Measure synchronization in hybrid quantum–classical systems

Haibo Qiu(邱海波), Yuanjie Dong(董远杰), Huangli Zhang(张黄莉), and Jing Tian(田静)

School of Science,Xi’an University of Posts and Telecommunications,Xi’an 710121,China

Keywords: measure synchronization,quantum measure synchronization,hybrid quantum–classical systems

1. Introduction

Synchronization is ubiquitous in nature. It appears in ecosystems, physical systems, chemical systems, and even social behavior, etc.[1–4]Up to now, various types of synchronization have been found. e.g.,phase synchronization,[5]complete synchronization,[6]lag synchronization,[7]projective synchronization,[8]and measure synchronization.[9]In the past decade,research interests of synchronization has been extended into quantum regime,and quantum synchronization has emerged as a new research field.[10–13]It has been shown that quantum synchronization is often related to quantum entanglement, or other kinds of quantum correlations. However,there are no general relations revealed yet,e.g.,in some cases,the quantum synchronization can manifest only in the classical correlations rather than quantum correlations.[14]

In 1999, the concept of measure synchronization (MS)was proposed by Hampton and Zannett for coupled classical Hamiltonian systems.[9]They revealed a new type of synchronous behavior arising in coupled classical Hamiltonian systems,of which the coupled systems experience a synchronization transition,from a state in which the subsystems visit different phase space regions to a state in which their orbits cover the same region of the phase space with identical invariant measures. Up to now,classical MS behavior have been revealed in different model systems,i.e.,φ4model,[15]Duffing model,[16]coupled bosonic Josephson junction[17,18]and more recently in non-dissipative Huygens’pendulum.[19]The classical concept of MS has been extended into quantum regime.We have revealed that quantum MS appears in coupled quantum many body systems,and quantum MS will manifest itself in a purely quantum mechanical way, i.e., collapses and revivals dynamics will get synchronized once the system reaching quantum MS.[20]More recently, quantum MS has also been found in a pair of Harper systems[21]and in between two scattering modes in Bose–Einstein condensates.[22]

Accompanied by the trend of extending classical concept of synchronization into quantum regime, there are also efforts in exploring intermediate regime in between classical and classical synchronization, i.e., quantum synchronization in semi-classical Kuramoto model,[23]and synchronization in quantum van de Pol oscillators.[24]Moreover,synchronization arising in coupled quantum–classical systems had been explored by one of the authors. Synchronous dynamics which termed as hybrid synchronization had been revealed based on the time correlation manifestations.[25]This kind of synchronization can be very strong, i.e., if the coupled classical system and quantum system have the exact correspondence in its initial conditions, the synchronization phenomenon can be as strong as complete synchronization.[25]

In this paper,we extend the investigation of measure synchronization into intermediate regime of quantum and classical dynamics,by exploring MS in a new scenario,of a system with coupled quantum–classical dynamics. This paper is organized as follows. Section 2 describes the coupled classicalquantum systems. Section 3 presents the results and an analysis of MS in the system. Finally,the results are summarized in Section 4.

2. Coupled quantum and classical Hamiltonian systems

In our previous investigation,[25]we have introduced a hybrid quantum–classical systems, which consists of one quantum Hamiltonian system ˆHacoupled with one classical Hamiltonian systemsHb. The coupled classical and quantum model systems can be written as ˆH= ˆHa+Hb+ ˆHabwith

where ˆa†L(R)(ˆaL(R)) are creation (annihilation) operators for theLorRmodes of the quantum subsystem.nb,L(R)andφbare dynamical variables for classical subsystem which denote the particle number ofL(R) mode and phase difference ofb. The Hamiltonian includes tunneling terms,proportional toJa(b),which in absence of any interaction induce periodic Rabi oscillations of the populations between the states. The contact interaction translates into terms with strength proportional toUa,UbandUab, which gauge theaa,bbandabcontact interactions. Notice ˆHabis the coupling term, which involve both classical and quantum variables, withUabdenoting the coupling strength in between the quantum subsystem and the classical subsystem.

To study the coupled quantum–classical dynamics, we solve the following set of equations. For quantum subsystem,the dynamics is governed by the time dependent Schr¨odinger equation as

where we have introduced the dimensionless ratios,Λa ≡NaUa/(2Ja),Λb ≡NbUb/(2Jb), and the new coefficient for coupling strengthΛab ≡NaUab/(2Jb).

The time dependence of ˆHstems from the time dependence ofzb,which can be seen as a classical parametric driving for theasubsystem. At the same time, we include the feedback effects ofaonbthrough the last term of Eq.(3). It is worth to note that the effect of the quantum fluctuations of ˆzais neglected in evaluating the dynamical evolution ofφb,as we approximate ˆzaby its expectation value.

Considering the above mentioned reciprocal influence in between quantum and classical subsystems, we find that the solutions of Eqs.(2)and(3)are numerically conserve the total energy of the hybrid system in all the calculations reported in this paper. Conservation of the total energy also implies transfer of energy between the quantum and classical subsystems.

