Instantaneous phase synchronization of two decoupled quantum limit-cycle oscillators induced by conditional photon detection

We show that conditional photon detection induces instantaneous phase synchronization between two decoupled quantum limit-cycle oscillators. We consider two quantum van der Pol oscillators without mutual coupling, each with an additional linearly coupled bath, and perform continuous measurement of photon counting on the output fields of the two baths interacting through a beam splitter. It is observed that in-phase or anti-phase coherence of the two decoupled oscillators instantaneously increases after the photon detection and then decreases gradually in the weak quantum regime or quickly in the strong quantum regime until the next photon detection occurs. In the strong quantum regime, quantum entanglement also increases after the photon detection and quickly disappears. We derive the analytical upper bounds for the increases in the quantum entanglement and phase coherence by the conditional photon detection in the quantum limit.

Instantaneous phase synchronization of two decoupled quantum vdP oscillators induced by conditional photon detection. Either in-phase or anti-phase coherence is induced after photon detection at detector P or M, respectively.

I. INTRODUCTION
Synchronization phenomena, first reported by Huygens in the 17th century, are widely observed in various areas of science and engineering, including laser oscillations, mechanical vibrations, oscillatory chemical reactions, and biological rhythms [1][2][3][4][5][6]. While synchronization of coupled or periodically driven nonlinear oscillators has been extensively investigated [1][2][3]7], oscillators that do not involve any interactions or periodic forcing can also exhibit synchronous behaviors when driven by common random forcing, such as consistency or reproducibility of laser oscillations and spiking neocortical neurons receiving identical sequences of random signals [8,9]. The common-noise-induced synchronization has been theoretically investigated for decoupled limit-cycle oscillators subjected, e.g., to common random impulses [10][11][12] and Gaussian white noise [13][14][15].
Recent developments in nanotechnology have inspired theoretical investigations of quantum synchronization , and the first experimental demonstration of quantum phase synchronization in spin-1 atoms [40] and on the IBM Q system [41] has been reported very recently. Many studies have analyzed coupled quantum nonlinear dissipative oscillators, for example, synchronization of quantum van der Pol (vdP) oscillators [16][17][18], synchronization of ensembles of atoms [19], synchronization of triplet spins [20], measures for quantum synchronization of two oscillators [21][22][23], and synchronization blockade [24,25]. The effects of quantum measurement backaction on quantum nonlinear dissipative oscillators have also been investigated as a unique feature of quantum systems, including improvement in the accuracy of Ramsey spectroscopy through measurement of synchronized atoms [26], measurement-induced transition between in-phase and anti-phase synchronized states [27], unraveling of nonclassicality in optomechanical oscillators [42], characterization of synchronization using quantum trajectories [28], and enhancement of synchronization by quantum measurement and feedback control [29].
In this study, inspired by the common-noise-induced synchronization of decoupled classical oscillators, we consider phase synchronization of two decoupled quantum oscillators induced by common backaction of quantum measurement. We consider two quantum van der Pol oscillators without mutual coupling, each with an additional linearly coupled bath, and perform continuous measurement of photon counting on the output fields of the two baths interacting through a beam splitter. It is demonstrated that the quantum measurement backaction of conditional photon detection common to both oscillators induces instantaneous phase synchronization of the oscillators.

II. MODEL
A schematic of the physical setup is depicted in Fig. 1. The stochastic master equation (SME) of the system can be expressed as where the natural frequency ω and the decay rates γ 1 , γ 2 , and γ 3 for negative damping, nonlinear damping, and linear damping, respectively, are assumed identical for both oscillators, N ± are two independent Poisson processes whose increments are given by dN ± = 1 with probability γ 3 Tr [L † ± L ± ]dt and dN ± = 0 with probability 1 − γ 3 Tr [L † ± L ± ]dt in each interval dt, where dN + = 1 and dN − = 1 represent the photon detection at detectors P and M in Fig. 1, respectively, and the reduced Planck constant is set to = 1. The SLH framework [43,44] has been used to describe the cascade and concatenate connections of the quantum system components in the derivation of the SME (1) [see Appendix A for the derivation of the SME (1)].

