# Criterion of quantum synchronization and controllable quantum synchronization based on optomechanical system

## Abstract

We propose a quantitative criterion to determine whether the coupled quantum systems can achieve complete synchronization or phase synchronization in the process of analyzing quantum synchronization. Adopting the criterion, we discuss the quantum synchronization effects between optomechanical systems and find that the error between the systems and the fluctuation of error are sensitive to coupling intensity by calculating the largest Lyapunov exponent of the model and quantum fluctuation, respectively. Through taking the appropriate coupling intensity, we can control quantum synchronization even under different logical relationship between switches. Finally, we simulate the dynamical evolution of the system to verify the quantum synchronization criterion and to show the ability of synchronization control.

###### pacs:

75.80.+q, 77.65.-j^{3}

## I Introduction

In recent years, synchronization effects of two or more interconnected classical systems have aroused comprehensive attention because synchronization phenomena are found widely in nature. For examples, Huygens found that two clocks with different swings at the initial time will appear synchronization with time evolvement; it was also observed that the fireflies glow synchronously and the oscillation of heart cells in human or animal will keep in step with each other. At the same time, synchronization effects exhibit inimitable application potential in many fields, such as the synchronous transmission of information in the Internet, the synchronous transmission and amplification of signals between coupled lasers, the encryption and decryption of signals using chaotic synchronization technology, and so on. To date, the synchronization of classical system has gradually become the investigation focus in the numerous fields. The groundbreaking work on theoretical exploration of classical synchronization is marked by Yamada and Fujisaka who put forward a criterion to judge synchronization behaviors through calculating the Lyapunov exponent of the coupling system and obtained synchronization conditions [1]. After this, Pecora and Carroll found synchronization phenomena in the electronic circuit and designed a circuit scheme of encrypted communications using synchronization techniques, which demonstrates the attractive application prospect of synchronization effects and arouses the intense research interest for synchronization theory and application [2]. Recently, many effective synchronization techniques have been proposed in order to achieve complete or phase synchronization of classical systems [3-7].

Naturally, it is expected to found a similar synchronization phenomenon in quantum systems in order to realize the synchronous transmission of quantum information or states due to its unique advantages of synchronization effects. However, it is difficult to define precisely some concepts which describe synchronization in quantum systems, like “tracks” and “errors”. Even some concepts used in classical dynamics are completely unsuitable to be used in quantum dynamics because of remarkable differences of two kinds of systems. Therefore, the relative research of quantum synchronization once is thought as unfeasible. The optomechanical system, as a representative of mesoscopic systems, has attracted widespread attention and systematic discussion recently [8-11,26]. Mesoscopic systems exhibit simultaneously both properties of classical and quantum system under certain conditions because the scale of the system is in-between macro-system and micro-system. So some phenomena, no matter what chaos behaviors and limit cycle in classical kingdom [12-14,23] or quantum entanglement and quantum coherent in quantum domain [15-18,23], have been observed in optomechanical systems, which provides reliable basis to expand synchronization theory from classical to quantum. At 2013, Mari et al. extended the classical synchronization concepts to the quantum system [19] and developed quantitative theory of synchronization for continuous variable systems evolving in the quantum regime. And in their work, two different measures quantifying the level of synchronization of coupled continuous variable are also introduced. Whereafter, some progress has been made in interrelated theories and experiments [23-25,27].

However in the general case, the synchronization effects are very sensitive to parameters of the systems, such as driving field, coupled intensity, and so on. Therefore, it is expected further to investigate and obtain a quantitative synchronization criterion in order to determine directly whether the synchronization can be realized. Meanwhile, the synchronization criterion can also be regarded as a necessary and sufficient condition of the synchronization effect, which means that the quantum coupling systems can be adjusted and controlled to satisfy the synchronization criterion and to realize the aim of quantum synchronization. Further, the controllability and practicability of quantum synchronization can be improved.

In this work, we present a general method for discussing synchronization effects in mesoscopic quantum systems. We introduce the first order and second order measurements to describe the expectation value and the fluctuation of error respectively and give the necessary conditions to estimate the presence of quantum synchronization effects. Using this theory, we design a model based on optomechanical system to realize logic control of quantum synchronization. Subsequently, we validate the criterion through the simulation.