For numerical simulation of the quantum subsystema, we employ the Crank–Nicolson method.[26]We choose Fock basis of theNa+1 dimensional space,|na,L,na,R〉={|Na,0〉,...,|0,Na〉},and the most generalaquantum state is written as

3. Measure synchronization

As a special synchronization phenomenon in coupled Hamiltonian system, measure synchronization was originally defined only for coupled identical Hamiltonian systems.[9]Being identical, the only difference in between the coupled Hamiltonian systems are their initial conditions.[9,15–20]Even though for coupled quantum Hamiltonian and classical Hamiltonian systems, ˆHaandHb, the two Hamiltonian systems are intrinsic different. Here we shall relax the requirement by setting the classical Hamiltonian system and quantum Hamiltonian system in exact correspondence except for their initial conditions. We setNa=Nb=30,Λa=Λb=0.1,Ja=Jb=1,and all the parameters with quantum–classical correspondence are the same. However,the initial conditions will be different for classical system and quantum system,here we choose initial conditions of quantum and classical systems without exact correspondence,i.e.,we choose〈za〉=0.6,whereaszb=0.1,and〈φa〉=φb=0. Moreover, for quantum system, we will choose the initial state as a coherent state, which is the most classical-like quantum state. It reads

Here cos(θa)=〈ˆza〉,as mentioned above that we have chosen〈ˆza〉=0.6,so we need to setθa=arccos(0.6)accordingly.

Our results reveal that measure synchronization(MS)occurs in coupled quantum–classical dynamics. The transition from non-MS states to MS states with increased couplingΛabis shown in Figs. 1(a)–1(d). Due to the difficulty in defining a proper phase space for the quantum system in correspondence with the classical phase space, we adopt the method

Fig. 1. Measure synchronization in hybrid quantum–classical systems characterized by the phase space domain covered during the evolution of each subsystem in the 3D phase space defined by pseudoangular momentum (jx,jy,jz). (a)Λab =0; (b)Λab =0.005; (c)Λab =0.01;(d)Λab =0.75. Here za(0)=0.6,zb(0)=0.1,θa(0)=θb(0)=0, andNa=Nb=30.

ForΛab=0, as shown in Fig. 1(a), there exists two distinctive phase space domains,one in green representing classical phase space domain and the other in red representing quantum phase space domain. It is worth noting that the classical phase space domain consists of a periodic orbit, meaning the classical dynamics is in periodic motion forΛab=0. However, for quantum dynamics, it is not a periodic motion.[20]As coupling strengthΛabincreases, the two phase space domains will come closer, as shown in Figs. 1(b) and 1(c) forΛab=0.005 and 0.01,respectively. As coupling strengthΛabincreases further,both phase space domains completely overlapped [e.g., Fig. 1(d) forΛab=0.75], which indicates occurrence of MS.[9,20]

The transition process from non-MS to MS is shown in Fig. 2, in which we plot the average energiesEa/(NaJa) and

whereEa(t)=〈Ψ(t)| ˆHa|Ψ(t)〉, andEb(t)=Hb, with|Ψ(t)〉the evolved quantum state.

Fig. 2. Average energiesEa(b)/(Na(b)Ja(b)) versus Λab, with N =30.Red for average energy of the quantum system, and green for average energy of classical system. In the insets we depict Ea(t)/(NaJa) and Eb(t)/(NbJb) for two specifci values of Λab =0.01 and Λab =0.75.The green and red dotted lines correspond to the classical prediction of Ref.[17].

The MS transition in coupled classical Hamiltonian systems is discontinuous as indicated by the average energy calculations.[15–17]This discontinuity allows us to define a critical point to characterize the MS transition. However, for the hybrid quantum–classical systems, the MS transition is continuous,which indicates the transition to MS is a crossover behavior. It is worth mentioning that MS transition in coupled quantum Hamiltonian systems is also a crossover behavior.[20]Besides, we have found that by increasing particle numbersN=Na=Nbwhile fixing other parameters and initial conditions, MS will occur at a smaller value ofΛab(e.g., forN=300,andΛc ≈0.27).And similar to MS transition in coupled classical Hamiltonian systems,extensive numerical simulations confirm that initial conditions have strong impact on MS transitions in hybrid quantum–classical systems. In general,more widely different choice of initial conditions for the two subsystems result in a larger value ofΛcfor the onset of MS.

The physical mechanism behind MS transition in coupled quantum–classical dynamics is revealed in Fig. 3. The condensed fractionn1versus time for three different coupling strength are presented in Fig. 3(a), withΛab= 0,0.01, and 0.75. We note thatn1(t) will be closer to its initial value 1, as coupled strength increased. At initial time,n1= 1 corresponds to the initial quantum state as a coherent state,which is the most classical-like of all quantum states. So in time evolution,n1(t)≈1 means quantum evolution is essentially coherent. The results indicate that as coupling strength increases, the quantum subsystem will behave more classically. Figure 3(b) shows the dispersion of population imbalanceσ2z=〈ˆz2a〉−〈ˆza〉2versus time for different coupling strength. It is found that with increased coupling strength,the dispersion is suppressed gradually,and it is much more closer to 0 as the coupled quantum–classical dynamics reaches MS.This indicates that as coupling strength in between the quantum subsystem and classical subsystem increases,the quantum fluctuations (as revealed by dispersion) for the quantum subsystem will be greatly suppressed. In summary, we find that the reason behind the appearance of MS in coupled quantum–classical dynamics is the fact that the quantum system will become more classical-like as a result of the coupling effects in between the quantum and classical systems.