III. WEAK QUANTUM REGIME
First, we numerically analyze the quantum SME (1) in the weak quantum regime. To characterize the degree of phase coherence between the two quantum vdP oscillators, we use the absolute value of the normalized correlator [27] as the order parameter, which is a quantum analog of the order parameter for two classical noisy oscillators [3]. The modulus |S 12 | takes values in 0 ≤ |S 12 | ≤ 1; |S 12 | = 1 when the two oscillators are perfectly phase-synchronized and |S 12 | = 0 when they are perfectly phase-incoherent. We also use the argument θ 12 to characterize the averaged phase difference of the two oscillators in order to distinguish in-phase and anti-phase coherence. We use the negativity N = ( ρ Γ1 1 − 1)/2 to quantify the quantum entanglement of the two oscillators, where ρ Γ1 represents the partial transpose of the system with respect to the subsystem representing the first oscillator and X 1 = Tr |X| = Tr √ X † X [45,46]. When the two oscillators are entangled with one other, N takes a nonzero value. We also observe the purity P = Tr [ρ 2 ].
Figures 2(a), 2(b), 2(c), and 2(d) plot the time evolution of |S 12 |, θ 12 , N , and P in the weak quantum regime, respectively, calculated for a single trajectory of the quantum SME (1). As shown in Fig. 2(a), |S 12 | instantaneously increases after the detection of a photon either at P or M, indicating that phase coherence of the two decoupled oscillators is induced by the conditional photon detection. After the photon detection, |S 12 | gradually decreases because the two oscillators converge to the desynchronized steady state of the SME (1) in the absence of photon detection, i.e., dN ± = 0.
In this regime, the nonlinear damping is not strong and the relaxation to the desynchronized state is relatively slow. Therefore, the subsequent photon detection typically occurs before the convergence to the desynchronized state and |S 12 | remains always positive. Figure 2(b) shows that θ 12 takes either θ 12 = 0 or θ 12 = π. This indicates that the two oscillators immediately attain in-phase coherence after the photon detection at P or anti-phase coherence after the photon detection at M. The negativity and purity are shown in Fig. 3(c) and 3(d), respectively, where the negativity is always zero and the purity takes small values between 0.03 and 0.05, indicating that the system is separable and mixed.

IV. STRONG QUANTUM REGIME
We next analyze the quantum SME (1) in a stronger quantum regime. Figures 3(a), 3(b), 3(c), and 3(d) show the evolution of |S 12 |, θ 12 , N , and P , respectively. As shown in Fig. 3(a), |S 12 | takes large values close to 1 immediately after the photon detection, indicating that instantaneous phase coherence also arises in this case. In this regime, the nonlinear damping is strong and the system quickly converges to the desynchronized steady state of the SME (1) when the detection does not occur, i.e., dN ± = 0. Therefore, the phase coherence quickly disappears and |S 12 | remains zero until the next photon detection occurs. Similar to Fig. 2 Fig. 3(b) shows that θ 12 takes either θ 12 = 0 or θ 12 = π. Thus, the two oscillators become in-phase coherent after the photon detection at P and anti-phase coherent after the photon detection at M. Remarkably, Figs. 3(c) and 3(d) show that non-zero negativity and purity with values between 0.5 and 0.6 are attained instantaneously after the photon detection, indicating that mixed entangled states are obtained in this case. However, the quantum entanglement quickly disappears as shown in the inset in Fig. 3