This paper is organized as follows: In Sec.II, the classical synchronization theory is briefly introduced. In Sec.III, the processing method of quantum mesoscopic synchronization is described and the quantitative criteria for determining quantum complete synchronization and quantum phase synchronization are proposed. In Sec.IV, a controllable quantum synchronization model base on optomechanical system is designed and the phase synchronization effect is discussed. Finally, the summary and the prospects are given in Sec.V.

## Ii Classical synchronization theory

Considering two classical coupled systems

(1) |

where and are state variables of two systems, and are couplings between systems, respectively. If the error when , the complete synchronization between classical systems is realized. If the phases and of and meet , the phase synchronization between the systems is obtained.

The classical synchronization criteria reported previously are mainly to analyze the stability of the error or phase error and to determine whether or can converge asymptotically to zero through calculating the largest Lyapunov exponent. In the next section, we will propose a criterion for quantum synchronization based on the classical synchronization theory.

## Iii Quantum synchronization criterion

In the Heisenberg picture, we use quadrature operators and to describe two coupled quantum systems (here ; ). Hence, the error operators and between the systems can be defined as follows

(2) |

It can be seen from Eq.(2) that the error operators and are physical quantities which describe the differences between the conjugate mechanical quantities of two systems. However, both and cannot be very small simultaneously due to the Heisenberg uncertainty relation. Therefore, it is required to consider synthetically both values of two error operators in the quantum synchronization measurement. For this reason, Mari et al. introduced the following figure of merit based on Eq.(2)

(3) |

Eq.(3) is used to gauge the level of quantum complete synchronization and its value is in the range .

Nevertheless, it is also difficult to discuss quantum synchronization by analyzing directly because it is not easy to give a definite criterion for judging the system synchronization or not. On the other hand, the calculation of is also complex in quantum systems. Fortunately, the mean value approximation is acceptable in a mesoscopic system, meaning that we can write a mesoscopic system operator in the form of , here is the expectation value of the operator at the moment and it can be regarded as a description of “classical properties”. represents the quantum fluctuation of the operator near its expectation value and the quantum effects of the system can be embodied in . For a mesoscopic system, is small but can not be ignored. Then, on the basis of Eq.(2), the error operators of the systems can be rewritten as follow

(4) |

where and .As discussed above, the quantum effects of the systems are embodied in . Hence, it is only needed to consider and in the discussion of the synchronization deviation influenced by quantum effects and Eq.(3) can be rewritten as

(5) |

It should be noted that in Eq.(5) is a measurement of the error operator’s quantum fluctuations near its expectation value. will equal and it is a synchronization measurement only if and . Therefore, we define as a second order measurement to reflect the differences between systems generated by the quantum noise even though the classical synchronization conditions are reached. Correspondingly, and can be defined as a first order measurement of quantum synchronization. Now that and satisfy the classical properties, the stability analysis method through calculating the largest Lyapunov exponent of the system can be used to determine whether and tend stably to zero. Thus, it is only demanded to control the stability of and , that is, we can control mesoscopic quantum synchronization instead of analyzing directly.

Similarly, in the discussion of quantum phase synchronization, the phase error operator can be defined as follow

(6) |

here . We introduce as the measurement of the quantum phase synchronization

(7) |

Similar to the discussion of complete quantum synchronization, let and rewrite Eq.(7):

(8) |

where should be a second order measurement as well under .

In summary, the synchronization effects in mesoscopic quantum systems can be discussed through the following steps:

a. Write the operator equations of system’s conjugate mechanical quantities in the Heisenberg picture, define the error operators and take them as the form of fluctuations near their expectation value, that is, .

b. Make stability analysis for and calculate the largest Lyapunov exponent of the error equations. If the largest Lyapunov exponent is less than zero, the evolution of can tend to zero stably after a certain time, whereas it may be ruleless oscillation.

c. If the largest Lyapunov exponent is less than zero, the following work is to discuss the magnitude of the noise () and to calculate and base on Eq.(5) and Eq.(8), respectively. Oppositely, if and keeps a constant but not zero, the synchronization between the quantum systems is achieved.

