Enhancement of quantum synchronization via continuous measurement and feedback control
EEnhancement of quantum synchronization via continuous measurement and feedbackcontrol
Yuzuru Kato ∗ and Hiroya Nakao Department of Systems and Control Engineering,Tokyo Institute of Technology, Tokyo 152-8552, Japan (Dated: September 14, 2020)We study synchronization of a quantum van der Pol oscillator with a harmonic drive and demon-strate that quantum synchronization can be enhanced by performing continuous homodyne measure-ment on an additional bath linearly coupled to the oscillator and applying feedback control to theoscillator. The phase coherence of the oscillator is increased by reducing quantum fluctuations viathe continuous measurement, whereas the measurement backaction inevitably induces fluctuationsaround the phase-locking point. We propose a simple feedback policy for suppressing measurement-induced fluctuations by adjusting the frequency of the harmonic drive, which results in enhancementof quantum synchronization. We further demonstrate that the maximum enhancement of quantumsynchronization is achieved by performing quantum measurement on the quadrature angle at whichthe phase diffusion of the oscillator is the largest and the maximal information of the oscillatorphase is extracted. ∗ Corresponding author: [email protected] a r X i v : . [ n li n . AO ] S e p Homodynedetection Quantumvan der Poloscillator LocaloscillatorLineardamping θ est θ θ est ControllerHarmonicdrive
FIG. 1. Enhancement of synchronization of a quantum vdP oscillator with a harmonic drive via the continuous homodynemeasurement and feedback control.
I. INTRODUCTION
Studies pertaining to synchronization of nonlinear oscillators began in the 17th century when Huygens first docu-mented the discovery of mutual synchronization between two pendulum clocks. Henceforth, synchronization phenom-ena have been widely observed in various fields of science and technology, e.g., laser oscillations, chemical oscillations,spiking neurons, chorusing crickets, and mechanical vibrations [1–5]. Furthermore, synchronization have also been an-alyzed in engineering applications, such as voltage standards [6], injection locking [7], phase-locked loops in electricalcircuits [8], and deep brain stimulation for the treatment of Parkinson’s disease [9].Experimental studies of synchronizing nonlinear oscillators have recently reached the micrometer and nanometerscales [10–19], and experimental demonstrations of quantum phase synchronization in spin-1 atoms [20] and on theIBM Q system [21] have been reported. Owing to these experimental developments, the theoretical analysis of quantumsynchronization has received significant attention [22–44], and these studies have revealed that quantum fluctuationsgenerally induce phase diffusion in quantum limit-cycle oscillators and disturb strict synchronization [22–26]. Toovercome this deleterious effect of quantum fluctuations on synchronization, Sonar [25] applied the squeezing effectand demonstrated that the entrainment of a quantum van der Pol (vdP) oscillator to a squeezing signal can suppressquantum fluctuations, and consequently enhance quantum synchronization.Measurement is one of the peculiar features in quantum systems; it changes the quantum state of a system depend-ing on the probabilistic outcomes [45, 46]. When knowledge about a system is indirectly obtained by continuouslymonitoring the output of a field environment interacting with an open quantum system, the system dynamics underthe measurement can be described by a continuous quantum trajectory, i.e., a stochastic evolution of the systemconditioned by the measurement outcomes [47, 48]. This continuous measurement framework facilitates the investi-gation of novel dynamical features of quantum measurement, such as state preparation [49, 50], dynamical creationof entanglement [51], and unveiling [52, 53] and controlling [54] the chaotic behavior of quantum systems. It is alsonotable that the experimental realization of continuous measurement has been investigated recently [55–57].Furthermore, the effect of continuous measurement of quantum limit-cycle oscillators has also been investigated, suchas measurement-induced transitions between in-phase and anti-phase quantum synchronization [29], enhancement ofnonclassicality in optomechanical oscillators via measurement [58], improvement in accuracy of Ramsey spectroscopythrough measurement of synchronized atoms [41], characterization of synchronization using quantum trajectories[30], and instantaneous quantum phase synchronization of two decoupled quantum limit-cycle oscillators induced byconditional photon detection [59]. However, to the best of our knowledge, the effect of continuous measurement onthe enhancement of quantum synchronization has never been discussed.In this study, we consider synchronization of a quantum vdP oscillator with a harmonic drive and demonstrate thatperforming continuous homodyne measurement on an additional bath linearly coupled to the oscillator and applying afeedback control to the oscillator can enhance quantum synchronization. We demonstrate that quantum fluctuationsdisturbing the phase coherence can be reduced by continuous homodyne measurement, and that the measurementbackaction inevitably induces stochastic deviations from the phase-locking point. We propose a simple feedback policythat can suppress the fluctuations by adjusting the frequency of the harmonic drive. Furthermore, we demonstratethat the maximal enhancement of quantum synchronization can be achieved by performing the measurement on thequadrature angle at which the phase diffusion of the oscillator is the largest and the maximal information on thephase of the oscillator can be obtained via the measurement. Using measurement and feedback control, the proposedmethod yields more significant enhancement of phase coherence than that achieved by optimizing the waveform of theperiodic amplitude modulation of the driving signal in the feed-forward setting analyzed in our previous study [27].
