1. Introduction

Entanglement as a unique quantum phenomenon is often considered in the microscopic world. However, many attempts towards searching entanglement in the macroscopic world have been made in recent years [1–10]. The macroscopic entanglement is not only a very important resources in potential applications in quantum technology, but also a clue that may reveal the boundary between classical and quantum realms [11,12]. With the development of experimental technique, optomechanical system [13–32]. become a hopeful candidate for testing the properties of macroscopic entanglement. A cavity field and a mechanical oscillator are coupled via the radiation pressure in the optomechanical system, thus a microscopic object (field) and a macroscopic object (mirror) are connected. Since the mass of the mirror can be huge [33], it paves a new way for testing macroscopic quantum effects and answering some of the most fundamental questions in modern physics.

A lot of theoretical and experimental studies of the optomechanical system have been done in the past two decades [13–32]. However, the evolution of the system is often investigated in the Markov regime. The traditional tool of analyzing the model mainly based on the Langevin equations or the master equations derived by using Markov approximation [19,34–36]. The memory effect from the environment are not taken into account [37–40]. However, recent experiment indicates the bath coupled to optomechanical system is usually non-Markovian [41]. Moreover, the non-Markovian bath has been proved to be helpful in the preservation of optomechanical entanglement [42]. With the development of the experiment technique on environment engineering [37], it could be possible to experimentally study the non-Markovian impact on the dynamics of optomechanical system. Therefore, it is highly valuable to develop a systematic approach to solve the optomechanical system in non-Markovian regime.

In this paper, we investigate an optomechanical system consisting of a F-P cavity and two movable mirrors embedded in a non-Markovian environment. Starting from the fully quantized Hamiltonian, we derive the fundamental dynamic equation of this model without any approximation. The technique we use to solve the model is a newly developed theoretical and numerical tool of solving non-Markovian dynamics of open quantum systems, which is called the non-Markovian quantum state diffusion (NMQSD) approach [43–56]. It is worth to note that deriving an exact non-Markovian master equation for an optomechanical system without linearization of the Hamiltonian was a challenging task by using path integral technique [57–59]. Sometimes, the coupling terms have to be linearized approximately [17]. However, in this paper, we show that it is possible to obtain the exact solution of this model without using the linearized Hamiltonian. Then, based on the solution we obtained, we investigate several interesting non-Markovian properties. We examine the macroscopic entanglement generation from a separable initial state, and find that the non-Markovian memory effect may enhance the entanglement generation between two macroscopic objects (mirrors). In order to obtain the largest entanglement, one need to control the memory time of the environment to be neither too big nor too small. Besides, the central frequency of the environment has also to be chosen carefully depending on the properties of the system. The sudden death and revival of the entanglement in this system are also investigated. It is shown that in some particular cases, the revival can be only observed in non-Markovian regime.

2. Model and solution

As it is shown in Fig. 1, the model we considered is an F-P cavity with two movable mirrors. In the quantized description of this model, the two movable mirrors can be treated as two quantum harmonic oscillators and the cavity field is described by a light field. Therefore, the total Hamiltonian of the system plus a non-Markovian environment is

 

Fig. 1. Sketch of the model. An F-P cavity with two movable mirrors is considered. The cavity field and two mirrors are coupled due to the radiation pressure.

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The environment we are considered is a common bosonic bath

In order to solve this model, we could introduce a stochastic state vector as $|\psi _{t}(z^{*})\rangle \equiv \langle z|\psi _{tot}(t)\rangle$, where $|z\rangle \equiv \prod _{i}|z_{i}\rangle$ is a collective Bargmann coherent representation for all the modes in environments. Applying the Schrödinger equation to the total system, the dynamic equation for the stochastic state vector $|\psi _{t}(z^{*})\rangle$ can be derived as [43–45]

The key to solving Eq. (5) is to replace the functional derivative $\frac {\delta }{\delta z_{s}^{\ast }}$ by a time-dependent operator $O$ as $\frac {\delta }{\delta z_{s}^{\ast }}|\psi _{t}(z^{*})\rangle =O(t,\;s,\;z^{\ast })|\psi _{t}(z^{*})\rangle$ [45] with the initial condition $O(t=s,\;z^{*})=L$. Then, the NMQSD equation can be rewritten as