## Iv Design of the controlled quantum synchronization model and quantum phase synchronization

A controlled quantum synchronization model is designed based on the quantum optomechanical system in order to check the validity of the above-mentioned quantitative criteria. In this model, we can realize quantum synchronization control through different logical relationship of the switches, shown in Fig.1.

Two coupled optomechanical systems are driven by laser and interact mutually through a phonon tunnel and a fiber which can be controlled by the open or close of the switches and . The Hamiltonian of the system can be given directly after a rotating approximation [20, 21] ().

(9) |

here and are the optical creation and annihilation operators for the system , and are the mechanical creation and annihilation operators. and are the optical detunings and the mechanical frequencies, respectively. is the optomechanical coupling constant and is the laser intensity which drives the optical cavities. is the intensity of the phonon tunnel and is coupling constant of the fiber.The switches and can change and values from zero to a positive constant by on and off. After considering the dissipative effects, the following quantum Langevin equations can be written in Heisenberg picture through the input-output properties

(10) |

here and are the optical and mechanical damping rates. and are the input bath operators, which satisfy and , where

Using the expectation value and quantum fluctuation to replace operators from Eq.(10), we can get following two equations.

The parts of expectation value are

(11) |

where and .

The parts of quantum fluctuation are

(12) |

The dynamic properties of the cavity and oscillator can be described by their own conjugate mechanical quantities, i.e.

(13) |

Substituting Eq.(13) into Eq.(12), Eq.(12) can be expressed in the matrix form

(14) |

where is a vector and means a input vector

. is time-dependent matrix.

(15) |

For the convenience of calculation, a covariance matrix is defined as

(16) |

and the evolution of matrix can be determined by Eq.(17):

(17) |

where is a diagonal noise correlation matrix defined by:

(18) |

At this point, the dynamical analysis of the model we used has already been finished and the system will evolution according to Eq.(11) and Eq.(14). Subsequently, we are going to discuss the synchronization effects between the systems by using those equations.

We will discuss the phase synchronization between the oscillators of the systems. Firstly, define the “classical” part (expectation value) of phase error on the basis of Eq.(6) [22].

(19) |

where

(20) |

The evolution of can be simulated numerically with simultaneous equations (11), (19) and (20) under the certain initial conditions and the largest Lyapunov exponent of the error can be calculated by following equation

(21) |

and in Eq.21 mean the disturbances of the phase errors when and , respectively, and can be seen as the eigenvalue of Jacobian matrix corresponding to . Because the phase synchronization is controlled by the intensity of the phonon tunnel and the coupling constant of the fiber, we calculate the largest Lyapunov exponent of phase error with the variation of and . In calculation, the damping rates and the intensity of driving field are assumed to be equal in both systems, but there are the differences in frequencies and initial conditions. The values of ,, and are taken as the same as Mari’s work so that the conclusion is more easy to be verified by the experiment. Moreover, we properly reduce the intensity of the driving field in order to highlight the coupling function in synchronization. Otherwise, too strong driving field will dilute the coupling effect, which leads the system in the “forced” synchronous effect. It can be known from Eq.9 and Eq.11 that the phonon coupling can directly influence mechanical oscillators, however, the photon coupling can only influence them indirectly by changing the light field in the cavities. For making two switches have similar ability to control synchronization, we reduce the intensity of phonon channel() and increase coupling constant of the fiber. Therefore, we calculate the Lyapunov exponent in the region of , and the calculation result is shown in Fig.2.

Instead of analyzing the largest Lyapunov exponent concretely, as it is discussed above, we can determine whether the system is in the synchronization state only by comparing the largest Lyapunov exponent with zero. Therefore, Fig.2 is redrawn in this form: some parameter regions where the largest Lyapunov exponent is greater than zero are projected and marked in red, contrary, the regions where the largest Lyapunov exponent is less than zero are marked in blue. Moreover, we also plot a curve of the Lyapunov exponent with a fixed in order to display it more clearly, shown in Fig.3.

