Instantaneous phase synchronization of two decoupled quantum limit-cycle oscillators induced by conditional photon detection
IInstantaneous phase synchronization of two decoupled quantum limit-cycle oscillatorsinduced by conditional photon detection
Yuzuru Kato ∗ and Hiroya Nakao Department of Systems and Control Engineering,Tokyo Institute of Technology, Tokyo 152-8552, Japan (Dated: September 18, 2020)We show that conditional photon detection induces instantaneous phase synchronization betweentwo decoupled quantum limit-cycle oscillators. We consider two quantum van der Pol oscillatorswithout mutual coupling, each with an additional linearly coupled bath, and perform continuousmeasurement of photon counting on the output fields of the two baths interacting through a beamsplitter. It is observed that in-phase or anti-phase coherence of the two decoupled oscillators instan-taneously increases after the photon detection and then decreases gradually in the weak quantumregime or quickly in the strong quantum regime until the next photon detection occurs. In the strongquantum regime, quantum entanglement also increases after the photon detection and quickly dis-appears. We derive the analytical upper bounds for the increases in the quantum entanglement andphase coherence by the conditional photon detection in the quantum limit. ∗ Corresponding author: [email protected] a r X i v : . [ n li n . AO ] S e p Quantum van der PoloscillatorLineardampling Beamsplitter Photondetector MPhotondetector P φ - φ PhotonPhoton ( a - a )/ √ ( a + a )/ √ a a FIG. 1. Instantaneous phase synchronization of two decoupled quantum vdP oscillators induced by conditional photondetection. 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 ofscience and engineering, including laser oscillations, mechanical vibrations, oscillatory chemical reactions, and biolog-ical rhythms [1–6]. While synchronization of coupled or periodically driven nonlinear oscillators has been extensivelyinvestigated [1–3, 7], oscillators that do not involve any interactions or periodic forcing can also exhibit synchronousbehaviors when driven by common random forcing, such as consistency or reproducibility of laser oscillations andspiking neocortical neurons receiving identical sequences of random signals [8, 9]. The common-noise-induced synchro-nization has been theoretically investigated for decoupled limit-cycle oscillators subjected, e.g., to common randomimpulses [10–12] and Gaussian white noise [13–15].Recent developments in nanotechnology have inspired theoretical investigations of quantum synchronization [16–39], and the first experimental demonstration of quantum phase synchronization in spin-1 atoms [40] and on the IBMQ system [41] has been reported very recently. Many studies have analyzed coupled quantum nonlinear dissipativeoscillators, for example, synchronization of quantum van der Pol (vdP) oscillators [16–18], synchronization of ensemblesof atoms [19], synchronization of triplet spins [20], measures for quantum synchronization of two oscillators [21–23], andsynchronization blockade [24, 25]. The effects of quantum measurement backaction on quantum nonlinear dissipativeoscillators have also been investigated as a unique feature of quantum systems, including improvement in the accuracyof Ramsey spectroscopy through measurement of synchronized atoms [26], measurement-induced transition betweenin-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 quantummeasurement and feedback control [29].In this study, inspired by the common-noise-induced synchronization of decoupled classical oscillators, we considerphase 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 coupledbath, and perform continuous measurement of photon counting on the output fields of the two baths interactingthrough a beam splitter. It is demonstrated that the quantum measurement backaction of conditional photon detectioncommon 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 canbe expressed as dρ = L ρdt + G [ L + ] ρ (cid:16) dN + − γ Tr [ L † + L + ρ ] dt (cid:17) + G [ L − ] ρ (cid:16) dN − − γ Tr [ L †− L − ρ ] dt (cid:17) , L ρ = (cid:88) j =1 , (cid:16) − i (cid:104) ωa † j a j , ρ (cid:105) + γ D [ a † j ] ρ + γ D [ a j ] ρ + γ D [ a j ] ρ (cid:17) ,L ± = 1 √ a ± a ) , D [ L ] ρ = LρL † − (cid:0) ρL † L + L † Lρ (cid:1) , G [ L ] ρ = LρL † Tr [