II. MODEL
We consider a quantum vdP oscillator subjected to a harmonic drive. A schematic diagram of the physical setupis shown in Fig. 1. We introduce an additional linear bath coupled to the oscillator, perform continuous homodynemeasurement of the output field from the oscillator to the bath, and apply a feedback control to adjust the frequencyof the harmonic drive (Fig.1).We denote by ω and ω d the frequencies of the quantum vdP oscillator and harmonic drive, respectively. Under theassumption that the Markov approximation can be employed, the stochastic master equation of the quantum systemin the coordinate frame rotating with the frequency ω d is written as [22, 23, 26] dρ = (cid:8) − i (cid:2) − (∆ + ∆ fb ) a † a + iE ( a − a † ) , ρ (cid:3) + γ D [ a † ] ρ + γ D [ a ] ρ + γ D [ a ] ρ (cid:9) dt + √ ηγ H [ ae − iθ ] ρdW,dY = √ ηγ Tr[( ae − iθ + a † e iθ ) ρ ] dt + dW, (1)with D [ L ] ρ = LρL † − ( L † Lρ + ρL † L ) , H [ L ] ρ = Lρ + ρL † − Tr [( L + L † ) ρ ] ρ, where D is the Lindblad form and H [ ae − iθ ]characterizes the measurement on the quadrature ae − iθ + a † e iθ , ρ is the density matrix representing the system state, a and a † denote the annihilation and creation operators ( † represents Hermitian conjugate), respectively, ∆ = ω d − ω is the frequency detuning of the harmonic drive from the oscillator, ∆ fb is the feedback control to adjust the frequencydetuning, i.e., the frequency of the harmonic drive, E is the intensity of the harmonic drive, γ , γ , and γ representthe decay rates for the negative damping, nonlinear damping, and linear damping, respectively, η is the efficiencyof the measurement (we set η = 1 when the measurement is performed, and η = 0 when it is not), θ specifies thequadrature angle of the measurement, W represents a Wiener process satisfying E [ dW ] = 0 and E [ dW ] = dt , Y isthe output of the measurement result, and the reduced Planck’s constant is set as (cid:126) = 1.In the following, we use the parameter settings such that the oscillator is synchronized with the harmonic driveand the Wigner distribution, a quasiprobability distribution [60], of the steady-state density matrix ρ ss of Eq. (1) isconcentrated around a stable phase-locking point along the limit-cycle orbit in the classical limit (see, e.g., Fig. 3(a))when the measurement is not performed ( η = 0).We set the feedback control ∆ fb as (see Appendix 1 for details)∆ fb = − K fb ( θ est − θ ) , (2)where K fb ( >
0) represents the feedback gain and θ = arctan (Tr [ pρ ss ] / Tr [ xρ ss ]) represents the locking phase inthe absence of the measurement, which is calculated as the angle between the expectation values of the positionoperator x = ( a + a † ) / p = − i ( a − a † ) / ρ ss of Eq. (1)without measurement, and θ est = arctan (Tr [ pρ est ] / Tr [ xρ est ]), which is chosen such that − π + θ ≤ θ est < θ + π ,represents the phase of the system calculated from the instantaneous state ρ est