Enhancement of quantum synchronization in optomechanical system by modulating the couplings

We study two coupled optomechanical systems interact mutually through an optical fiber and a phonon tunneling, which are controlled by two switches K1 and K2. Compared to the quantum synchronization of the optomechanical systems without modification, we find that a proper periodic modification by different switches setups can achieve a better quantum synchronization. In addition, we also analyze the robustness of the periodically modified system against the bath’s mean temperature or the oscillators’ frequency difference, respectively.


Introduction
As a very fascinating phenomenon in classical physics, synchronization has played an important role in the research on many fundamental issues, e.g. neuron networks [1,2], chemical reactions [3], heart cells [4], etc. With the progress in communications technology, physics system is required to ensure a higher degree of synchronization in order to guarantee the reliable information transmission. Therefore, the synchronization has been widely applied in the field of control and communication.
According to the previous studies, there are two ways to realize the synchronization between the two mechanical oscillators. One way is the mutual exchange of energy between the two oscillators [5,35,36], and the other way is controlling the system by external fields [35][36][37][38][39]. The two mechanical oscillators will trend to synchronize after a long period of time and synchronization is very sensitive to many parameters in optomechanical systems, such as coupled intensity, driving field, temperature and so on. In particular, it has been found that the quantum synchronization can be enhanced by periodically modulating some parameters of the system [40,41]. Farace et al proposed that periodic modifications can be applied to improve the quantum effects in optomechanical systems [40]. In addition, Mari et al and Chen et al proved that modifications can also enhance the quantum entanglement in optomechanical systems, respectively [42,43]. Therefore, we want to study whether quantum synchronization can be enhanced by time-periodic modulations.
In this paper, based on the previous work [44] which used the Mari's method, we study the quantum synchronization controlled by different logical relationships of two switches (open or closed) of coupling between two mechanical oscillators. We realize two cases by controlling the switches: one is that the two mechanical oscillators are indirectly coupled [37,45,46] and the other is that the two mechanical oscillators are Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. directly coupled [5,35,36]. By including the periodically modulating cavity detunings, the quantum synchronization can be greatly enhanced.
This paper is organized as follows. In section 2, we introduce the model and the methods, especially the method of measuring the quantum synchronization. In section 3, periodic cavity-modulation schemes of two cases are proposed and the quantum synchronization degree is analyzed. Furthermore, we compare the stability of cavity-modulation schemes by studying the robustness of the bath's mean temperature and the eigenfrequency difference. Conclusions are given in section 4.

