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 [513], 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 [1536]. Moreover, many theoretical and experimental progresses have been achieved in the field of optomechanics, such as ground-state cooling and strong mechanical squeezing [3753].

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 [1619]. 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 [5456]. 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 ($\hbar =1$)

$$\begin{aligned} H & = {\displaystyle \sum_{j=1}^{2}}[\Delta_{0}{\textstyle a_{j}^{\dagger}}a_{j}+\frac{\omega_{m}}{2}{\textstyle (p_{j}^{2}+{\textstyle q_{j}^{2}}})-\bar{g}{\textstyle a_{j}^{\dagger}}a_{j}q_{j}+iE{(}{\textstyle a_{j}^{\dagger}-a_{j})}\\ & \quad +i\Lambda(e^{i\theta}e^{-i\Omega t}{\textstyle a}_{j}^{\dagger2}-e^{-i\theta}e^{i\Omega t}a_{j}^{2})]+\lambda({\textstyle a_{1}^{\dagger}a_{2}+a_{1}{\textstyle a_{2}^{\dagger}})},\end{aligned}$$
where $\Delta _{0}=\omega _{c}-\omega _{l}$ and $\Omega /2=\omega _{p}-\omega _{l}$ are the detunings of the cavity and the degenerate photons with respect to the driving laser; $a_{j}$ and ${{\textstyle a_{j}^{\dagger }}}$ are the annihilation and creation operator of the cavity modes, and $q_{j}$ and $p_{j}$ are the dimensionless position and momentum operators for the movable mirrors, satisfying the standard canonical commutation relation $[q_{j},\:p_{j}]=i$. The first two terms in Eq. (1) describe the free components of cavity and mechanical modes, respectively; and the third term describes the nonlinear coupling between the cavity and mechanical modes caused by the radiation pressure, in which $\bar {g}=\omega _{c}x_{ZPF}/L$ is the single-photon coupling strength and $x_{ZPF}=\sqrt {\hbar /(2m\omega _{m})}$ is the zero-point fluctuation of the mirror. The fourth term represents the interaction between cavity mode and laser field, and the fifth term gives the coupling between the cavity mode and OPA. The last term is the linear coupling between the two cavities.

 figure: Fig. 1.

Fig. 1. Schematic diagram of our system to entangle distant mechanical modes. Two optomechanical cavities are driven by classical laser fields, and the coupling between the two distant cavities is realized by an optical fiber. Each cavity contains an OPA pumped as shown, and the amount degree of entanglement between the two mechanical modes can be generated by the especially tuned OPA.

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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

$$\begin{aligned} \dot{q}_{1} & = \omega_{m}p_{1},\quad \dot{q}_{2}\;=\;\omega_{m}p_{2},\\ \dot{p}_{1} & = -\omega_{m}q_{1}-\gamma_{m}p_{1}+\bar{g}{\textstyle a_{1}^{\dagger}a_{1}}+\xi_{1}(t),\\ \dot{p}_{2} & = -\omega_{m}q_{2}-\gamma_{m}p_{2}+\bar{g}{\textstyle a_{2}^{\dagger}a_{2}}+\xi_{2}(t),\\ \dot{a}_{1} & = -(\kappa+i\Delta_{0})a_{1}+i\bar{g}a_{1}q_{1}+E+2\Lambda e^{-i(\Omega t-\theta)}{\textstyle a_{1}^{\dagger}}-i\lambda a_{2}+\sqrt{2\kappa}{\textstyle a_{1}^{in}(t),}\\ \dot{a}_{2} & = -(\kappa+i\Delta_{0})a_{2}+i\bar{g}a_{2}q_{2}+E+2\Lambda e^{-i(\Omega t-\theta)}{\textstyle a_{2}^{\dagger}}-i\lambda a_{1}+\sqrt{2\kappa}{\textstyle a_{2}^{in}(t),}\end{aligned}$$
where ${\textstyle a_{j}^{in}}$ is the zero-mean vacuum cavity input noise operator satisfying the auto-correlation function $\langle {\textstyle a_{j}^{in\dagger }(t)a_{j}^{in}(t^{'})}\rangle =\delta (t-t^{'})$ [67], and ${\textstyle \xi _{j}(t)}$ is the thermal noise operator acting on the mechanical oscillator. With high mechanical quality factor $Q\equiv \omega _m/\gamma _m \gg 1$, the Markovian approximation can be applied to the thermal noise, leading to [68] $\langle \xi _{j}(t)\xi _{j}(t^{'})+\xi _{j}(t^{'})\xi _{j}(t)\rangle /2 \simeq \gamma _{m}(2n_{m}+1)\delta (t-t')$, where $n_{m}=[\exp (\hbar \omega _{m}/k_{B}T)-1]^{-1}$ is the mean thermal excitation number in the movable mirror, and $k_{B}$ is the Boltzmann constant and $T$ is the thermal temperature of the mechanical oscillators.

