## Abstract

We report an efficient mechanism to generate mechanical entanglement in a two-cascaded cavity optomechanical system with optical parametric amplifiers (OPAs) inside the two coupled cavities. We use the especially tuned OPAs to squeeze the hybrid mode composed of two mechanical modes, leading to strong macroscopic entanglement between the two movable mirrors. The squeezing parameter as well as the effective mechanical damping are both modulated by the OPA gains. The optimal degree of mechanical entanglement therefore depends on the balanced process between coherent hybrid mode squeezing and dissipation engineering. The mechanical entanglement is robust to strong cavity decay, going beyond simply resolved sideband regime, and is resistant to reasonable high thermal noise. The scheme provides an alternative way for generating strong macroscopic entanglement in cascaded optomechanical systems.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Quantum entanglement is a fundamental phenomenon providing the crucial resources for quantum information processing and quantum communication [1]. Ranging from microscopic to mesoscopic physical system such as atomic [2], photonic [3] and artificial qubits [4], quantum entanglement has been demonstrated. Recently, the generation of quantum entanglement in macroscopic physical system has attracted many attentions [5–13], for it may help to figure out the boundary between the classical and quantum worlds [14]. Cavity optomechanics, which is based on the interaction between light and mechanical resonators by radiation pressure, can provide a good platform for studying quantum entanglement at macroscopic scales [15–36]. Moreover, many theoretical and experimental progresses have been achieved in the field of optomechanics, such as ground-state cooling and strong mechanical squeezing [37–53].

Entanglement between the two basic components (i.e. the cavity and mechanical modes) of a standard cavity optomechanical system is naturally envisioned and is widely studied [16–19]. Vitali *et al.* have shown that the stationary entanglement between the two modes can be generated with blue-detuned cavity driving laser based on the linear conversion Hamiltonian, which is then generalized to the red-detuned driving regime excluding the rotating wave approximation [16,17]. Moreover, a squeezed hybrid mode composed of the cavity and mechanical modes can be generated by a periodically modulated driving laser, leading to a significantly improved optomechanical entanglement [18]. In addition, quantum entanglement has also been studied by extending the standard optomechanical system to a hybrid scenario. For example, placing an atomic ensemble in the optical cavity, one can study entanglement between the mechanical mode and the atomic ensemble [19]. In a hybrid optomechanical system involving two vibrating mirrors, one can investigate the macroscopic entanglement between two mechanical oscillators. Such macroscopic mechanical entanglement can be used to test the fundamental principles of quantum mechanics and to improve ultrahigh-precision measurements [54–56]. Many schemes have been proposed to entangle two mechanical oscillators, such as by injecting a pair of entangled light beams into two independent optical cavities [26], by using squeezed light to drive a double-cavity setup [27,28] and by utilizing entanglement swapping [29,30]. In order to realize macroscopic entanglement between distant mirrors, Joshi et al. [31] have put forward a physically different setup, where two different Fabry-Perot cavities are coupled by an optical fiber and each cavity has a movable mirror. However, the obtained stationary mechanical entanglement is small and is limited to low thermal temperature [31]. An improved scheme was then proposed by Chen et al. [32], in which the hybrid mode formed by two mechanical modes can be squeezed by using a periodically modulated driving laser, giving rise to strong mechanical entanglement [18].

An optical parametric amplifier is generally used to induce a squeezed cavity mode [57]. In the context of optomechanical systems, an OPA can improve optomechanical cooling [58], and modify the normal-mode splitting behavior of the coupled movable mirror and the cavity field [59], as well as enhance the optomechanical interaction strength into the single-photon strong-coupling regime [60]. Moreover, it could also be used to induce genuine tripartite entanglement [61], and to generate or enhance mechanical squeezing [62,63]. In this paper, we propose an alternative scheme for entangling two mirrors in two fiber-coupled optomechanical cavities, with two OPAs in each of them. We find squeezing of the hybrid mechanical mode can be generated by the especially tuned OPAs, leading to strong entanglement between the distant mirrors. In difference from previous schemes [27,28,32], our scheme does not apply modulated cavity driving or additional squeezed light to drive the two cavities. Compared with the original scheme by Joshi et al. [31], OPAs can greatly enhance the mechanical entanglement with more feasible experimental parameters. Moreover, the mechanical entanglement is more robust to thermal noise and cavity decay with the assistant of OPAs. In addition, we find the optimal frequency for the laser pumping the OPAs, that maximize the mechanical entanglement. Our scheme can be potentially realized with the state-of-the-art experimental setup.