of Eq. (1) with measurement, whichis conditioned on the measurement record. The feedback control above can actually suppress the fluctuations of thesystem state around the phase-locking point as will be shown in the next section.To evaluate the phase coherence of the quantum vdP oscillator, we use the order parameter [29, 32] S = | S | e iφ = Tr [ aρ ] (cid:112) Tr [ a † aρ ] , (3)which is a quantum analog of the order parameter for a single classical noisy oscillator [2, 3]. The absolute value | S | quantifies the degree of phase coherence and assumes the values in 0 ≤ | S | ≤
1, where | S | = 1 when the oscillatorstate is perfectly phase-coherent and | S | = 0 when the state is perfectly phase-incoherent. Note that φ representsthe average phase value of the oscillator. III. RESULTS
Numerical simulations of Eq. (1) are performed. In Sections III A, III B, and III C, we set the parameter values in thesemiclassical regime, (∆ , γ , γ , E ) /γ = (0 . , . , . , √ . K fb /γ = 1 when we apply the feedback control. In Secs. III D,we discuss the applicability of the proposed method in the quantum regime with parameter values (∆ , γ , γ , E ) /γ =(0 . , . , . , √ . K fb /γ = 7 .
5. In Sections III A, III B, andIII D, we set θ = 0 for the quadrature of measurement and, in Section III C, the effect of varying θ is analyzed. Wealways set the initial state of the simulation as the vacuum state, i.e., ρ = | (cid:105)(cid:104) | . A. Without feedback control
We first consider the case without feedback control, i.e., K fb = 0, in the semiclassical regime. When the measure-ment is performed, we calculated the average values over 300 trajectories obtained by the numerical simulations ofEq. (1) from the same initial state ( ρ = | (cid:105)(cid:104) | ) because the system trajectories behaved stochastically. The averageresults are compared with the results in the case without measurement when the system trajectory of Eq. (1) isdeterministic.Figures 2(a), 2(b), 2(c), and 2(d) show the trajectories of the absolute values of the order parameter | S | quantifyingthe degree of phase coherence, the purity P = Tr [ ρ ], the expectation value of the position operator (cid:104) x (cid:105) = Tr [ xρ ],and the expectation value of the momentum operator (cid:104) p (cid:105) = Tr [ pρ ], respectively. Note that these expectation valuesare fluctuating in the case with measurement.As shown in Fig. 2(a), the average value of the order parameter | S | with measurement is larger than the (determin-istic) value of | S | without measurement, e.g., | S | = 0 .
859 with measurement and | S | = 0 .
737 without measurementat t = 250, signifying the phase coherence increased on average due to the continuous homodyne measurement. Theincrease in the purity is evident in Fig. 2(b); the average values of the purity P with measurement are larger than thestationary value of P without measurement, e.g., P = 0 .
258 with measurement and P = 0 .
