r ? ) W ss ( x, p ) t /
10 0 p / p z p f ˆ r ˆ r ? ss h ˆ r ? 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 (ˆ 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: γ /γ = 0 . , F/γ = 4 , and ∆ /γ = 1 . . Within the effective model synchronization attracts the system’s dynamics towards the stable fixed point β ss . We can capturethe dynamics using small deviations around β ss . A natural choice are deviations in radial direction, δR , and in phase direction, δφ , similar to the classical treatment. We can define corresponding operators ˆ r = cos ( φ ss )ˆ x/x zpf + sin ( φ ss )ˆ p/p zpf in radialdirection and perpendicular to it, ˆ r ⊥ = sin ( φ ss )ˆ x/x zpf − cos ( φ ss )ˆ p/p zpf . With this, deviations of the phase can be approximatedvia δφ ≈ −(cid:104) ˆ r ⊥ (cid:105) /R ss such that the full phase is given by φ ( t ) ≈ φ ss − (cid:104) ˆ r ⊥ (cid:105) ( t ) /R ss . We show the phase as a function oftime in Fig. S2(c). The system shows underdamped phase motion, i.e. a few damped oscillations can be observed in the fulland effective quantum model, as well as in the classical simulation. It is consistent with the corresponding Wigner densities,that the trajectories of the full and effective quantum model are damped towards a different steady state. Only the steady stateof the effective quantum model and the classical equations are equal by construction. Note that the relation of (cid:104) ˆ r ⊥ (cid:105) to thephase deviations δφ is only accurate if the deviations are small. In Fig. S2(d) we show the variance V ar (ˆ r ⊥ ) as a functionof detuning. For small ∆ synchronization works best, i.e. the Wigner density is more confined in phase space and thus theresulting variance is small. Deviations between the full and effective quantum model appear with increasing detuning. Then,synchronization becomes weaker and the full model can develop a less ellipse-like Wigner density. Approaching the Hopfbifurcation the variance within the effective model blows up, signalling the break-down of the model. The full model showsan increasing variance, which is consistent with the synchronization becoming weaker and the Wigner density becoming moresmeared out. DETAILS ON THE SQUEEZING
In the main text we derived the squeezing Hamiltonian of our effective model, Eq. (4) of the main text, and discussed theasymmetry of the squeezing ellipses in Fig. 2(b) of the main text. Due to the quadratic Hamiltonian, the state is fully characterizedby its covariance matrix σ ij = Tr [ˆ ρ eff { ˆ X i , ˆ X j } / with the quadratures ˆ X = ( δ ˆ b + δ ˆ b † ) / √ and ˆ X = − i ( δ ˆ b − δ ˆ b † ) / √ (same definition as in the main text). The equation of motion for the covariance matrix can be expressed in the following form, ˙ σ = M σ + σM (cid:62) + D, (S5)0with the matrices M = (cid:18) i ( r − r ∗ ) + γ / − γ | β ss | r + r ∗ − ∆ r + r ∗ + ∆ − i ( r − r ∗ ) + γ / − γ | β ss | (cid:19) (S6) D = (cid:18) γ / γ | β ss | γ / γ | β ss | (cid:19) . (S7)Here we used r = iγ β ss / for brevity. The steady-state solution to this equation of motion, ˙ σ = 0 , can be analyticallyobtained, resulting in a × matrix σ that depends only on system parameters and the classical steady-state amplitude β ss .Then, the eigenvalues of the covariance matrix σ can be calculated and analyzed. The ratio of these eigenvalues determinesthe asymmetry of the squeezing ellipses discussed in the main text. However, also the absolute amount of squeezing can beanalyzed by comparing to the size of the vacuum state. If any direction of the squeezing ellipse becomes smaller than the widthof the vacuum state this is referred to as squeezing below shot noise. To this end, we calculate the shot-noise covariance matrix σ sn ij = Tr [ | (cid:105)(cid:104) | { ˆ X i , ˆ X j } / , which is diagonal and has λ sn = 1 / as doubly degenerate eigenvalue. We compare this to thesmallest eigenvalue min( λ cov ) of the covariance matrix of the synchronized VdP oscillator. Squeezing below shot noise occursfor values min ( λ cov ) < / .Interestingly, the synchronized VdP oscillator does feature squeezing below shot noise at small