The paper is organized as follows. In Sec. 2, we introduce the setup with two coupled cavity optomechanical systems and give linearized dynamical equations for the system. In Sec. 3, we solve the dynamics of the classical mean values, and derive analytical solutions for the mean values and prove that in the long time limit the system will acquires a period. In Sec. 4, we derive the squeezing hybrid mechanical mode by the help of OPAs and give the method of measuring mechanical entanglement. In Sec. 5, we give the numerical simulation results of mechanical entanglement. Finally, we summarize the results in Sec. 6.

## 2. Model

We consider a physical setup consists of two identical optomechanical subsystems coupled by an optical fiber [31], as shown in Fig. 1. Each individual subsystem is essentially a Fabry-Perot cavity with an OPA inside and has one movable and one fixed mirror. The fixed mirror is partially transmitting, and then the movable mirror is totally reflecting. We assume that each cavity only has a single optical mode with frequency $\omega _{c}=2\pi c/L$ and decay rate $\kappa =\pi c/(2FL)$, where $L$ is the equilibrium length and $F$ is the finesse of each cavity. The movable mirrors are considered as quantum mechanical harmonic oscillators, with mass $m$ , frequency $\omega _{m}$, and damping rate $\gamma _{m}=\omega _{m}/Q$, where $Q$ is the mechanical quality factor of the movable mirrors. The coupled optomechanical system envisioned can be experimentally realized with quantum optical devices such as coupled toroid microcavities or waveguide-coupled Fabry-Perot cavities [64,65]. We assume the coupling strength of the two cavities is $\lambda$. Both the optomechanical cavities are driven by a common external driving laser with driving frequency $\omega _{l}$ and strength $E=\sqrt {2\kappa P/(\hbar \omega _{l})}$, where $P$ is the input laser power. Another two laser fields pump the second-order nonlinear optical crystals, i.e. the OPA, at frequency $2\omega _{p}$, thus degenerate down-converted photons with frequency $\omega _{p}$ by OPA can be generated. The OPA gain $\Lambda$ depends on the pumping power, and the phase of the pump fields are assumed to be $\theta$. In the rotating frame $\omega _{l}a_{j}^{\dagger }a_{j}$, the corresponding Hamiltonian reads ($\hbar =1$)

Due to fluctuation-dissipation processes, the dynamics of such system subject to thermal environment should be described by the quantum Langevin equations (QLEs) [66]: $\partial O/\partial t=i[H,O]+N-H_\textrm {diss}$, where $N$ is the quantum noise operator, $H_\textrm {diss}$ characterizes the dissipation, and $O=p_{j}, q_{j}, a_{j}$ denote any operators for the whole system. The set of nonlinear QLEs is then given by

When the system is strongly driven by the external driving laser, one can adopt the standard linearization technique to the QLEs in Eq. (2) and rewrite each Heisenberg operator as $O=\langle O(t)\rangle +\delta O\,\,(O=q_{j},\,p_{j},\,a_{j})$. The detailed discussion about the classical mean values $\langle O(t)\rangle$ is in the following section, and the linearized QLEs for the quantum fluctuations $\delta O$ are as following

## 3. Classical dynamics of system

Since the evolution of quantum fluctuation operators in Eq. (3) depends on $\langle O(t)\rangle$, the classical mean values should be firstly solved in order to investigate the quantum dynamics of the system. Assuming that the two optomechanical cavities are driven by classical laser fields of the same strengths, the coupling between the two distant cavities is reciprocal, and the OPAs are pumped by a common laser beam, we can find $\langle q_{1}(t)\rangle =\langle q_{2}(t)\rangle =\langle q(t)\rangle$, $\langle p_{1}(t)\rangle =\langle p_{2}(t)\rangle =\langle p(t)\rangle$, $\langle a_{1}(t)\rangle =\langle a_{2}(t)\rangle =\langle a(t)\rangle$, which have been numerically verified by showing the synchronized dynamics of the two subsystems [see Fig. 2(b)]. Using the well-known mean-field assumption $\langle a^{\dagger }(t)a(t)\rangle \simeq \left |\langle a(t)\rangle \right |^{2}$ and $\langle a(t)q(t)\rangle \simeq \langle a(t)\rangle \langle q(t)\rangle$, and then the equations of motion for the classical mean values can be described by the following set of nonlinear differential equations