Model and methods
In order to show the enhancement of the quantum synchronization [37] via the periodic modification, we choose a controlled quantum synchronization model which is designed based on the quantum optomechanical system as shown in figure 1. Here, we focus on a single-mode electromagnetic field coupling to the mechanical motion of the moving mirror via the radiation-pressure coupling in each cavity. Without loss of generality, we assume that the two optomechanical cavities are identical and driven by the same lasers with intensity E. The couplings between the two optomechanical systems are realized by a fiber with coupling constant λ [36,45] and a phonon tunnel with intensity μ [5,35,36], which are controlled by the switches K 1 and K 2 [35], respectively. Therefore, the quantum synchronization control can be realized through different logical relationships of the two switches K 1 and K 2 . The switches K 1 and K 2 can change λ and μ values from zero to a positive constant by turning the switches on and off [35]. To better the performance of the scheme, periodic modulations on the eigenfrequency of each cavity [47] are introduced.
In the rotating frame, the Hamiltonian of this model can be written as (ÿ = where Δ j =ω L −ω cj refers to the detuning of the laser driving to the cavity mode [5,[35][36][37], and ω mj are the mechanical eigenfrequencies assumed to be slightly different with each other. The operators a j † and a j are the creation and annihilation operators for the optical field, satisfying a a , ( j = 1, 2). q j and p j are dimensionless position and momentum operators of the j-th mechanical oscillator satisfying q p i , g is the optomechanical coupling constant. The piezoelectric transducer can convert the electrical signal to mechanical vibration. We can use it to achieve the periodic modifications of cavity lengths, so as to realize the periodic modifications on the cavity frequencies and the detunings [47]. ε j is the amplitude of modulations we appended on cavities, and Ω j is the frequency of the modulations correspondingly. In calculation, the damping rates and the intensity of the driving field are assumed to be equal in both cavities, while the frequencies and initial conditions of the two cavities are different. Compare to the former relevant work [5], we properly reduce the intensity of the driving field in order to highlight the coupling function in synchronization, since a toostrong driving field will weaken the coupling effect and lead to the 'forced' synchronous effect.
Next, we take dissipation effects into consideration. The quantum Langevin equations (QLE) of our system in Heisenberg picture [48,49] can be written as Here, κ is the decay rate of the optical cavities, and m g represents the damping rate of the mechanical oscillators which is inversely proportional to quality factor Q a . j in is the radiation vacuum input noise operator which satisfies the autocorrelation function a t a t a t a t t t ] is the mean phonon number of the mechanical bath which is determined by the environment temperature T [50][51][52].
Since it is difficult to solve equation (2) analytically, we introduce the mean-field approximation to simplify the calculations by rewriting the system variable operators as the sum of a c number mean value and a small fluctuation near the mean value, i.e.
Because of the presence of random noise, the mean values are much bigger than the fluctuations. Therefore, by substituting equation (3) into (2), the QLE equation can be approximately divided into the zero-order mean value terms and the first-order quantum fluctuation terms. Then we get two different sets of equations. One for the expectation values reads, And one for the quantum fluctuation reads,    With the evolution of the expectation values and the quantum fluctuations, we next study the quantum synchronization in the mean field treatment. The error operators q t -( ) and p t -( ) of the position and momentum operators operators q t j ( ) and p t j ( ) between the two mechanical oscillators can be defined as follows, It is easy to find that the quantum complete synchronization [5] will be realized when both q t -( ) and p t -( ) asymptotically vanish as they evolve. However, these two error operators can not have zero values simultaneously due to the Heisenberg uncertainty relation. Hence, the quantum complete synchronization has a bound which is distinct from the classical synchronization. For this reason, we introduced the following figure of merit [5] based on S t q t p t 11 to measure the quantum complete synchronization. This is a valid metric for genuine quantum synchronization since both of the influence of quantum fluctuation and nonlocal quantity are considered in this synchronization measure. The classical synchronization (even in quantum system) and genuine quantum synchronization can also be effectively distinguished. Following the Heisenberg uncertainty relation q t p t which sets a universal limit for the complete synchronization of the two CV systems. While, for a classical theory, S c does not have this upper bound. In practice, a small value of S t c ( ) can have two possible origins: the mean values of q t -( ) and p t -( ) are not exactly zero which is the same as the classical synchronization, or the fluctuations of such operators are large. If we want to investigate only the quantum effects on the synchronization, the error operators can be redefined as the relative ones: Obviously, the bound of equation (12) still holds for this relative measure which has the range S t 0, 1 ]. In our mean-field treatment (3), the relative error operator (13) takes the form Since the quantum fluctuation part of the QLE is stochastic, we introduce the correlation matrix to calculate the quantum fluctuations of the system variables and obtain the value of the synchronization measure. The fluctuations in the stable regime will also evolve to an asymptotic zero-mean Gaussian state. The covariance matrix we use here is defined as Its evolution satisfies the linear ordinary differential equation [53,54] C SC CS N, is the noise correlation satisfying N t t n t n t n t n t Hence the synchronization measure can be written as By numerically solving equations (4), (6), (17), the value of the synchronization measure S t c ¢ ( ) can be derived. Besides, we also calculate the time-averaged synchronization measure to include different parameters setups. It can be verified that the system is asymptotically stable according to R−H criterion [55], since all the eigenvalues of the coefficient matrix S will be negative after a temporary evolutionary process. Hence there will be a stable limit cycle solution representing a periodic oscillation.