169 without measurementat t = 250 sufficiently after the initial relaxation.We note that the observed increase in | S | or P is an average effect; the values of these quantities for a singletrajectory of Eq. (1) with the measurement fluctuates significantly and occasionally take smaller values than thosewithout measurement, as shown by the dark gray lines in Figs. 2(a) and 2(b). We also note that the increase inthe purity implies the reduction in the phase diffusion of the oscillator (See Appendix 2). Owing to the increase inphase coherence by the measurement, the measurement backaction inevitably induces fluctuations in the system statearound the phase-locking point. It is evident from Figs. 2(c) and 2(d) that 10 trajectories of (cid:104) x (cid:105) and (cid:104) p (cid:105) obtained bysimulating Eq. (1) with measurement (gray lines) exhibit strong fluctuations based on the measurement outcomes.The increase in phase coherence by the measurement is also observed in the Wigner distribution. Figure 3(a) showsthe steady-state Wigner distribution obtained from Eq. (1) without measurement ( ρ converges to a steady state inthis case), and Figs. 3(b), 3(c), and 3(d) show the instantaneous Wigner distributions at t = 250 of three trajectoriesobtained by simulating Eq. (1) with measurement ( ρ behaves stochastically in this case). Comparing Figs. 3(b), 3(c),and 3(d) with Fig. 3(a), increase in phase coherence by the continuous homodyne measurement is observed from thestrongly concentrated Wigner distributions. We also observe that the location of the distribution differs by trajectorybecause the measurement backaction randomly disturbs the system state based on the measurement outcomes. B. With feedback control
As presented in Section III A, we observed that the measurement increases phase coherence but induces fluctuationsin the system state around the phase-locking point simultaneously. To suppress the fluctuations of the system state,we introduce the feedback control expressed in Eq. (2).Figures 4(a), 4(b), 4(c), and 4(d) show the trajectories of | S | , P , (cid:104) x (cid:105) , and (cid:104) p (cid:105) , respectively. The feedback controlis applied from t = 100. As shown in Fig. 4(a), the average order parameter | S | with measurement takes largervalues than | S | without measurement, e.g., | S | = 0 .
889 with measurement and | S | = 0 .
737 without measurement t t t t < p >< x > P (d)(c) (b)(a) η =1 η =0 η =1 η =0 η =1 η =0 η =1 η =0 | S | FIG. 2. Measurement-induced increase in phase coherence without feedback control in semiclassical regime. (a) Orderparameter | S | . (b) Purity P . (c) Expectation values of position operator (cid:104) x (cid:105) . (d) Expectation values of momentum operator (cid:104) p (cid:105) . For the case with measurement ( η = 1), average values of results calculated from 300 trajectories are shown by red linesand 10 out of 300 individual trajectories are shown by gray lines, with the dark one representing a single realization of thetrajectory. For the case without measurement ( η = 0), the results of a single trajectory are shown by blue lines. at t = 250. We also see in Fig. 4(b) that the average values of P with measurement are larger than those withoutmeasurement, e.g., P = 0 .
275 with measurement and P = 0 .
169 without measurement at t = 250.The role of the feedback control is evident from Figs. 4(c) and 4(d), where 10 trajectories of (cid:104) x (cid:105) and (cid:104) p (cid:105) obtainedby simulating Eq. (1) with measurement are shown (gray lines). The fluctuations around the phase-locking point aresuppressed by the feedback control that is turned on after t = 100. We note that we used the same sequences of theWiener increments in the numerical simulations of Eq. (1) in the case without feedback control.The average values of (cid:104) x (cid:105) and (cid:104) p (cid:105) (red lines) are smaller than those for the case without measurement (blue lines).This can be explained as follows. The backaction induces strong fluctuations in (cid:104) x (cid:105) because the measurement isperformed on 2 x = ( a + a † ) with θ = 0. Without