## 4. Nonlocal mechanical mode squeezing and entanglement measurement

#### 4.1 Squeezing of nonlocal mechanical mode

To find the set of optimal parameters for generating mechanical entanglement, it is better to represent all the operators in the linearized Hamiltonian Eq. (4) by bosonic operators. For the mechanical fluctuations $\delta b_{j} = (\delta q_{j}+i\delta p_{j})/\sqrt {2},\delta b_{j}^{\dagger } = (\delta q_{j}-i\delta p_{j})/\sqrt {2}$. Meanwhile, with the definition of the effective coupling $G(t)=\sqrt {2}\bar {g}\langle a(t)\rangle$ and the effective detuning $\Delta (t)=\Delta _{0}-g\langle q(t)\rangle$, by Eq. (4) the linearized Hamiltonian can be rewritten as

#### 4.2 Measurement of mechanical entanglement

To support the qualitative discussion, the mechanical entanglement measured by logarithmic negativity will be numerically investigated based on the linearized QLEs Eq. (3). We first introduce the the amplitude and phase fluctuations for the two cavity modes $\delta x_{j}=(\delta a_{j}^{\dagger }+\delta a_{j})/\sqrt {2}$ and $\delta y_{j}=({\delta a_{j}-\delta a_{j}^{\dagger }})/i\sqrt {2}$, and their input noises $x_{j}^{in}(t)=[\delta a_{j}^{in\dagger }(t)+\delta a_{j}^{in}(t)]/\sqrt {2}$ and $y_{j}^{in}(t)= [\delta a_{j}^{in}(t)-\delta a_{j}^{in\dagger }(t)]/i\sqrt {2}$. The linearized QLEs for the quantum fluctuation in Eq. (3) can be rewritten as $\dot {u}(t)=M(t)u(t)+n(t)$, in which $u(t)=[\delta q_{1},\delta p_{1},\delta q_{2},\delta p_{2},\delta x_{1},\delta y_{1},\delta x_{2},\delta y_{2}]^{T}$, and the drift matrix

The full dynamics of the system described by $\dot {V}(t)$ and Eqs. (5) can be straightforwardly solved by numerical simulation. In a long time limit, the periodicity of the classical solutions indicates $M(t+\tau )=M(t)$, which according to the Floquet theorem leads to $V(t)=V(t+\tau )$ [18]. The mechanical entanglement can be measured by the logarithmic negativity $E_{N}$ [73,74], which can be easily calculated from the reduced $4\times 4$ CM $V_{m}(t)$ for the two mechanical modes. Capturing from the full $8\times 8$ CM $V(t)$ by just keeping the first four rows and columns, the detailed information about $V_{m}(t)$ is given $V_{m}(t) =\left [A,C;C^{T},B\right ]$, with $A$, $B$ and $C$ being the $2\times 2$ sub-block matrices of $V_{m}(t)$. The logarithmic negativity $E_{N}$ is given by

with $\eta =(1/\sqrt {2})[\Sigma -\sqrt {\Sigma ^{2}-4\det V_{m}}]^{1/2}$ and $\Sigma =\det A+\det B-2\det C$.## 5. Numerical results and discussions

In this section, we numerically calculate the logarithmic negativity $E_{N}$ given by Eq. (19) to show mechanical entanglement under the effect of the OPAs. We first examine the effect of the parametric phase $\theta$ on the mechanical entanglement $E_{N}$ and show $E_{N}$ as a function of time $t$ for different parametric phases $\theta =0$, $\pi /2$, $\pi$, see Fig. 4. In the long time limit, the dynamically mechanical entanglement $E_{N}$ also possesses the same period $\tau$ as that of the classical values. Note that the time-dependence of $E_{N}$, i.e. the amplitude or periodicity does not change for different parametric phases $\theta$. This is fair since the mechanical entanglement is determined by the squeezing parameter $r$, as pointed out by the analytical analysis in Sec. 4. Therefore, without loss of generality, the parametric phase $\theta =0$ is set in the following.