LρL † ] − ρ, (1)where the natural frequency ω and the decay rates γ , γ , and γ for negative damping, nonlinear damping, and lineardamping, respectively, are assumed identical for both oscillators, N ± are two independent Poisson processes whoseincrements are given by dN ± = 1 with probability γ Tr [ L †± L ± ] dt and dN ± = 0 with probability 1 − γ 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 (cid:126) = 1. The SLH framework [43, 44] has been used to describethe cascade and concatenate connections of the quantum system components in the derivation of the SME (1) [seeAppendix 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 ofphase coherence between the two quantum vdP oscillators, we use the absolute value of the normalized correlator [27] S = | S | e iθ = Tr [ a † a ρ ] (cid:113) Tr [ a † a ρ ]Tr [ a † a ρ ] (2)as the order parameter, which is a quantum analog of the order parameter for two classical noisy oscillators [3]. Themodulus | S | takes values in 0 ≤ | S | ≤ | S | = 1 when the two oscillators are perfectly phase-synchronized and | S | = 0 when they are perfectly phase-incoherent. We also use the argument θ to characterize the averaged phasedifference of the two oscillators in order to distinguish in-phase and anti-phase coherence. We use the negativity N = ( (cid:13)(cid:13) ρ Γ (cid:13)(cid:13) − / ρ Γ represents the partialtranspose of the system with respect to the subsystem representing the first oscillator and (cid:107) X (cid:107) = 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 ].Figures 2(a), 2(b), 2(c), and 2(d) plot the time evolution of | S | , θ , 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 | instantaneouslyincreases after the detection of a photon either at P or M, indicating that phase coherence of the two decoupledoscillators is induced by the conditional photon detection. After the photon detection, | S | gradually decreasesbecause the two oscillators converge to the desynchronized steady state of the SME (1) in the absence of photondetection, 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 | remains always positive. Figure 2(b) shows that θ takes either θ = 0 or θ = π . This indicates that the twooscillators immediately attain in-phase coherence after the photon detection at P or anti-phase coherence after thephoton detection at M. The negativity and purity are shown in Fig. 3(c) and 3(d), respectively, where the negativityis always zero and the purity takes small values between 0 .
03 and 0 .
05, indicating that the system is separable andmixed.The phase coherence of the two oscillators can also be captured by using the Hushimi Q distribution of the phasedifference θ = φ − φ [47] between the two oscillators, Q( θ ), calculated by introducing the two-mode Q distribution [48] Q ( α , α ∗ , α , α ∗ ) = π (cid:104) α , α | ρ | α , α (cid:105) with R j e iφ j = α j ( j = 1 ,
2) and integrating over R , R , and φ + φ .Figures 2(e) and 2(f) show Q ( θ ) of the system states immediately after the first photon detection at the detectors Pand M, respectively. The peak of Q ( θ ) occurs at θ = 0 in Fig. 2(e) and at θ = π in Fig. 2(f), clearly indicating thatin-phase and anti-phase coherence of the two oscillators are induced by the conditional photon detection. P | S | ttt (a)(b)(c)(d) θ π /2 π θ π /2 3 π /2 2 ππ θ π /2 3 π /2 2 ππ (e)(f) Q ( θ ) Q ( θ ) FIG. 2. Results in the weak quantum regime. The parameters are ( ω, γ , γ ) /γ = (0 . , . , .
1) with γ = 1. (a-d): Timeevolution of (a) absolute value of the normalized correlator | S | , (b) average phase value θ , (c) negativity N , and (d) purity P . (e,f): Q distributions Q ( θ ) immediately after the first photon detection at (e) P ( t = 4 .
51) and (f) M ( t = 0 . 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 theevolution of | S | , θ , N , and P , respectively. As shown in Fig. 3(a), | S | takes large values close to 1 immediatelyafter 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 | remains zero until the next photon detection occurs.Similar to Fig. 2(b), Fig. 3(b) shows that θ takes either θ = 0 or θ = π . Thus, the two oscillators becomein-phase coherent after the photon detection at P and anti-phase coherent after the photon detection at M. Remark-ably, Figs. 3(c) and 3(d) show that non-zero negativity and purity with values between 0 . . Q ( θ ) of the system states immediately after the first photon detection at the detectors P and M,respectively. The Q distributions are peaked at θ = 0 and θ = π , clearly indicating that in-phase and anti-phasecoherence of the two oscillators are induced also in this case. P θ | S | tt t (b)(c)(d) (a) t π /2 π θ π /2 3 π /2 2 ππ θ π /2 3 π /2 2 ππ (e)(f) Q ( θ ) Q ( θ ) FIG. 3. Results in the strong quantum regime. The parameters are ( ω, γ , γ ) /γ = (0 . , , .