The advantage of using this stochastic method is that one can only simulate dynamics of pure state trajectories $|\psi _{t}(z^{*})\rangle$. The requirement for computational resource is reduced from $N^{2}$ to $N$ comparing to solving the density matrix. Alternatively, one can also derive an exact master equation based on the NMQSD Eq. (5). The exact master equation can be achieved straightforwardly by following the method in [46,52] with the exact $O$ in Eq. (7). It is worth to note that it is usually very difficult to use path integral to derive an exact master equation for such a model with the couplings $a^{\dagger }aq_{1}$ and $a^{\dagger }aq_{2}$ (not linearized form). However, the exact master equation can be easily derived in NMQSD approach. This reflects another advantage of the NMQSD approach, i.e., deriving master equation without linearization of the optomechanical Hamiltonian. Moreover, in the NMQSD approach, there is also a systematic way to derive a perturbative master equation up to certain order [44,52]. If we only consider the noise free $O$ operator $O^{(0)}(t,\;s)=\sum _{i=1}^{5}f_{i}(t,\;s)O_{i}$ as an approximate solution, the calculation will be hugely reduced. Meanwhile, the accuracy of this approximation is remarkable in many cases [53]. Generally, the correction from the $f_{6}$ term only contribute to $g_{i}^{4}$, the fourth order (or higher) of $g_{i}$. When $g_{i}$ is smaller than $\omega _{1}$ and $\omega _{2}$, the noise term $f_{6}$ is always negligible. Eventually, taking time-derivative to $\rho =M\{|\psi _{t}\rangle \langle \psi _{t}|\}$ and using Eq. (6), the master equation can be obtained as

The non-Markovian corrections can be clearly shown in Eq. (19). But, in the numerical simulation, one can use the exact NMQSD Eq. (6) to obtain the density matrix $\rho =M\{|\psi _{t}\rangle \langle \psi _{t}|\}$ by taking statistical average. Or, one can also derive an exact master equation by following the method in [46,52].

3. Numerical simulations and discussion

In order to study the entanglement between two mirrors, we first simply review entanglement measure for two mode Gaussian states. We define a vector $\xi =(q_{1},\;p_{1},\;q_{2},\;p_{2})$ containing the momentum operator $p$ and position operator $q$ for two modes, then, the properties of the two-mode Gaussian state is fully determined by the variance matrix $V_{\alpha \beta }=\langle (\Delta \xi _{\alpha }\Delta \xi _{\beta }+\Delta \xi _{\beta }\Delta \xi _{\alpha })/2\rangle$, where $\Delta \xi _{\alpha }=\xi _{\alpha }-\langle \xi _{\alpha }\rangle$. Finally, the entanglement measure called logarithmic negativity [62–64]. is defined as

In the numerical calculation, a particular correlation function $\alpha (t,\;s)$ is needed. Here, we emphasize again that all of our derivations are independent on the choice of correlation function $\alpha (t,\;s)$. Equations (5) and (19) are applicable to arbitrary correlation functions, either predicted by theoretical calculation or directly measured in experiments. For example, one can consider the Ohmic noise $\alpha (t,\;s)=\hbar \int _{0}^{\infty }d\omega J(\omega )\left [\coth \left (\frac {\hbar \omega }{k_{B}T}\right )\cos [\omega (t-s)]-i\sin [\omega (t-s)]\right ]$ which is temperature dependent. It could be useful in a future research if one would like to investigate the temperature impact. However, in this paper, we choose the Ornstein-Uhlenbeck (O-U) noise

There are two reasons we chose the O-U noise. First, an arbitrary correlation function $\alpha (t,\;s)$ can be decomposed into a summation of several O-U noises as $\alpha (t,\;s)=\alpha _1(t,\;s)+\alpha _2(t,\;s)+\cdots$. Second, it is easy to observe the transition from Markov regime to non-Markovian regime. In the O-U correlation function, the parameter $1/\gamma$ indicates the memory time of the environment. If $\gamma$ is large enough, the correlation function $\alpha (t,\;s)$ is close to a $\delta$-function, which represent the environment is very close to a Markov environment. While the parameter $\gamma$ is small, non-Markovian properties can be observed.