If and are in the blue regions of Fig.3(a), the largest Lyapunov exponent is less than zero, indicating the evolution of the phase error tends to zero stably after a certain time and the systems reach the synchronization. By contrast, the largest Lyapunov exponent is greater than zero while and are in the red regions and the systems are not synchronous because the phase error tends to random oscillations. Therefore, it can be seen from Fig.3(a) that the systems will not synchronize when two systems are not coupled( and =0). Once there is coupling between systems, however, the red regions will be replaced gradually by blue area with the increasing of and . We expect to control the synchronization by the switches and together, i.e. it will be happened only if two switches meet the logic “AND”. Then we choose the parameters according to the following principles: the largest Lyapunov exponent is greater than zero when and as well as and , but it must be less than zero as and at the same time. According to this, the appropriate parameters can be found in and recurring to the help of Fig.3 and within the range the systems will achieve phase synchronization only if switches and are closed synchronously. (It is worth to note that alone point with different color from ones around can be ignored as an error.) In other words, we can control synchronization effect with switches and by using above characteristics.

In the above discussion, we give a range of parameters and but not exact values. This is because the “classical” error can tend to zero for all parameters in that range. Whereas for quantum synchronization, not only the expectation value of errors tending to zero is necessary, but also the error fluctuations as small as possible. For the sake of reaching a perfect synchronization, we need to select appropriate values of and and to ensure a minimum quantum fluctuations. It is the reason why we calculate the second order measurement of the quantum phase synchronization.

For the calculating of , the matrix defined in Eq.(16) is transformed firstly as

(22) |

here and , . can be obtained by substituting the matrix into Eq.(17) and can be expressed as

(23) |

Time-averaged is further calculated in order to show directly the size of quantum fluctuation under the different parameters.

(24) |

and the calculation result is shown in Fig.4.

Although the phase expectation values of every system tend to be equal, the quantum fluctuations between the systems under different parameters still influence the perfection of quantum phase synchronization, as shown in Fig.4. The fluctuation of system error will be reduced to minimum extent while taking and , which draws the conclusion that the best effect of synchronization has been reached.

The dynamical evolution of the system is simulated here to test the validity of our criterion. Before the simulation we let and and assume that the initial phase error between the systems is . The remaining parameters are same as ones used in Fig.2. The simulation results are illustrated in Fig.5 in which the unit of ordinate is .

It can be known from Figs.5(a)(c) that the synchronization between systems will not be achieved as long as any a switch of and is opened. Upon further inspection, we notice that two different systems will never achieve phase synchronization in the other parameters (,, and ) but the same when couples disappear. Only when two switches are both closed, the systems can realize synchronization, shown as Fig.5(d). This result is identical with our analysis and it also verifies the quantum synchronous criterion proposed in our work. By the way, the logical relationship “AND” of two switches is taken as an example in Fig.3, however, the logical relationships “OR” or “exclusive-OR” between the switches can also be selected to realize the quantum synchronization of the systems by adjusting appropriate parameters of and .

## V Conclusions

In this paper, we investigate quantum synchronization effects and present the quantitative criteria of complete synchronization and phase synchronization between quantum systems. Further, we realize the quantum phase synchronization between coupled optomechanical systems by using our criterisa. Through calculating the largest Lyapunov exponent, we find that the systems will not reach synchronization unless switches and are closed synchronously (satisfying the logic relation “AND”) when the parameter values are taken at the ranges and . At the same time, we obtain the information that the fluctuation of system error will reduce to minimum while and by calculating the second order measurement of the quantum phase synchronization. Finally, the dynamical evolution of the system is simulated in order to test the validity of our criterion under above parameters. Since the concrete quantum synchronization criteria have been proposed and the control theory of quantum synchronization effects is simple and efficient in the work, other designers can set different synchronization conditions to satisfy themselves aims. We believe that our work can bring certain application values in quantum communication, quantum control and quantum logical gates.

### Acknowledgements

This research was supported by the National Natural Science Foundation of China (Grant No 11175033) and the Fundamental Research Funds for the Central Universities (DUT13LK05).

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### Footnotes

- Corresponding author. E-mail:lichong@dlut.edu.cn
- Corresponding author. E-mail:hssong@dlut.edu.cn
- preprint: Physical Review Letters