Main results
We firstly discuss the quantum synchronization of the system without any modification. In our model, the two coupled optomechanical systems can be modulated by the switches K 1 and K 2 . Thus, there are four cases as follows: (a) K 1 and K 2 are both open at the same time. In this case, the quantum synchronization can not be generated because there is no coupling between the two optomechanical systems [35]. (b)K 1 is closed and K 2 is open. In this case, our model can be simplified to two optomechanical systems that interact through an optical fiber [37,45,46] and the two mechanical oscillators are indirectly coupled. The quantum synchronization is not good whatever the value of parameter λ is [45]. (c)K 1 is open and K 2 is closed. In this case, our model can be reduced to two optomechanical systems interacting through a phonon channel and the two mechanical oscillators are directly coupled, which is just the model Mari proposed [5]. The quantum synchronization is also not perfect, S t c ( ) can only reach 0.15 at the best [5,35,36]. (d) K 1 and K 2 are both closed at the same time. The quantum synchronization can be improved, but it is still not good enough (The maximum value of S p is not more than 0.15 in what Li et al studied) and the process is too fussy [35].
Therefore, we focus on the cases (b) and (c), and add periodic cavity-modification on each optomechanical system to improve the degree of the quantum synchronization. The piezoelectric transducer can convert the electrical signal to mechanical vibration. We can use it to achieve the periodic modifications of cavity lengths [47], so as to realize the periodic modifications on the cavity frequencies and the detunings. In the following discussion, we choose the parameters that can be achieved experimentally [44] and we will explore whether the quantum synchronization of the system can be further improved by periodically modifying the cavities.

Modulation of cavity detunings with K 1 closed
For the situation that K 1 is closed and K 2 is open, the periodic modifications to the two cavities are applied at the same time , ). As shown in figure 2, we plot mean value of quantum synchronization S t c ( ) as a function of ε and Ω for a double cavity-mode modulation. The most parameter regions of S t c ( ) is low; conversely, the regions where the cavity-modulation frequency Ω is an integral multiple of m1 w (such as 2 or 3) [35,40] are so different with the increase of the cavity-modulation strength ε. The reason is that energy can be transferred from external modulations to mechanical oscillators more simply and the peak positions may change from Ω=2, 3 to other integers depending, e.g., on the value of ε [35]. Besides, when Ω exceeds a critical value, the modulation effect may fail suddenly, i.e., S t c ( ) become invariant [35]. It can be found that when Ω=3, ε=0.9, S t c ( ) has the maximum value around 0.4, the degree of quantum synchronization has been greatly improved than previous works.
To further prove our conclusion, we examine in figure 3. As shown in figure 3(a), we fix Ω=3, ε=0.9 and show the solution of equation (4) for the mean values Q t ( ) and P t ( ) of the two mechanical oscillators' position and momentum. The evolution tends indeed to an asymptotic periodic orbit and the two limit cycles tend to be consistent. From figures 3(b) and (c), we can see that the system reaches the stable state after a transient Figure 2. Evolution of the mean value of the quantum synchronization S t c ( ) versus the cavity-modulation frequency Ω and the cavitymodulation strength ε. We set ω m1 =1 as a reference unit of frequency, the other physical parameters that have been used in the simulation are ω m2 =1.005, Δ j =ω mj , g=0.005, μ=0, λ=0.05, κ=0.15, E=100, and γ m =0.005.
evolution. The evolution of the mean values Q 1 (t) and Q 2 (t) tends to be coincident as shown in figure 3(b). Meanwhile, the evolution of the mean values P 1 (t) and P 2 (t) also gradually tends to be coincident as shown in figure 3(c). From figure 3(d), we can see that the system reaches a steady state finally and the value of S t c ( ) is around 0.4. To sum up, we found that the degree of quantum synchronization can be enhanced by introducing the periodic cavity-modifications and choosing appropriate parameters.