5) with γ = 1. (a-d): Timeevolution of (a) absolute value of the normalized correlator | S | , (b) averaged phase value θ , (c) negativity N , and (d) purity P . (e,f): Q distributions Q ( θ ) immediately after the first photon detection at (e) P ( t = 16 .
8) and (f) M ( t = 0 . V. QUANTUM LIMIT
From the previous numerical results, it is expected that the maximum quantum entanglement is attained in thequantum limit, i.e., γ → ∞ . In this limit, we can map the quantum vdP oscillator to an analytically tractabletwo-level system with basis states | (cid:105) and | (cid:105) [17], and transform the SME (1) to dρ = L q ρdt + G [ L q + ] ρ (cid:16) dN + − γ Tr [ L q † + L q + ρ ] dt (cid:17) + G [ L q − ] ρ (cid:16) dN − − γ Tr [ L q †− L q − ρ ] dt (cid:17) , L q ρ = (cid:88) j =1 , (cid:0) − i (cid:2) ωσ + j σ − j , ρ (cid:3) + γ D [ σ + j ] ρ + (2 γ + γ ) D [ σ − j ] ρ (cid:1) , L q ± = 1 √ σ − ± σ − ) , (3)with σ − j = | (cid:105)(cid:104) | j and σ + j = | (cid:105)(cid:104) | j representing the lowering and raising operators of the j th system ( j = 1 , | (cid:105) γ −→ | (cid:105) γ −→ | (cid:105) can be regarded as | (cid:105) γ −→ | (cid:105) when γ → ∞ .The steady state of Eq. (3) without detection, i.e., dN ± = 0, can be analytically obtained, which is given by adiagonal matrix ρ pre = diag ( ρ pre , ρ pre , ρ pre , ρ pre ) with ρ pre = ( k − √ k + 2 k + 9 + k − k + 92 k ,ρ pre = 3 √ k + 2 k + 9 − k − k ,ρ pre = − ( k + 3) √ k + 2 k + 9 + k + 4 k + 92 k . (4)Note that only a single parameter k = γ /γ 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 ρ pos ± = ρ pos | (cid:105)(cid:104) | + ρ pos (cid:18) | (cid:105) ± | (cid:105)√ (cid:19) (cid:18) (cid:104) | ± (cid:104) |√ (cid:19) (5)with ρ pos = − √ k + 2 k + 9 + k + 9 k (cid:0) √ k + 2 k + 9 − k − (cid:1) ,ρ pos = ( k + 3) √ k + 2 k + 9 − k − k − k (cid:0) √ k + 2 k + 9 − k − (cid:1) . (6)Using this result, we can explicitly calculate the normalized correlator S and the Q distribution of the phasedifference between the two oscillators.In this case, the correlator S of the states ρ pos ± immediately after the photon detection always takes S = ± k (and then quickly decays). The Q distribution for ρ pos ± can also be calculated as (similarcalculation for the Wigner distribution of the phase difference has been performed in [17]) Q ( θ )[ ρ pos ± ] = 12 π ± ρ pos cos θ . (7)These results qualitatively agree with the corresponding results in the strong quantum regime in Fig. 3. It is notablethat the dependence of the phase coherence on k can be captured by the peak height of Q ( θ ) but not by the normalizedcorrelator S in the quantum limit. Indeed, the element ρ pos | (cid:105)(cid:104) | in Eq. (5) affects Q ( θ ) (through ρ pos = 1 − ρ pos )in Eq. (7), whereas it does not affect the value of S .The above result indicates that the degree of phase coherence is better quantified by the peak height of Q ( θ ) ratherthan S in strong quantum regimes. This is because S is defined as a quantum analog of the order parameterfor the coherence of classical noisy oscillators, which is quantitatively correct only in the semiclassical regime. Thisobservation is also important in interpreting the results in the weak and strong quantum regimes shown in Figs. 2and 3, where Q ( θ ) in the weak quantum regime (Fig. 2) are more sharply peaked than those in the strong quantumregime (Fig. 3), while | S | in Fig. 2 takes smaller values than that in Fig. 3. Thus, S may not work well forcomparing phase coherence between different quantum regimes.In the quantum limit, the symmetric superpositions | S (cid:105) = ( | (cid:105) + | (cid:105) ) / √ | A (cid:105) = ( | (cid:105) − | (cid:105) ) / √ Q ( θ )[ | S (cid:105)(cid:104) S | ] = π + cos θ and Q ( θ )[ | A (cid:105)(cid:104) A | ] = π − cos θ are peaked at θ = 0 and θ = π , respectively. As | A (cid:105) and | S (cid:105) are dark stateswith respect to L q + and L q − , i.e., L q + | A (cid:105) = 0 and L q − | S (cid:105) = 0, the photon detection at the detector P annihilates theanti-phase-synchronized state | A (cid:105) and creates