detuning and sufficiently largeforcing. As shown in Fig. S3, for a fixed detuning, squeezing becomes stronger if the external force F is increased, eventuallydropping below the shot noise value. In combination with Fig. 2(b) of the main text we conclude that approximately the radialdirection is squeezed. Since squeezing below the shot noise level occurs mainly at small detuning, it occurs mostly within theregime of overdamped phase motion, but can reach into the underdamped regime as well. However, approaching the classicalHopf bifurcation, i.e., the instability of the effective model, by increasing the detuning ∆ , the squeezing necessarily decreasessince the Wigner density smoothly transforms back into a (circular) limit cycle.This parameter dependence of the absolute squeezing can also be directly explained from the squeezing Hamiltonian, Eq. (4)of the main text. At first sight squeezing depends on the steady state of the system, i.e., β ss , and thus has an intricate dependenceon all parameters. However, we generally observe that large forcing F leads to large values of | β ss | . Notably, the squeezingHamiltonian does not depend on this absolute value, but on the complex value β ss instead. Investigating the steady state Wignerdensity of the synchronized VdP oscillator, we observed in Fig. 1(b) of the main text a crucial dependence on the detuning:Although the value | β ss | does slightly decrease with ∆ , the more important effect is a rotation in phase space, correspondingto a change of the synchronization phase. Thereby β ss transforms from an almost real quantity to an almost purely imaginaryquantity, thus significantly decreasing the real part of β ss even if its modulus would be conserved completely. Therefore, we canconclude that large squeezing appears if the force is sufficiently large compared to the detuning such that the synchronizationphase (the peak position of the phase distribution) is close to the ideal value of or π . . . . . . . . . . . / F / f o r c e detuning . . min( cov ) FIG. S3. (color online).
Squeezing.
Minimum of the eigenvalues λ cov of the covariance matrix σ , as a function of the detuning ∆ andforcing F . Values smaller than . indicate squeezing below shot noise. Note that squeezing is strongest at small detuning and large forcing.Parameters: γ /γ = 0 . . time . . t ⌦ e↵ / ⇡ = 0 . t ⌦ e↵ / ⇡ = 0 (a) (b) p / p z p f x/x zpf x/x zpf p / p z p f W ( x , p , t ) t ⌦ e↵ / ⇡
06 06 6 x/x zpf x/x zpf x/x zpf (c) (d) (e) steady state t ⌦ e↵ / ⇡ = 19 . t ⌦ e↵ / ⇡ = 111 . p / p z p f (f)(g)
50 150 250100 200 time t ⌦ e↵ / ⇡ FIG. S4. (color online).
Long time evolution of coherent synchronization dynamics.
Wigner densities W ( x, p, t ) of (a) an initial superpositionstate | Ψ( t = 0) (cid:105) ∼ ( | β ss + 2 (cid:105) + | β ss − (cid:105) ) , (b)-(d) several snapshots at later times, and (e) the final steady state. The underdamped syn-chronization phase dynamics rotates the state around the classical steady-state solution (yellow cross). The interference fringes vanish due todephasing and a classical mixture of displaced states remains, still rotating around the classical steady state. The displaced states eventuallymerge to the steady state on a timescale set by the damping. (e) and (f) show cuts of the Wigner densities, i.e. W ( p, x = 0 , t ) , as a functionof time. These time evolutions clearly show that dephasing and relaxing into the steady state occur on separate timescales. Parameters as inFig. 3(a)-(c) of the main text: γ /γ = 0 . , F/γ = 1 . × , and ∆ /γ = 7 × . LONG TIME EVOLUTION OF COHERENT SYNCHRONIZATION DYNAMICS