3.1 Non-Markovian properties and entanglement generation

First, we will investigate how does the memory time of the environment affect the entanglement generation. In order to observe the entanglement generation, we choose a separable two-mode coherent state $|1\rangle |1\rangle$ as the initial state, and then focus on the the time evolution of the entanglement between two mirrors. In the upper-right inset of Fig. 2, we plot the time evolution of the logarithmic negativity $E_{N}$ for different values of $\gamma$. When $\gamma$ is large (for example, $\gamma =6$), the environment is close to a Markov environment, the memory effect is very weak. In this case, there is no obvious entanglement generation. In contrast, when $\gamma$ is relatively small (for example, $\gamma$ is in the range from $0.25$ to $1$), entanglement generation can be observed in the early stage of the time evolution. The fact that the entanglement generation can be only observed in non-Markovian case shows that the memory effect is essential to the entanglement generation. Interestingly, the relation between the maximum entanglement generation and the memory time is subtle. In order to obtain a better entanglement generation, one need to control the memory time to be neither too big nor too small. Actually, the smallest $\gamma$ which represents the highest memory effect does not generate the largest entanglement, while the largest entanglement is generated in the case that the memory effect is relatively weaker($\gamma \approx 0.9$ as it is indicated by the peak of the red curve in the main plot of Fig. 2).

 

Fig. 2. Maximum entanglement generation for different correlation time. The inset at the upper-right corner is the real-time entanglement dynamics, while the main plot is the maximum entanglement generation (during the entire evolution) as a function of $\gamma$. The initial state is chosen as a separable state $|1\rangle |1\rangle$. The parameters are chosen as $\omega _{1}=\omega _{2}=\omega =1$, $G_{1}=G_{2}=1$, $\Omega =\frac {2}{3}\pi$, $\kappa _{1}=\kappa _{2}=1$.

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Another parameter $\Omega$ in the generalized O-U correlation function is also essential to the entanglement generation. Mathematically, in Eq. (22), $\Omega$ typically determines the oscillating speed of the correlation function. Physically, $\Omega$ describes the central frequency of the environment. The major difference between Markov and non-Markovian cases is that there will be a back reaction effect in non-Markovian case. It can be regarded as an external driven force to the system. As we know, when the frequency of the driven force is equal to the intrinsic frequency of the system, the amplitude of the oscillation of the system reach its maximum value. This is called the resonance phenomenon in classical mechanics. Therefore, in analogy to the classical resonance, the quantum entanglement generation will achieve the maximum value when the central frequency of the environment “matches” certain properties of the system.

This can be understood in a limiting case that the environment only contains a single mode, which is definitely a very strong non-Markovian environment. It is well known that when the frequency of the single mode in the environment matches the intrinsic frequency of the system, the dynamics of the system will be coherently driven by the back reaction of the environment. From this simple example, one could understand why the back reaction from the environment is essential to the dynamics of the system. In Fig. 3, the condition for maximum entanglement generation is plotted. In order to achieve maximum entanglement generation, the parameters $\omega$ and $\Omega$ have to be chosen in an approximate linear relation. This somehow reflects the resonance condition, namely the environmental frequency should match the intrinsic frequency (could be one of the normal mode) of the system. However, in our model, the environment is too complicated to be analyzed in details, and the influence of the environment includes many aspects other than the central frequency. Therefore, the relation between $\omega$ and $\Omega$ is not exactly linear in Fig. 3. Nevertheless, our numerical results explicitly show that in order to achieve the maximum entanglement, the non-Markovian properties of the environment have to be carefully chosen depending on the properties of the system. It is true that controlling the non-Markovian properties is a challenge in experiments. However, recent experiment about environment engineering technique shed a new light on mimic non-Markovian environment. Besides, some artificial non-Markovian environment are controllable. For example, the pseudomode [68] plays the role of an effective non-Markovian environment, and the properties of this non-Markovian environment can be controlled by adjusting pseudomode couplings.

 

Fig. 3. Maximum entanglement generation in parametric space. The position of the red circle makers indicate the maximum entanglement is achieved for the given set of parameters $\omega$ and $\Omega$. For example, the last marker shows that when $\omega /G=4$, the maximum entanglement appears at $\Omega /G\approx 8$. In the plot, we choose the symmetric case $\omega _{1}=\omega _{2}=\omega$, $G_{1}=G_{2}=G$, $\kappa _{1}=\kappa _{2}=1$, and $\gamma =0.4$.