Modulation of cavity detunings with K 2 closed
Now we consider that switch K 1 is open and the switch K 2 is closed, the two coupled optomechanical systems interact mutually through a photon channel. The quantum synchronization of the systems without applying any modification is not very good. As discussed in the previous section 3.1, we apply the same periodic modifications to the two cavities , ). As shown in figure 4, the most of the parameter regions are blue and the mean value of the quantum synchronization S t c ( ) is low. However, one common feature of situation (b) and situation (c) is that optimal quantum synchronization occurs when Ω is an integral multiple of m1 w (such as 2 or 3) [35,40] before a critical value. It can be found that when Ω=3, ε=0.9, S t c ( ) has its maximum value around 0.6, the quantum synchronization degree has greatly improved than previous work.
We examine in figure 5 to further prove our conclusion. As shown in figure 5(a), we set Ω=3, ε=0.9 and show the evolution of the mean values Q t ( ) and P t ( ) which are position and momentum of the two oscillators, respectively. The evolution trends to an asymptotic periodic orbit (i.e. the two limit cycles tend to be consistent).  From figures 5(b) and (c), the system reaches the stable state after the transient evolution, because the evolution of Q 1 (t) and Q 2 (t) gradually tends to be coincident and the evolution of P 1 (t) and P 2 (t) does the same. From figure 5(d), we can see that the system reaches a steady state in the end and S t c ( ) rises toward the value around 0.6, which is obviously a better result. From the above discussions, we found that the degree of quantum synchronization can be enhanced by introducing the periodic cavity-modifications and choosing appropriate parameters.

Comparison of the two cases
According to the previous study, we apply the same periodic modulations to the cavities in two cases and the quantum synchronization has been greatly improved. As we known, it is important for a good modification scheme to keep the synchronized system relatively stable with the increase of the bath's mean temperature T and the oscillators' eigenfrequency difference m D . In other words, we want to measure the robustness of the modulation scheme we proposed. It is significant to examine how S t c ( ) reduces until negligible, because a large decrease of S t c ( ) may be resulted from a slight increase T and m D .
We plot S t c ( ) versus the bath's mean temperature T in figure 6 and the oscillators' eigenfrequency difference m D in figure 7 for the optimal modulations in both cases. That is, each point represents the maximal value of S t c ( ), for a given value of T and m D , obtained by choosing the optimal values of ε and Ω. As T is gradually increased from k b /ÿω m1 to 10k b /ÿω m1 (k b is the Boltzmann constant) shown in figure 6(a), the two curves show a gradual downward trend and the degree of decline is different. It is obvious that the trend of decline in the  w as shown in figure 6(b), the trend of decline in the second case is still larger. It is clear that the optimal modulation in the first case results in a better quantum synchronization than that in the second case with the increase of T.
As shown in figure 7 figure 2). Now we compare the system stability in the two cases by changing m D and find that the difference of two lines is distinct.
Though the downtrend of S t c ( ) in both cases is small when m D is increased from 0 to 0.01 . However, the trend of decline in the second case is larger when we increase S t c ( ) from 0.01 to 0.025. Especially, when Δ m is 0.025, S t c ( ) is about 0.068 in the second case and it is almost impossible for the system to achieve synchronization at that time. Conversely, S t c ( ) declines slightly in the first case. Only when Δ m is greater than 0.025 or more, there will be a large decline. Therefore, It is obvious that the optimal modulation in the first case results in a better quantum synchronization than that in the second case with the increase of m D .

Conclusion and disscussion
In summary, we have studied a coupled optomechanical system in which the couplings are controlled by two switches and analyzed the different scheme of the switches' setups. The degree of quantum synchronization in the system without any modification is found to be bad. After detailed analysis and comparison, we find that appropriate modulations on cavity detunings can enhance the quantum synchronization in our system. Besides, two types of switches setups with periodic modification are compared. An examination of the robustness of S t c ( ) against T and m D shows that the first case (K 1 off and K 2 on) is more appealing in achieving a preferable quantum synchronization behavior than the second case (K 1 on and K 2 off). Compare to the former studies [35] (as K 1 is closed and K 2 is open) and [5](as K 1 is open and K 2 is closed), the periodically modulating cavity detunings greatly enhance the quantum synchronization and are robust against thermal fluctuations and the eigenfrequency difference. All in all, we believe that our scheme is effective and can be used to improve the degree of quantum synchronization under different conditions.

ORCID iDs
H Geng https://orcid.org/0000-0002-9616-8566 H D Liu https:/ /orcid.org/0000-0001-8863-5331 Figure 7. Evolution of the mean value of the quantum complete synchronization S t c ( ) versus the eigenfrequency difference of the mechanical oscillators Δ m =ω m2 −ω m1 for an optimal modulation in both caseses. The red line represents the first case and the blue line represents the second case. Relevant parameters expect ε and Ω are the same as in figure 2.