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

$$\begin{aligned} \delta\dot{q}_{1} & = \omega_{m}\delta p_{1},\quad \delta\dot{q}_{2}\;=\;\omega_{m}\delta p_{1},\\ \delta\dot{p}_{1} & = -\omega_{m}\delta q_{1}-\gamma_{m}\delta p_{1}+\bar{g}[\langle a_{1}(t)\rangle\delta a^{\dagger}+\langle a_{1}(t)\rangle^{*}\delta a_{1}]+\xi_{1}(t),\\ \delta\dot{p}_{2} & = -\omega_{m}\delta q_{2}-\gamma_{m}\delta p_{2}+\bar{g}[\langle a_{2}(t)\rangle\delta a^{\dagger}+\langle a_{2}(t)\rangle^{*}\delta a_{2}]+\xi_{2}(t),\\ \delta\dot{a}_{1} & = -[\kappa+i\Delta_{j}(t)]\delta a_{1}+i\bar{g}\langle a_{1}(t)\rangle\delta q_{1}+2\Lambda e^{-i(\Omega t-\theta)}\delta a_{1}^{\dagger}-i\lambda\delta a_{2}+\sqrt{2\kappa}a_{1}^{in}(t),\\ \delta\dot{a}_{2} & = -[\kappa+i\Delta_{j}(t)]\delta a_{2}+i\bar{g}\langle a_{2}(t)\rangle\delta q_{2}+2\Lambda e^{-i(\Omega t-\theta)}\delta a_{2}^{\dagger}-i\lambda\delta a_{1}+\sqrt{2\kappa}a_{2}^{in}(t).\end{aligned}$$
where $\Delta _{j}(t)=\Delta _{0}-g\langle q_{j}(t)\rangle$ is the effective cavity detuning modulated by the mechanical motion, and the nonlinear terms $\delta a_{j}^{\dagger }\delta a_{j}$ and $\delta a_{j}\delta q_{j}$ have been safely neglected [16]. As a result, the intrinsic interactions for the quantum operators can be described by the linearized Hamiltonian, given by
$$\begin{aligned} H_{\textrm{lin}} & = {\displaystyle \sum_{j=1}^{2}}\{\Delta_{j}(t)\delta{\textstyle a_{j}^{\dagger}}\delta a_{j}+\frac{\omega_{m}}{2}{\textstyle (\delta p_{j}^{2}+{\textstyle \delta q_{j}^{2}}})-\bar{g}[\langle a_{j}(t)\rangle^{*}\delta a_{j}+\langle a_{j}(t)\rangle\delta a_{j}^{\dagger}]\delta q_{j}\\ & \quad +i\Lambda[e^{-i(\Omega t-\theta)}\delta a_{j}^{\dagger2}-e^{i(\Omega t-\theta)}\delta a_{j}^{2}]\}+\lambda(\delta a_{1}^{\dagger}\delta a_{2}+\delta a_{1}\delta a_{2}^{\dagger}).\end{aligned}$$