feedback control, (cid:104) x (cid:105) , which fluctuates near the phase-locking point,occasionally exhibits a large increase along the limit-cycle trajectory to the clockwise direction. This large increasein (cid:104) x (cid:105) is suppressed by the feedback control, which results in a smaller average value of (cid:104) x (cid:105) . Although the backactionis weaker for (cid:104) p (cid:105) , the suppression of large increase in (cid:104) x (cid:105) by the feedback control results in a smaller average value of (cid:104) p (cid:105) .The effect of feedback control for suppressing the fluctuations of the system state is also evident from the Wignerdistribution of the system. Figure 5(a) shows the steady-state Wigner distribution of Eq. (1) without measurement,whereas Figs. 5(b), 5(c), and 5(d) show three realizations of the Wigner distributions at t = 250 of Eq. (1) withmeasurement. Comparing Figs. 5(b), 5(c), and 5(d) with Figs. 3(b), 3(c), and 3(d), it is clear that the fluctuations -5 -2.5 0 2.5 5-5-2.502.55 (d)-5 -2.5 0 2.5 5-5-2.502.55 (a) -5 -2.5 0 2.5 5-5-2.502.55 (b)-5 -2.5 0 2.5 5-5-2.502.55 (c) x p x p x p x p FIG. 3. Wigner distributions of system without feedback control in semiclassical regime. (a) Wigner distribution of steadystate of Eq. (1) without measurement. (b,c,d) Wigner distributions of three different trajectories of Eq. (1) with measurementat t = 250. around the phase-locking point are suppressed effectively by the feedback control.The results above indicate that enhancement of synchronization, i.e., larger phase coherence and smaller fluctuationsaround the phase-locking point, can be achieved via continuous measurement and feedback control. C. Dependence on measurement quadrature
Thus far, we have fixed θ , the quadrature of the measurement, at 0. Next, we consider the effect of varying θ onthe enhancement of quantum synchronization.Figures 6(a) and 6(b) show the average values of | S | and P at t = 250 for 0 ≤ θ ≤ π , respectively, which arecalculated from 400 trajectories of Eq. (1) with measurement and the feedback control. For comparison, we also showthe values of | S | and P for the steady state of Eq. (1) without the measurement. The maximum values of | S | and P are achieved at θ = 2 . θ = 3 . θ isorthogonal to θ , and performing the measurement on the quadrature specified by this θ extracts the maximuminformation regarding the oscillator phase. Hence, the maximum reduction in quantum fluctuations and enhancementin synchronization are attained at the quadrature angle. | S | t t P η =1 η =0 η =1 η =0 t t < p >< x > η =1 η =0 η =1 η =0 (d)(c) (b)(a) FIG. 4. Measurement-induced enhancement of quantum synchronization with feedback control in semiclassical regime.Feedback control is applied from t = 100. (a) Order parameter | S | . (b) Purity P . (c) Expectation values of the positionoperator (cid:104) x (cid:105) . (d) Expectation values of the momentum operator (cid:104) p (cid:105) . For the case with measurement ( η = 1), average values ofresults calculated from 300 trajectories are shown by red lines and 10 out of 300 individual trajectories are shown by gray lineswith the dark one representing a single realization of the trajectory. For the case without measurement ( η = 0), the results ofa single trajectory are shown by blue lines. D. Applicability in stronger quantum regime
Finally, we discuss the enhancement of quantum synchronization via continuous measurement and feedback controlin a stronger quantum regime. The parameters are shown at the beginning of Section III. Figures 7(a), 7(b), 7(c),and 7(d) show the trajectories of | S | , P , (cid:104) x (cid:105) , and (cid:104) p (cid:105) , respectively. The feedback control is applied after t = 100.As shown in Fig. 7(a), the average order parameter | S | with measurement takes larger values than | S | withoutmeasurement, e.g., | S | = 0 .
687 with measurement and | S | = 0 .
586 without measurement at t = 250. We also see inFig. 7(b) that the averaged values of P with measurement are larger than the values of P without measurement, e.g., P = 0 .
340 with measurement and P = 0 .