the in-phase synchronized state | S (cid:105) with S = 1 ( θ = 0), while thephoton detection at the detector M annihilates | S (cid:105) and creates | A (cid:105) with S = − θ = π ).Figures 4(a), 4(b), and 4(c) show the dependence of the elements ρ pre and ρ pos and Q ( θ )[ ρ pos + ] on k , respectively(we only plot Q ( θ )[ ρ pos + ] because Q ( θ )[ ρ pos − ] = Q ( θ + π )[ ρ pos + ]). As shown in Fig. 4(a), ρ pre and ρ pre take larger valueswhen k is smaller. When k → ρ pre and ρ pre approach the supremum values, ρ pre → and ρ pre → ( ρ pre → ),corresponding to the completely incoherent steady state of the two decoupled quantum vdP oscillators in the quantumlimit, i.e., ρ pre → ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) ⊗ ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ), where Q ( θ ) is uniform [16, 17]. Therefore, ρ pos approachesthe supremum value, ρ pos → ( ρ pos → ), as shown in Fig. 4(b), and Q ( θ )[ ρ pos + ] exhibits the maximum peak as shownin Fig. 4(c), indicating that the maximum phase coherence is obtained. In the opposite limit, k → ∞ , ρ pre convergesto the two-mode vacuum state ρ pre → | (cid:105)(cid:104) | , i.e., ρ pre , ρ pre → ρ pre → ρ pos → ρ pos →
1) andthe uniform distribution Q ( θ )[ ρ pos + ] → π . ρ ρ ρ ρ ρ kk θkk k ρ , , ρ , C P π /2 3 π /2 2 ππ k → k =50 Q ( θ ) (c)(a) (b)(d) (e) (f) preprepre pospos p o s p r e FIG. 4. Dependence of the results on the parameter k = γ /γ of the two-level system in the quantum limit γ → ∞ . (a)Elements of ρ pre . (b) Elements of ρ pos . (c) Q ( θ )[ ρ pos + ] distributions of the phase difference of two oscillators for k → k = 0 . , , ,
10 (gray lines from the red line to the blue line), and k = 50 (blue line) are shown. (d) Concurrence. (e)Negativity. (f) Purity. In addition to the negativity N and purity P , the quantum entanglement of the density matrix ρ pos in Eq. (5)can also be quantified using the concurrence [49], C = max (0 , λ − λ − λ − λ ), where λ , λ , λ , and λ are thesquare roots of the eigenvalues of ρ ˜ ρ with ˜ ρ = ( σ y ⊗ σ y ) ρ ∗ ( σ y ⊗ σ y ) in decreasing order. The concurrence C takes anon-zero value when the two oscillators are entangled with each other ( C ∈ [0 ,
1] by definition).Figures 4(d), 4(e), and 4(f) show the dependence of C , N and P on k for ρ pos ± , respectively. Note that C , N , and P take the same values for both ρ + and ρ − . In the limit k → C , N , and P approach the upper bounds as C → , N → √ − , and P → . In the opposite limit k → ∞ , these values converge as C → N →
0, and P →
1, whichcorresponds to those quantities for the two-mode vacuum states.Photon detection occurs less frequently when k is smaller, because the probability of the photon detection in theinterval dt at detectors P or M is given by k Tr (cid:104) L †± L ± (cid:105) γ dt . Therefore, on average, infinitely-long observation timeis required before the photon detection to approach the upper bounds for the degree of phase coherence and quantumentanglement in the limit k → VI. CONCLUSION
We have analyzed two decoupled quantum van der Pol oscillators and demonstrated that quantum measurementbackaction of conditional photon detection induces instantaneous phase synchronization of the oscillators. In-phaseor anti-phase coherence between the oscillators has been observed instantaneously after the photon detection, whichdecays 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 thequantum limit, we analytically obtained the upper bounds for the increases in the quantum entanglement and phasecoherence.Recently, physical implementations of the quantum vdP oscillator with ion trap systems [16, 17] and optomechan-ical systems [18, 31] have been discussed. The additional linearly coupled bath and photon detectors can also beintroduced [48, 50]. The physical setup considered in the present study does not require explicit mutual couplingbetween the oscillators. Therefore, it can, in principle, be implemented by using existing experimental methods andprovide a method for generating phase-coherent states of quantum limit-cycle oscillators.