V. QUANTUM LIMIT
From the previous numerical results, it is expected that the maximum quantum entanglement is attained in the quantum limit, i.e., γ 2 → ∞. In this limit, we can map the quantum vdP oscillator to an analytically tractable two-level system with basis states |0 and |1 [17], and transform the SME (1) to with σ − j = |0 1| j and σ + j = |1 0| j representing the lowering and raising operators of the jth system (j = 1, 2), respectively, because the transition |1 The steady state of Eq. (3) without detection, i.e., dN ± = 0, can be analytically obtained, which is given by a diagonal matrix ρ pre = diag (ρ pre 0 , ρ pre 1 , ρ pre 1 , ρ pre 2 ) with Note that only a single parameter k = γ 3 /γ 1 specifies the elements of the matrix, where we assume k > 0, namely, the photon detection occurs with a non-zero probability. The states ρ pos ± = L q ± ρ pre L q † ± /Tr [L q ± ρ pre L q † ± ] immediately after the photon detection occurs at the detector P (ρ pos + ) and M (ρ pos − ) can be represented by a density matrix Using this result, we can explicitly calculate the normalized correlator S 12 and the Q distribution of the phase difference between the two oscillators. In this case, the correlator S 12 of the states ρ pos ± immediately after the photon detection always takes S 12 = ±1 irrespective of the value of k (and then quickly decays). The Q distribution for ρ pos ± can also be calculated as (similar calculation for the Wigner distribution of the phase difference has been performed in [17]) These results qualitatively agree with the corresponding results in the strong quantum regime in Fig. 3. It is notable that the dependence of the phase coherence on k can be captured by the peak height of Q(θ) but not by the normalized correlator S 12 in the quantum limit. Indeed, the element ρ pos 0 |00 00| in Eq. (5) affects Q(θ) (through ρ pos 1 = 1 − ρ pos 0 ) in Eq. (7), whereas it does not affect the value of S 12 .
The above result indicates that the degree of phase coherence is better quantified by the peak height of Q(θ) rather than S 12 in strong quantum regimes. This is because S 12 is defined as a quantum analog of the order parameter for the coherence of classical noisy oscillators, which is quantitatively correct only in the semiclassical regime. This observation is also important in interpreting the results in the weak and strong quantum regimes shown in Figs. 2 and 3, where Q(θ) in the weak quantum regime (Fig. 2) are more sharply peaked than those in the strong quantum regime (Fig. 3), while |S 12 | in Fig. 2 takes smaller values than that in Fig. 3. Thus, S 12 may not work well for comparing phase coherence between different quantum regimes.
Photon detection occurs less frequently when k is smaller, because the probability of the photon detection in the interval dt at detectors P or M is given by k Tr L † ± L ± γ 1 dt. Therefore, on average, infinitely-long observation time is required before the photon detection to approach the upper bounds for the degree of phase coherence and quantum entanglement in the limit k → 0.

VI. CONCLUSION
We have analyzed two decoupled quantum van der Pol oscillators and demonstrated that quantum measurement backaction of conditional photon detection induces instantaneous phase synchronization of the oscillators. In-phase or anti-phase coherence between the oscillators has been observed instantaneously after the photon detection, which decays gradually in the weak quantum regime or quickly in the strong quantum regime until the next photon detection. In the strong quantum regime, short-time increase in the quantum entanglement has also been observed. In the quantum limit, we analytically obtained the upper bounds for the increases in the quantum entanglement and phase coherence.
Recently, physical implementations of the quantum vdP oscillator with ion trap systems [16,17] and optomechanical systems [18,31] have been discussed. The additional linearly coupled bath and photon detectors can also be introduced [48,50]. The physical setup considered in the present study does not require explicit mutual coupling between the oscillators. Therefore, it can, in principle, be implemented by using existing experimental methods and provide a method for generating phase-coherent states of quantum limit-cycle oscillators.

Appendix A: SLH framework
In this Appendix, we derive the SME (1) using the SLH framework to describe cascade and concatenate connections of the quantum system components [43,44]. In this framework, the parameters in the time evolution of a quantum system ρ are specified by G = (S, L, H), with where S is the scattering matrix with operator entries satisfying S † S = SS † = I n , L is a coupling vector with operator entries, and H is a self-adjoint operator referred to as the system Hamiltonian. We denote by I n an identity matrix with n dimensions. With these parameters, the time evolution of the system obeys the master equation where S is involved in the calculation of the cascade and concatenation products and has an important role in determining the forms of H and L of the whole network system consisting of the system components. This specification of parameters is based on Hudson-Parthasarathy's work [52]. The cascade product (Fig. 5(a)) of G 1 = (S 1 , L 1 , H 1 ) and G 2 = (S 2 , L 1 , H 2 ) is given by and the concatenation product (See Fig. 5(b)) of G 1 and G 2 is given by Our aim is to derive the SME (1) of the physical setup depicted in Fig. 1 [43,44]. To this end, we denote G QV DP j as the parameters of the jth quantum vdP oscillator with an additional linearly coupled bath The concatenate connection of G QV DP 1 and G QV DP where we have changed the order of the elements in L for simplicity of notation.
In this study, we consider a 50:50 beam splitter. The parameters of the beam splitter G BS for the output fields of the two baths are where we denote by O nm a zero matrix with the dimensions n × m. The cascading connection of the two above-mentioned components is given by Using transformation D[ a1+a2 √ 2 ]ρ + D[ a1−a2 √ 2 ]ρ = D[a 1 ]ρ + D[a 2 ]ρ, the quantum master equation (A2) with the parameters given in Eq. (A8) gives dρ = L 0 ρdt of the SME (1). Then, using the quantum filtering theory [53,54], SME (1) can be obtained.