266 without measurement at t = 250.As shown from the results above, both | S | and P increase on average with measurement even in this quantumregime. The suppression of the measurement-induced fluctuations by the feedback control is shown in Fig. 7(c),where 10 trajectories of (cid:104) x (cid:105) obtained by simulating Eq. (1) with measurement are shown (gray lines). We see thatthe fluctuations in (cid:104) x (cid:105) around the phase-locking point are suppressed by the feedback control which is turned on after t = 100. However, in Fig. 7(d) where 10 trajectories of (cid:104) p (cid:105) obtained by simulating Eq. (1) with measurement areshown (gray lines), the fluctuations in (cid:104) p (cid:105) still remain and can be even stronger after the feedback control is turnedon at t = 100. We note that the fluctuations in (cid:104) p (cid:105) become smaller on average but (cid:104) p (cid:105) also exhibits occasional bursty -5 -2.5 0 2.5 5-5-2.502.55 (d)-5 -2.5 0 2.5 5-5-2.502.55 (a) -5 -2.5 0 2.5 5-5-2.502.55 (b)-5 -2.5 0 2.5 5-5-2.502.55 (c) x p x p x p x p FIG. 5. Wigner distributions of system with feedback control in semiclassical regime. (a) Wigner distribution of steady stateof Eq. (1) without measurement. (b,c,d) Wigner distributions of three different trajectories of Eq. (1) with measurement at t = 250. Feedback control is applied from time t = 100. increases when the feedback control is applied. This is because the feedback control induces more localized stateswith stronger phase coherence than the case without feedback, and measurement-induced fluctuations of such statesyield larger variations in (cid:104) p (cid:105) .These results are also observed from the Wigner distribution of the system. Figure 8(a) shows the steady-stateWigner distribution of Eq. (1) without measurement and Figs. 8(b), 8(c), and 8(d) show three realizations of theWigner distributions at t = 250 of Eq. (1) with measurement, respectively. The fluctuations around the phase-locking point are suppressed effectively by the feedback control, and the enhancement of quantum synchronization isachieved, in Figs. 8(b) and 8(c). However, the measurement-induced fluctuation remains and the phase coherence ofthe oscillator is decreased in Fig. 8(d).These results indicate that quantum synchronization is enhanced only probabilistically in the strong quantumregime considered here. From the numerical results shown in Figs. 7 and 8, we empirically obtain a probability ofsuccess approximately 80 percents for the enhancement of quantum synchronization, namely, Wigner distributions attime t = 250 are strongly localized around the phase-locking point θ for approximately 80 percent of the trajectories.In this regime, because of the strong quantum fluctuations, the feedback control occasionally fails to suppress themeasurement-induced fluctuations and enhance quantum synchronization.We also note that the strong quantum fluctuations lead to the weaker enhancement of synchronization. This isevident in the improvement of | S | = 0 .
687 from | S | = 0 .
586 by a factor 0 . / .
586 = 1 .
172 in the quantum regime,which is smaller than the improvement of | S | = 0 .
889 from | S | = 0 .
737 by a factor 0 . / .
737 = 1 .
206 in thesemiclassical regime in Figs. 4 and 5. More detailed and systematic numerical analysis in the strong quantum regimeis the subject of future study. π /2 3 π /2 2 ππ π /2 3 π /2 2 ππθ P η =1 η =0 η =1 η =0 | S | FIG. 6. Dependence of results on measurement quadrature. (a) Order parameter | S | . (b) Purity P . Order parameter andpurity averaged over 400 trajectories with measurement at t = 250 (red lines) are compared with those for a single trajectorywithout measurement (blue lines). Phase-locking point θ (a solid black vertical line) and points orthogonal to the phase-lockingpoint θ + π/ IV. CONCLUSION
We considered synchronization of a quantum van der Pol oscillator with a harmonic drive. We demonstrated thatintroducing an additional linear bath coupled to the system and performing continuous homodyne measurement ofthe bath can increase the phase coherence of the system. We also proposed a simple feedback policy for suppressingthe fluctuations in the system state around the phase-locking point by adjusting the frequency of the harmonicdrive, and achieved the measurement-induced enhancement of synchronization. We further demonstrated that themaximum enhancement of synchronization is achieved when we perform measurement on the quadrature angle atwhich the phase diffusion of the oscillator is maximized and the maximum information regarding the oscillator phaseis attained. Finally, we demonstrated that the enhancement of quantum synchronization via continuous measurementand feedback control can be achieved with a high probability of success even in the stronger quantum regime.The proposed system can, in principle, be implemented using the current experimental setups; synchronization ofa quantum vdP with a harmonic drive can be experimentally implemented using optomechanical systems [23] or iontraps [22], and the feedback control can be implemented by adjusting the frequency of the harmonic drive using themeasurement outcomes. Quantum measurement, an essential feature in quantum systems, helps us resolve the issueof quantum fluctuations that disturb strict quantum synchronization and is important for the realization and futureapplications of quantum synchronization in the evolving field of quantum technologies.