Acknowledgments.-
The numerical simulations are performed by using the QuTiP numerical toolbox [51]. Weacknowledge JSPS KAKENHI JP17H03279, JP18H03287, JPJSBP120202201, JP20J13778, and JST CREST JP-
Concatenate Connection(a) (b)Cascade Connection
G G G G G G
FIG. 5. (a) Cascade and concatenate connection of the two system components G and G . MJCR1913 for financial support.
Appendix A: SLH framework
In this Appendix, we derive the SME (1) using the SLH framework to describe cascade and concatenate connectionsof the quantum system components [43, 44]. In this framework, the parameters in the time evolution of a quantumsystem ρ are specified by G = ( S , L , H ), with S = S · · · S n ... ... ... S n · · · S nn , L = L ... L n , (A1)where S is the scattering matrix with operator entries satisfying S † S = SS † = I n , L is a coupling vector withoperator entries, and H is a self-adjoint operator referred to as the system Hamiltonian. We denote by I n an identitymatrix with n dimensions.With these parameters, the time evolution of the system obeys the master equation dρdt = − i [ H, ρ ] + n (cid:88) i =1 D [ L i ] ρ, (A2)where S is involved in the calculation of the cascade and concatenation products and has an important role indetermining the forms of H and L of the whole network system consisting of the system components. This specificationof parameters is based on Hudson-Parthasarathy’s work [52].The cascade product (Fig. 5(a)) of G = ( S , L , H ) and G = ( S , L , H ) is given by G (cid:47) G = (cid:18) S S , L + S L , H + H + 12 i (cid:16) L † S L − L † S † L (cid:17)(cid:19) , (A3)and the concatenation product (See Fig. 5(b)) of G and G is given by G (cid:1) G = (cid:18)(cid:18) S S (cid:19) , (cid:18) L L (cid:19) , H + H (cid:19) . (A4)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 DPj as the parameters of the j th quantum vdP oscillator with an additional linearly coupled bath G QV DPj = I , √ γ a † j √ γ a j √ γ a j , ωa † j a j . (A5)The concatenate connection of G QV DP and G QV DP is G QV DP (cid:1) G QV DP = I , √ γ a † √ γ a † √ γ a √ γ a √ γ a √ γ a , (cid:88) j =1 , ωa † j a j , (A6)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 ofthe two baths are G BS = I O O (cid:32) √ − √ √ √ (cid:33) , , , (A7)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 G QV DP (cid:1) G QV DP (cid:47) G BS = I O O (cid:32) √ − √ √ √ (cid:33) , √ γ a † √ γ a † √ γ a √ γ a √ γ a − a √ √ γ a + a √ , (cid:88) j =1 , ωa † j a j . (A8)Using transformation D [ a + a √ ] ρ + D [ a − a √ ] ρ = D [ a ] ρ + D [ a ] ρ , the quantum master equation (A2) with theparameters given in Eq. (A8) gives dρ = L ρdt of the SME (1). Then, using the quantum filtering theory [53, 54],SME (1) can be obtained. [1] A. T. Winfree, The geometry of biological time (Springer, New York, 2001).[2] Y. Kuramoto,
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