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

$$\begin{aligned} \langle\dot{q}(t)\rangle & = \omega_{m}\langle p(t)\rangle,\\ \langle\dot{p}(t)\rangle & = -\omega_{m}\langle q(t)\rangle-\gamma_{m}\langle p(t)\rangle+\bar{g}|\langle a(t)\rangle|^{2},\\ \langle\dot{a}(t)\rangle & = -[\kappa+i(\Delta_{0}+\lambda)]\langle a(t)\rangle+i\bar{g}\langle a(t)\rangle\langle q(t)\rangle+E+2\Lambda e^{-i(\Omega t-\theta)}\langle a(t)\rangle^{*}.\end{aligned}$$
Equation (5) is nonlinear, but it can be solved numerically. In Fig. 2, we plot the asymptotic evolution of the cavity mode mean value by numerical simulation. One can find that in the phase space the evolutionary trajectories of $\langle a(t)\rangle$ finally converge into a limit cycle with the periodicity $\tau =2\pi /\Omega$, see Figs. 2(a) and (b). Note that, in Fig. 2 we actually consider the weak cavity driving regime, where the classical asymptotic dynamics in the phase space evolve towards the simply unique circles with periodicity $\tau$. For stronger drivings, the system may undergo highly nonlinear dynamics, such as period-doubling behavior, quasi-periodic dynamics, which are avoided under our consideration here and are numerically checked for each of our plots in the paper. Under the weak cavity driving condition and assuming that the system is far away from the optomechanical instabilities and multistabilities, one can then perform a double series expansion of the asymptotic solutions $\langle O(t)\rangle$ ($O=p,q,a$) in the power of the weak optomechanical coupling $\bar {g}$ and the frequency $\Omega$ [18,70]
$$\begin{array}{ccl} \left\langle O(t)\right\rangle=\sum\limits_{n=-\infty}^{\infty}O_ne^{in{\Omega}t}=\sum\limits_{j=0}^{\infty}\sum\limits_{n=-\infty}^{\infty}O_{n,\,j}e^{in\Omega t}\bar{g}^{j}. \end{array}$$
Substituting Eq. (6) into Eq. (5), we can easily obtain the following recursive formulas about the time-independent coefficients $O_{n,\,j}$. The corresponding zeroth-order ($j=0$) formulas are
$$\begin{aligned} p_{n,0} & = 0,\;q_{n,0}\;=\;0,\\ a_{0,0} & = \frac{[\kappa-i(\Delta_{0}+\lambda-\Omega)]E}{[\kappa-i({\Delta}_{0}+\lambda-\Omega)][\kappa+i(\Delta_{0}+\lambda)]-4\Lambda^{2}},\\ a_{-1,0} & = \frac{2\Lambda e^{i\theta}E}{[\kappa+i(\Delta_{0}+\lambda-\Omega)][\kappa-i(\Delta_{0}+\lambda)]-4\Lambda^{2}},\\ a_{n,0} & = 0,\qquad n\neq0,-1.\end{aligned}$$
For $j\geq 1$,
$$\begin{aligned} q_{n,j} & = \omega_{m}\sum_{k=0}^{j-1}\sum_{m=-\infty}^{\infty}\frac{a_{m,k}^{*}a_{n+m,j-k-1}}{\omega_{m}^{2}-(n\Omega)^{2}+i\gamma_{m}n\Omega},\\ p_{n,j} & = \frac{in\Omega}{\omega_{m}}q_{n,j},\\ a_{n,j} & = \frac{2\Lambda e^{i\theta}a_{-n-1,j}^{*}}{\kappa+i(\Delta_{0}+\lambda+n\Omega)}+i\sum_{k=0}^{j-1}\sum_{m=-\infty}^{\infty}\frac{a_{m,k}q_{n-m,j-k-1}}{\kappa+i(\Delta_{0}+\lambda+n\Omega)}. \end{aligned}$$
As shown in Fig. 2 and Fig. 3, we can see that the analytical results (green dashed) fit well with the the numerical results (blue solid). In all the calculations, we truncated the series in Eq. (8) to the terms with subscripts $|n|\leq 1$ and $j\leq 8$. The coefficients $O_{n}$ in Eqs. (6), (7), and (8), will be helpful for seeking the optimal parameters for studying quantum dynamics and the mechanical entanglement, see discussion later. In addition, owing to $\gamma _{m}\ll \kappa$ , it takes longer time for the phase space trajectories of the mechanical mode mean value $\langle q(t)\rangle$ and $\langle p(t)\rangle$ to approach to the limit cycle given by the analytical solutions, see Fig. 3.

 figure: Fig. 2.