Acknowledgments.-
Numerical simulations are performed using the QuTiP numerical toolbox [61]. The authorsgratefully thank N. Yamamoto for the stimulating discussions. We acknowledge JSPS KAKENHI JP17H03279,JP18H03287, JPJSBP120202201, JP20J13778, and JST CREST JP-MJCR1913 for financial support.0 | S | t t P η =1 η =0 η =1 η =0 (b)(a) t t < p >< x > η =1 η =0 η =1 η =0 (d)(c) FIG. 7. Measurement-induced enhancement of quantum synchronization with feedback control in quantum regime. Feedbackcontrol is applied from t = 100. (a) Order parameter | S | . (b) Purity P . (c) Expectation values of position operator (cid:104) x (cid:105) . (d)Expectation values of momentum operator (cid:104) p (cid:105) . For the case with measurement ( η = 1), averaged values of results calculatedfrom 300 trajectories are shown by red lines, and 10 out of 300 individual trajectories are shown by gray lines with the darkone representing a single realization of the trajectory. For the case without measurement ( η = 0), results of a single trajectoryare shown by blue lines.
1. Feedback policy
We discuss the feedback policy for suppressing measurement-induced fluctuations around the phase-locking point.To understand the core idea of the feedback policy with a simple model, we consider the system described in Eq. (1)without the linear coupling to the bath, i.e. γ = 0. We also assume that the system is in the semiclassical regime andthe oscillator dynamics can be described by a semiclassical stochastic differential equation (SDE) whose deterministicpart possesses a stable limit-cycle solution. We can then apply the semiclassical phase reduction [26] to obtain anapproximate one-dimensional SDE for the phase variable of the oscillator and use the standard classical methods forthe phase equation [1–5] to analyze synchronization dynamics of the oscillator driven by a periodic forcing.When the quantum noise is sufficiently weak and the classical limit can be taken, the deterministic phase equationfor the oscillator is expressed as (see also the next section) [23, 26] dφdt = ∆ + ∆ fb + (cid:114) γ γ E sin φ. (4)When | ∆ + ∆ fb | ≤ (cid:113) γ γ E , there exists a stable fixed point of Eq. (4), which corresponds to the phase-locking point1 -5 -2.5 0 2.5 5-5-2.502.55 (d)-5 -2.5 0 2.5 5-5-2.502.55 (a) -5 -2.5 0 2.5 5-5-2.502.55 (b)-5 -2.5 0 2.5 5-5-2.502.55 (c) x p x p x p x p FIG. 8. Wigner distributions of system with feedback control in quantum regime. (a) Wigner distribution of steady stateof Eq. (1) without measurement. (b,c,d) Wigner distributions of three different trajectories of Eq. (1) with measurement at t = 250. Feedback control is applied from time t = 100. of the system with the harmonic driving signal under the feedback control, satisfying φ fb = − arcsin (cid:18) ∆ + ∆ fb E (cid:114) γ γ (cid:19) . (5)The fixed point when the feedback control is turned off, i.e., ∆ fb = 0, is expressed as φ = − arcsin (cid:18) ∆ E (cid:114) γ γ (cid:19) . (6)Figure 9 shows a schematic diagram of the feedback policy for suppressing fluctuations around the phase-lockingpoint. As shown in Fig. 9, when θ est > θ , the feedback control is ∆ fb = − K fb ( θ est − θ ) < φ fb < φ .Similarly, when θ est < θ , we obtain φ fb > φ . Therefore, the feedback control shifts the locking phase from φ to φ fb , which is opposite to the direction from θ to θ est , and is expected to suppress the fluctuations of the systemaround the phase-locking point.