Fig.3. Coupling effects on evolution of two relevant quantum properties. Panels(a)and(b)depict the dispersion of the condensed fraction n1 and the population imbalance σ2z, respectively. Λab =0, 0.01, and 0.75 for both panels. All other parameters are the same as in Fig.1.

Fig. 4. Absolute value of the frequency spectrum z(ω) (Fourier transform of the average population imbalances), for different values of the coupling strength Λab =0/0.005/0.01/0.22/0.37/0.75. The frequency spectrum of the quantum result, 〈ˆza〉(ω) is given in the left panels.zb(ω)is depicted in the right panels. The frequency spectra are obtained from time series up to TMax=732tRabi.

To analyze time correlated dynamics behind measure synchronization in hybrid quantum–classical systems,the Fourier decomposition of the evolution of the population imbalance is presented in Fig.5. The frequency spectra of these signals are for different values ofΛab.In the uncoupled case the quantum signal is found to have several peaks around the frequencyω ≃1.04ωRabi.The different peaks reflect the multi-frequency nature of quantum dynamics. In contrast, the classical signal is found to have only one peak, as a result of period motion ofzb(t). As we couple the two subsystems, the spread in theasubsystem disappears and two prominent frequencies appear both foraandb. ForΛab=0.22 andΛab=0.37(fourth and fifth row)the Fourier decompositions of both signals are very similar, showing two large peaks at the same frequencies for the two signals, however, a closer scrutiny show that the amplitudes of the two peaks are a bit different. Keeping increasing the coupling strength, forΛab=0.75, we have the same Fourier structure in both cases, signaling the appearance of measure synchronization. This feature is shared by measure synchronization in two coupled classical Hamiltonian system,

where two subsystems get frequency-locked with very few frequencies being involved.[19]In contrast,for quantum measure synchronization,we have found that the frequency locking involves multiple frequencies.

At last,we investigate the many body effects on the coupled quantum–classical dynamics. Here we fix the coupling strengthΛab=0.4, and only change particle numbers, withN=6/10/20/80. The initial conditions are the same as in Fig. 1. It is found that with increasing particle numbers, the dispersion will be suppressed as a result,as shown in Fig.4(a).Meanwhile, the condensed fraction will be more close to its initial value as particle number increases. Compared with Fig.3,we notice the increased coupling strength and increased particle numbers have similar effects. That is, as commented above, to make the quantum subsystem more classical-like.The two results shown in here are quite natural considering the effects of particle number on quantum many body dynamics of ˆHaall alone. AsN →∞,〈Ψ| ˆHa|Ψ〉can approach its“classical”limit,which in the quantum–classical correspondence,isHb.

Fig.5. Many body effects on evolution of two relevant quantum properties. Panels(a)and(b)depict the condensed fraction n1,and the dispersion of the population imbalance σ2z,respectively. Particle numbers N=6,10,20 and 80 for both panels. The coupling strength Λ =0.4 is fixed. All other parameters are the same as in Fig.1.

In the end, we have shown how the MS has emerged in the coupled quantum–classical dynamics. As a comparison,hybrid synchronization(HS)can also exist in the same model system.[25]The differences between these two synchronization concepts lie in two aspects. Firstly,MS is a synchronization in the sense of spatial coincidence,[28]and it is a special synchronization characterized by phase space domain with invariant measure.[9]On the other hand, HS is a synchronization in the sense of time coincidence, and it is a general term for synchronization characterized by its strong time correlation in coupled quantum–classical dynamics. The time correlations can be as strong as complete synchronization,while in term of time correlation, MS is much weaker compared with complete synchronization. Secondly, there are dramatic difference in setting the initial conditions and parameters for hybrid synchronization and measure synchronization. For hybrid synchronization, we had required the quantum subsystem and classical subsystem to have the initial conditions with exact correspondence,[25]while for measure synchronization,the initial conditions can not have exact correspondence, and must be different.

4. Conclusion

In this paper, we have introduced the concept of measure synchronization in hybrid quantum–classical systems. It is found that MS can be induced by increasing the coupling strength in the hybrid quantum–classical system. However,the MS transition being found is a crossover behavior rather than a critical behavior. Similar feature also appears in quantum measure synchronization.[20]For the physical reason behind MS, we have revealed that as coupling strength in the hybrid quantum–classical systems increase,the quantum subsystem will behave more classical-like. And this effects leads to measure synchronization mimicking measure synchronization in coupled classical Hamiltonian systems. It is further shown that, with coupling strength increasing, the frequency spectra of the MS states share similar results as classical MS states.

Acknowledgements

Project supported by the National Natural Science Foundation of China (Grant No. 11402199), the Natural Science Foundation of Shaanxi Province,China(Grant Nos.2022JM-004 and 2018JM1050), and the Education Department Foundation of Shaanxi Province,China(Grant No.14JK1676).