In the section ’Quantum coherence’ in the main text we discuss that the synchronization dynamics can preserve quantumcoherence for a significant number of oscillations of the system. Here, in Fig. S4, we want to show how an initially preparedsuperposition state loses coherence and finally relaxes to the synchronized steady state. We numerically simulate the full quantummodel to stress that this behaviour, expected due to a sufficiently small dephasing rate obtained from our effective quantummodel, can indeed be observed (although quantitative deviations occur). The beginning of this time evolution is also shown anddescribed in Fig. 3(a)-(c) of the main text, but will be repeated here for completeness.Starting with a superposition state, Fig. S4(a), the Wigner density shows interference fringes with negativities. The synchro-nization dynamics described by the full master equation (1) of the main text, leads to rotations around the classical steady state(yellow cross), Fig. S4(b). The frequency of these oscillations is, approximately, given by Ω eff determined from the effectivemodel. Due to dephasing, the interference fringes start to fade out, Fig. S4(c), (f) and (g), and disappear after many oscillationsof the system. Fig. S4(c) shows a snapshot after almost oscillations, where the clear interference fringes have disappearedand only a small area of slightly negative Wigner density values remains. However, the system is still far from its steady state,because the timescale set by the dephasing can be vastly different from the timescale set by the damping rate. In the exampleshown here, the state becomes a classical mixture of two displaced states first. Those displaced states merge at much later times,Fig. S4(d), and finally form the synchronized steady state of the system, Fig. S4(e). Fig. S4(f) and (g) show cuts along themomentum axis of the Wigner densities as a function of time. The damping, i.e. the relaxing towards the steady state, can beviewed best in the long time evolution Fig. S4(f). To clearly see the dephasing, i.e. the loss of coherence in form of vanishingnegativities in the Wigner density, Fig. S4(g) zooms into the first part of the long time evolution.2 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, the squeezing Hamiltonian, and write down the quantum Langevine equations, δ ˙ˆ b = i ∆ δ ˆ b − Γ2 δ ˆ b − γ β ss δ ˆ b † + √ Γ ˆ ξ , (S8) δ ˙ˆ b † = − i ∆ δ ˆ b † − Γ2 δ ˆ b † − γ β ∗ ss δ ˆ b + √ Γ ˆ ξ † . (S9)Here the noise operators ˆ ξ and ˆ ξ † represent white noise, fulfilling (cid:104) ˆ ξ † ( t ) ˆ ξ ( t (cid:48) ) (cid:105) = ¯ nδ ( t − t (cid:48) ) and (cid:104) ˆ ξ ( t ) ˆ ξ † ( t (cid:48) ) (cid:105) = (¯ n + 1) δ ( t − t (cid:48) ) and ¯ n = 1 / (4 γ | β ss | /γ − is obtained from the dissipation rates of Eq. (3) in the main text, i.e. we identified γ ≡ ¯ n Γ and γ | β ss | ≡ (¯ n + 1)Γ . Eqs. (S8) and (S9) are easily solved in Fourier space where the problem simplifies to finding the inverseof a × -matrix. Choosing the convention δ ˆ b ( ω ) = (cid:82) + ∞−∞ dte iωt δ ˆ b ( t ) and δ ˆ b † ( ω ) = (cid:82) + ∞−∞ dte − iωt δ ˆ b † ( t ) we find δ ˆ b ( ω ) = [ − i ( ω − ∆) + Γ / √ Γ∆ − ( ω + i Γ / − γ | β ss | ˆ ξ ( ω ) − − γ β ss √ Γ∆ − ( ω + i Γ / − γ | β ss | ˆ ξ † ( − ω ) ,δ ˆ b † ( ω ) = − − γ β ∗ ss √ Γ∆ − ( ω + i Γ / − γ | β ss | ˆ ξ ( ω ) + [ − i ( ω + ∆) + Γ / √ Γ∆ − ( ω + i Γ / − γ | β ss | ˆ ξ † ( − ω ) . Within the effective model the fluctuation spectrum S eff ( ω ) = (cid:82) + ∞−∞ dte iωt (cid:104) δ ˆ b † ( t ) δ ˆ b (0) (cid:105) = (cid:82) + ∞−∞ dω (cid:48) π (cid:104) δ ˆ b † ( − ω ) δ ˆ b ( ω (cid:48) ) (cid:105) can beobtained from this solution by evaluating the relevant noise correlators. We find S eff ( ω ) = Γ γ | β ss | + ¯ n Γ (cid:2) (Γ / + γ | β ss | + ( ω + ∆) (cid:3)(cid:104) ( ω − (cid:112) ∆ − γ | β ss | ) + (Γ / (cid:105) (cid:104) ( ω + (cid:112) ∆ − γ | β ss | ) + (Γ / (cid:105) . (S10)This spectrum features peaks close to ω = ± (cid:112) ∆ − γ | β ss | ≡ Ω eff . [1] J. Kurths, A. Pikovsky, and M. Rosenblum, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press,2001).[2] H. W. Broer, B. Hasselblatt, and F. Takens,
Handbook of Dynamical Systems - Volume 3 (North Holland, 2010). S ( ! ) frequency !/ S ( ! ) S e↵ ( ! ) full modeleffective model FIG. S5. (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 S eff ( ω ) calculated analytically from the effective quantum model (dashed green lines).The three curves show the spectrum for different detuning ∆ /γ = 0 (black), ∆ /γ = 2 × (blue), and ∆ /γ = 5 × (red). Theeffective spectrum features two (possibly degenerate) peaks close to ± Ω effeff