Fig. 2. (a) The phase space trajectories of $\langle a(t)\rangle$ from $t=0$ to $t=900\tau$. The numerical result (blue solid) simulated by Eq. (5) agrees well with the analytical result (green dashed) simulated by Eq. (6). (b) Time evolution of the modulus of the cavity-mode mean values for the time interval [$895\tau$, $900\tau$]. The synchronization of $|\langle a_1(t)\rangle |$ (thick orange solid) and $|\langle a_2(t)\rangle |$ (thin blue solid) implies that the classical dynamics of the two subsystems can be effectively described by a set of common reduced equations of motion Eq. (5). The analytical solution $|\langle a(t)\rangle |$ (thin green dashed) fits well with the numerical results $|\langle a_1(t)\rangle |$ and $|\langle a_2(t)\rangle |$, which further proves the validity of Eq. (5). Here the time unit $\tau =2\pi /\Omega$. The system parameters are [69] (in units of $\omega _{m}$): $(E,g,\gamma _{m},\Delta _{0},\lambda )/\omega _{m}=(1.5\times 10^{5},4\times 10^{-6},2\times 10^{-3},3,2)$, cavity decay $\kappa /\omega _{m}=0.2$, $n_{m}=0$, and $n_{a}=0$. As for OPA $\Lambda /\kappa =0.3$, $\theta =0$ and $\Omega /\omega _{m}=2$.

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 figure: Fig. 3.

Fig. 3. Phase space trajectories of the classical mean values $\langle q(t)\rangle$ and $\langle p(t)\rangle$ for time intervals (a) $[0,20\tau ]$, (b) $[980\tau ,1000\tau ]$, [$1980\tau$, $2000\tau$ ], and (d) [$2980\tau$, $3000\tau$]. Again, the blue solid and green dashed lines are obtained from numerical and analytical solutions, respectively, and the parameters are the same as in Fig. 2.

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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