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3.2 Entanglement generation and coherence in initial states

Second, we study how does the coherence in the initial state affect the entanglement generation between two mirrors. We choose the initial states of the two mirrors as the following form

 

Fig. 4. Entanglement dynamics for initial state $(c_{1}|1\rangle +c_{2}|-1\rangle )\otimes |1\rangle$. The color reflects the value of the variable of $E_{N}$ which is an indicator of entanglement. The parameters are chosen as $\omega _{1}=\omega _{2}=\omega =1$, $G_{1}=G_{2}=1$, $\gamma =0.4$, $\Omega =\pi /3$, $\kappa _{1}=\kappa _{2}=1$.

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3.3 Entanglement sudden death and revival

At last, we will investigate the disentanglement process for different memory time. From Fig. 5, we found that the entanglement sudden death and rebirth [71–73] with the initial state $|\psi _{ini}\rangle \propto (|1\rangle |1\rangle +|-1\rangle |-1\rangle )$, which is known to be the “cat state”. In the time evolution, the indirect coupling between two mirrors can produce an entanglement revival with the help of the non-Markovian back-reaction. The revival strength significantly depends on the memory time $1/\gamma$. The transition from non-Markovian regime to Markov regime is clearly shown in Fig. 5. There are two critical memory time $1/\gamma _{c_{1}}$ and $1/\gamma _{c_{2}}$, dividing the parameter space into three regions. In region I ($\gamma <\gamma _{c_{1}}$), the environment is strong “non-Markovian” since the memory time is very long. In this region, there is no sudden death during the evolution time plotted in the figure (For a long-time limit, the entanglement may also disappear due to the dissipation). In region II ($\gamma _{c_{1}}<\gamma <\gamma _{c_{2}}$), the memory effect is moderate and the environment is gradually changing from non-Markovain to Markov. In this region, an entanglement revival after the sudden death can be observed. The strength of the revival becomes weaker with the increasing of $\gamma$. Finally in region III ($\gamma >\gamma _{c_{2}}$), the memory effect is weak, the environment is close to Markov, so that no revival can be observed.

 

Fig. 5. Entanglement dynamics ($E_{N}$) for different memory time. The initial state is chosen as the “cat state” $|\psi _{ini}\rangle \propto (|1\rangle |1\rangle +|-1\rangle |-1\rangle )$. The parameters are chosen as $\omega _{1}=\omega _{2}=\omega =1$, $G_{1}=G_{2}=1$, $\Omega =0$, $\kappa _{1}=\kappa _{2}=1$.

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Clearly, some novel phenomenon like the entanglement revival can be uniquely observed in non-Markovian regime ($\gamma _{c_{1}}<\gamma <\gamma _{c_{2}}$). The revival may be induced by a competition between the decay process and the non-Markovian back-reaction. When the memory time is longer, namely the back-reaction is strong, it is possible to produce a revival before all the information is lost into the environment. Otherwise, if the decay is much stronger than the back-reaction ($\gamma >\gamma _{c_{2}}$), the entanglement may be entirely lost before the back-reaction taking effect. In the past decade, the conditions for entanglement sudden death or revival have been studied in a large amount of papers. A lot of factors are investigated, such like initial states, system coupling strengths etc.. Our result in Fig. 5 indicates that the properties of the environment (memory effect for instance) are also crucial factors for entanglement sudden death and revival. Sometimes, it may cause qualitative difference (with or without revival) on the entanglement dynamics other than merely quantitative difference.

4. Conclusion

In this paper, we investigate the non-Markovian dynamics of an optomechanical system consisting of a cavity and two movable mirrors. By using the NMQSD approach, an exact solution of the dynamics is derived in the non-Markovian regime. It paves the way to study the optomechanical system beyond the Markov approximation in the future. According to the non-Markovian master equation, we discuss the macroscopic entanglement in the system. We study the entanglement generation induced by the memory effect of the non-Markovian environment. It is worth to note that the relation between entanglement generation and the non-Markovian properties is subtle. One need to control the memory time to be neither too short nor too long in order to obtain a desirable entanglement generation. Also, in order to achieve the maximum entanglement, the phase factor $\Omega$ has also to be chosen carefully depending on the properties of the system. The entanglement generation also depends on the presence of quantum coherence in the initial states. At last, we study the non-Markovian effect induced entanglement sudden death and revival for different memory time. We found that the revival can be ONLY observed in non-Markovian case. The examples we discussed in the paper show that the non-Markovian dynamics of optomechanical system can be quite different from the Markov ones. It is our hope that the systematic approach developed in this paper could become a powerful tool for theoretical researchers, so that more and more novel features of the optomechanical system can be found in future research.