Due to the periodicity of the dynamical entanglement, one can take the maximum of $E_{N}$ in a time period to quantity the degree of mechanical entanglement [18,70], namely, $E_{N,max}=\lim \limits_{t\to \infty }\textrm {max}\{E_{N}[t,t+\tau ]\}$. In Fig. 5, we show the maximum of mechanical entanglement $E_{N,max}$ as a function of the detuning $\Omega /\omega _{m}$. It can be seen that the optimal modulation frequency $\Omega _\textrm {opt}/\omega _{m}=1.994$ is close to but slightly less than $2\omega _{m}$, which is due to the fact that the effective cavity detuning $\Delta (t)$ is slightly modified by the optomechanical coupling. Moreover, we show the influence of the driving strength $E$ and the cavity dissipation rate $\kappa$ on the mechanical entanglement, see Fig. 6. For a fixed cavity decay rate $\kappa$, the squeezing parameter $r=\frac {2\Lambda |g|^{2}}{\kappa ^{2}-4\Lambda ^{2}}$ has a linear response to the input laser power for $|g|\propto E$, thus the mechanical entanglement $E_{N,max}$ can be enhanced as $E$ increases. However, the reduced model fails when the adiabatic approximation condition $\kappa \gg |g|$ is broken, and the increasing effective mechanical decay rate $\Gamma _{m}$ may prevent the system from being further squeezed. Therefore, the mechanical entanglement $E_{N,max}$ first reaches the maximum and then goes down. In addition, when the driving strength $E/\omega _{m}=1.5\times 10^{5}$ and the ratio of OPA gain to cavity decay $\Lambda /\kappa =0.3$ are fixed, the mechanical entanglement $E_{N,max}$ can be enhanced by increasing the cavity decay $\kappa /\omega _{m}$. This is because when $\kappa \gg |g|$ is satisfied, the adiabatic model can be well built, leading to efficient generation of mechanical entanglement. However, while $\kappa$ becomes larger, then the effective squeezing parameter becomes too small to further increase the mechanical entanglement, which vanishes for $\kappa$ over some threshold. In order to revive the mechanical entanglement, the OPA gain should be enlarged by increasing the pumping strength, as shown in Fig. 7. It shows that the mechanical entanglement $E_{N,max}$, with the optima strongly depend on the cavity decay rate, can be retrieved even though the system stays far from resolved sideband regime.

For a fixed cavity decay rate, mechanical entanglement emerges as the ratio $\Lambda /\kappa$ surpasses the threshold, which shifts to the larger side for a more dissipative cavity, and can be seen as the signature of quantum phase transition [34]. For a fixed cavity decay rate and an increasing $\Lambda /\kappa$, the entanglement can be enhanced due to the increase of the squeezing parameter $r\sim (\Lambda /\kappa )/(1-4\Lambda ^{2}/\kappa ^{2})$, and is otherwise limited by the increasing effective damping rate $\Gamma _{m}$. Therefore, there exists an optimal ratio of $\Lambda /\kappa$ to obtain the maximal degree of mechanical entanglement $E_{N,max}$ due to the balance between coherent hybrid mode squeezing and modified dissipation [see Fig. 7]. Another merit by applying OPAs in this setup is that it does not require strong cavity-cavity coupling and intensive cavity driving, which in the original scheme [31], works in the parameter regime: $\lambda /\omega _{m}=20$ and $E/\omega _{m}=1\times 10^{7}$. These parameter conditions may be hard to access in experiments. Moreover, the mechanical entanglement obtained in [31] is weak and is vulnerable to the cavity decay $\kappa /\omega _{m}$ and thermal noise, see Fig. 8(a). In contrast, we find that by introducing OPAs to such setup, the strong mechanical entanglement can be achieved with $E/\omega _{m}=1.5\times 10^{5}$, $\lambda /\omega _{m}=2$ and $\kappa /\omega _{m}=0.2$. The parameter regime is easier to reach in state-of-the-art experimental setups. Finally we consider the effect of a finite thermal temperature. In Fig. 8(b), we plot the evolution of $E_{N}$ for different thermal phonon number $n_{m}=0$, $0.2$ for the parametric gain $\Lambda /\kappa =0.49$. It shows that the mechanical entanglement can resist a reasonable thermal noise with the assistance of the OPAs.

## 6. Conclusions

In conclusion, we have demonstrated that the mechanical entanglement can be improved by introducing the OPAs into the fiber-coupled optomechanical system. It is found that the strong mechanical entanglement is induced by the squeezing of the hybrid mode formed by the two mechanical modes, which can be implemented by using especially tuned OPAs. On the other hand, the OPAs effectively modify the mechanical damping, in analogous to quantum reservoir engineering, determined simultaneously by the OPAs gain and cavity decay rate. Thus, it allows us to achieve strong mechanical entanglement by compromising the two processes, with a broadening parameter conditions, such as the un-resolved sideband regime and a relatively high thermal temperature, our scheme provides an alternative method for improving the quantum entanglement of two distant mechanical oscillators.

## Funding

National Natural Science Foundation of China (11674060, 11705030, 11774058, 11874114); Natural Science Foundation of Fujian Province (2017J01401); Natural Science Foundation of Fujian Province (2019J01219).

## Acknowledgments

H. W gratefully acknowledges the Qishan fellowship of Fuzhou University.

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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