$$\begin{aligned} H_\textrm{lin} & = {\displaystyle \sum_{j=1}^{2}}\{\Delta(t)\delta{\textstyle a_{j}^{\dagger}}\delta a_{j}+\omega_{m}\delta b_{j}^{\dagger}\delta b_{j}-\frac{1}{2}[G(t)^{*}\delta a_{j}+G(t)\delta a_{j}^{\dagger}](\delta b_{j}+\delta b_{j}^{\dagger})\\ & \quad +i\Lambda[e^{-i(\Omega t-\theta)}\delta a_{j}^{\dagger2}-e^{i(\Omega t-\theta)}\delta a_{j}^{2}]\}+\lambda(\delta a_{1}^{\dagger}\delta a_{2}+\delta a_{1}\delta a_{2}^{\dagger}).\end{aligned}$$
By introducing the nonlocal operators $A_{1}=(\delta a_{1}+\delta a_{2})/\sqrt {2}, A_{2}=(\delta a_{1}-\delta a_{2})/\sqrt {2}, B_{1}=(\delta b_{1}+\delta b_{2})/\sqrt {2}, B_{2}=(\delta b_{1}-\delta b_{2})/\sqrt {2}$, the Hamiltonian in Eq. (9) can be expressed as the sum of two uncoupled Hamiltonian, i.e. $H_\textrm {lin}=\Sigma _{j=1,2}H_{j}^\textrm {lin}$, where
$$\begin{aligned}H_{j}^\textrm{lin} & = {\displaystyle \Delta^{'}(t)A_{j}^{\dagger}A_{j}+\omega_{m}B_{j}^{\dagger}B_{j}}-\frac{1}{2}[G^{*}(t)A_{j}+G(t)A_{j}^{\dagger}](B_{j}+B_{j}^{\dagger})\\ & \quad +i\Lambda[e^{-i(\Omega t-\theta)}A_{j}^{\dagger2}-e^{i(\Omega t-\theta)}A_{j}^{2}], \end{aligned}$$
and the effective detunings of nonlocal cavity modes $A_{j}$ are modified as $\Delta _{1}(t)=\Delta (t)+\lambda$ and $\Delta _{2}(t)=\Delta (t)-\lambda$. Based on the analytical results for the classical mean values, the effective coupling can be approximately expressed as $G(t)=g_{0}+g_{1}e^{-i\Omega t}$, and the effective detuning of each cavity mode $a_{j}$ is given by $\Delta (t)=\Delta _{0}-\delta _{0}-\delta _{1}(e^{-i\Omega t}+e^{i\Omega t})$. Note that in the following we then consider the parameter regime with $|g_{0}|,|g_{1}|,\delta _{0},\delta _{1}\ll \omega _{m},\Omega$ being well satisfied. Setting the cavity-cavity coupling $\lambda =2\omega _{m}$ and the cavity detuning $\Delta _{0}=3\omega _{m}$, and neglecting the rapid oscillating terms $\sim e^{\pm i\Omega t}$, we can then obtain $\Delta _{1}^{'}(t) =\Delta (t)+\lambda \simeq 5\omega _{m}-\delta _{0},\; \Delta _{2}^{'}(t) =\Delta (t)+\lambda \simeq \omega _{m}-\delta _{0}$. In the rotating frame with respect to the free terms ${\displaystyle \Delta _{j}^{'}(t)A_{j}^{\dagger }A_{j}+\omega _{m}B_{j}^{\dagger }B_{j}}$, the Hamiltonian $H_{j}^\textrm {lin}$ becomes
$$\begin{aligned} H_{j}^\textrm{lin} & = \{-\frac{1}{2}[g_{1}^{*}e^{i(\Omega-\Delta_{j}^{'}-\omega_{m})t}B_{j}+g_{0}^{*}e^{-i(\Delta_{j}^{'}-\omega_{m})t}B_{j}^{\dagger}]A_{j} -\frac{1}{2}[g_{0}^{*}e^{-i(\Delta_{j}^{'}+\omega_{m})t}B_{j}\\ & \quad +g_{1}^{*}e^{i(\Omega-\Delta_{j}^{'}+\omega_{m})t}B_{j}^{\dagger}]A_{j}+i\Lambda e^{i\theta}e^{-i(\Omega-2\Delta_{j}^{'})t}A_{j}^{\dagger2}\}+ \textrm{H.c.}.\end{aligned}$$
Choosing $\Omega =2\omega _{m}-\delta _{0}$ and ignoring rapid oscillating terms, Eq. (11) reduces to
$$\begin{aligned} H_{1}^\textrm{lin} & \simeq 0,\\ H_{2}^\textrm{lin} & \simeq -\frac{1}{2}\{[g_{1}^{*}B_{2}+g_{0}^{*}e^{i\delta_{0}t}B_{2}^{\dagger}]A_{2}+\textrm{H.c.}\}+i\Lambda(e^{i\theta}e^{-i\delta_{0}t}A_{2}^{\dagger2}-e^{-i\theta}e^{i\delta_{0}t}A_{2}^{2}).\end{aligned}$$