2. Relationship between the phase diffusion and purity
We discuss the relation between the phase diffusion and purity of the quantum vdP oscillator when the measurementis absent. We consider the system described in Eq. (1) without the linear coupling to the bath, i.e., γ = 0; additionally,we assume that the system is in the semiclassical regime and driven by the weak perturbation. The system can thenbe approximately described by a SDE of the phase variable of the oscillator by using the semiclassical phase reductiontheory [26].2 stable φ fb φ y = -(∆ +∆ fb ) y = -∆∆ fb = - K fb ( θ est - θ ) < > θ est θ y unstable φ fb φ < y = FIG. 9. Schematic diagram of feedback policy for suppressing fluctuations around phase-locking point. Feedback control shiftsphase-locking point from φ to φ fb that is opposite to the direction from θ to θ est . We introduce the following rescaled quantities: γ = σγ γ (cid:48) , ∆ + ∆ fb = γ (∆ (cid:48) + ∆ (cid:48) fb ) , E = (cid:15)γ E (cid:48) / √ σ, dt (cid:48) = γ dt, dW (cid:48) = √ γ dW with dimensionless parameters γ (cid:48) , ∆ (cid:48) , ∆ (cid:48) fb , and E (cid:48) of O (1). We set 0 < σ (cid:28) < (cid:15) (cid:28) dφ = (cid:16) ∆ (cid:48) + ∆ (cid:48) fb + (cid:15) (cid:112) γ (cid:48) E (cid:48) sin φ (cid:17) dt (cid:48) + (cid:112) σD dW (cid:48) , (7)with D = γ (cid:48) .We first evaluate the phase diffusion of the oscillator based on the effective diffusion constant of Eq.(7) [62], D eff ∝ (cid:104) exp( v ( φ ) / ( σD )) (cid:105) φ (cid:104) exp( − v ( φ ) / ( σD )) (cid:105) φ ) , (8)where the potential v ( φ ) is given by v ( φ ) = − (cid:82) φφ (∆ (cid:48) + ∆ (cid:48) fb + (cid:15) (cid:112) γ (cid:48) E (cid:48) sin φ (cid:48) ) dφ (cid:48) with a reference phase point φ and (cid:104)·(cid:105) φ = π (cid:82) π ( · ) dφ . When σ is sufficiently small, using the saddle-point approximation (cid:104) exp( v ( φ ) / ( σD )) (cid:105) φ ≈ exp( v max / ( σD )) and (cid:104) exp( − v ( φ ) / ( σD )) (cid:105) φ ≈ exp( − v min / ( σD )), the effective diffusion constant can be approxi-mated as D eff ≈ v max − v min ) / ( σD )) , (9)where v max and v min are the maximum and minimum values of the potential v ( φ ), respectively (see [63] for details),and the constant factors are omitted.We next evaluate the purity. Using semiclassical phase reduction theory [26], the density matrix can be approx-imately reconstructed from the phase equation as ρ ≈ (cid:82) π dφP ( φ ) | α ( φ ) (cid:105) (cid:104) α ( φ ) | , where α ( φ ) = (cid:113) σγ (cid:48) exp( iφ ) isthe system state at φ on the classical limit cycle in the phase space of the P representation [60] and P ( φ ) is thesteady-state probability distribution of the Fokker-Planck equation for the phase variable given by ([3], Chapter 9) P ( φ ) ∝ (cid:90) π dφ (cid:48) exp (cid:20) v ( φ (cid:48) + φ ) − v ( φ )) σD (cid:21) . (10)3Because the size of the limit cycle is O (1 / √ σ ), i.e., α ( φ ) = O (1 / √ σ ), when σ is sufficiently small, the purity can beevaluated by using saddle-point approximation as P = Tr ( ρ ) ≈ (cid:90) π dφP ( φ ) (cid:90) π dφ (cid:48) P ( φ (cid:48) ) exp( −| α ( φ ) − α ( φ (cid:48) ) | ) ≈ (cid:90) π dφP ( φ ) ≈ (cid:90) π dφ (cid:18)(cid:90) π dφ (cid:48) exp (cid:20) v ( φ (cid:48) + φ ) − v ( φ )) σD (cid:21)(cid:19) ≈ (cid:90) π dφ (cid:18) exp (cid:20) v max − v ( φ )) σD (cid:21)(cid:19) ≈ exp (cid:20) v max − v min ) σD (cid:21) , where the constant factors are omitted. The effective diffusion constant D eff can then be approximately representedas D eff ∝ P ) / , (11)which indicates that a higher purity results in a smaller phase diffusion of the oscillator. [1] A. T. Winfree, The geometry of biological time (Springer, New York, 2001).[2] Y. Kuramoto,
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