For $\delta _{0}\ll |g_{0}|,\Lambda$, the slow varying terms $g_{0}^{*}e^{i\delta _{0}t}B_{2}^{\dagger }A_{2}$ and $-i\Lambda e^{i\theta }e^{i\delta _{0}t}A_{2}^{2}$ can be further treated as constants $g_{0}^{*}B_{2}^{\dagger }A_{2}$ and $-i\Lambda e^{i\theta }A_{2}^{2}$, respectively. Thus, we finally arrive at
$$H_{2}^\textrm{lin} \simeq [-\frac{1}{2}(g_{1}B_{2}^{\dagger}+g_{0}B_{2})A_{2}^{\dagger}+\textrm{H.c.}]+i\Lambda(e^{i\theta}A_{2}^{\dagger2}-e^{-i\theta}A_{2}^{2}),$$
where the first part corresponds to a nonlocal squeezed mode $B_{2}$, implying that the distant mechanical entanglement can exist. To see it explicitly, we introduce an Bogoliubov mode defined as $\beta \equiv B_{2}\cosh r_{1}+e^{i\phi }B_{2}^{\dagger }\textrm {sinh}r_{1}$ with $\tanh r_{1}=|g_{1}|/|g_{0}|$, $\phi =\phi _{1}-\phi _{0}$ [10,23], and the Hamiltonian $H_{2}^\textrm {lin}$ can be rewritten as
$$H_{2}^\textrm{lin} = -g\beta A_{2}^{\dagger}-g^{*}\beta^{\dagger}A_{2}+i\Lambda(e^{i\theta}A_{2}^{\dagger2}-e^{-i\theta}A_{2}^{2}).$$
with $g=e^{i\phi _{0}}\sqrt {|g_{0}|^{2}-|g_{1}|^{2}}/2$ , $g_{0}=|g_{0}|e^{i\phi _{0}}$ and $g_{1}=|g_{1}|e^{i\phi _{1}}$. Note that the OPAs make effects on mechanical entanglement mainly in two aspects. Firstly, the squeezed mode $\beta$ arises from the OPA pumping, which further modulates the system by the coupling strength $g_{1}$. However, it is limited by the parameter condition $|g_{1}|\ll |g_{0}|$, corresponding to $|a_{-1,0}|\ll |a_{0,0}|$, see Eq. (7). Second, the leading contribution by the OPAs can be witnessed by including just the damping of the mode $\beta$ and the dissipation of the mode $A_{2}$ and excluding the thermal noise
$$\begin{aligned} \dot{\beta} & = ig^{*}A_{2}-\frac{\gamma_{m}}{2}\beta,\\ \dot{A}_{2} & = -\kappa A_{2}+ig\beta+2\Lambda e^{i\theta}A_{2}^{\dagger}.\end{aligned}$$
In the weakly effective coupling regime $\kappa \gg |g|$, the adiabatic approximation can be made to the hybrid cavity mode $A_{2}$, i.e. $\dot {A}_{2}=0$, leading to $A_{2}=\frac {1}{\kappa ^{2}-4\Lambda ^{2}}(i\kappa g\beta -i2\Lambda e^{i\theta }g^{*}\beta ^{\dagger })$. And substituting it into Eq. (15), we finally have
$$\begin{aligned} \dot{\beta} & = ig^{*}\frac{1}{\kappa^{2}-4\Lambda^{2}}(i\kappa g\beta-i2\Lambda e^{i\theta}g^{*}\beta^{\dagger})-\frac{\gamma_{m}}{2}\beta,\\ & = -(\frac{\kappa|g|^{2}}{\kappa^{2}-4\Lambda^{2}}+\frac{\gamma_{m}}{2})\beta+\frac{2\Lambda e^{i\theta}g^{*2}}{\kappa^{2}-4\Lambda^{2}}\beta^{\dagger}.\end{aligned}$$
Equation (16) proves that the squeezing of the Bogoliubov mode $\beta$, and therefore the mechanical entanglement between the original mechanical modes $b_{1}$ and $b_{2}$ can be generated or further enhanced by cavity parametric interactions. Moreover, the first term in Eq. (16) can be regarded as the effective damping rate $\Gamma _{m}=\frac {\kappa |g|^{2}}{\kappa ^{2}-4\Lambda ^{2}}+\frac {\gamma _{m}}{2}$ of the Bogoliubov mode $\beta$, and the effective interaction Hamiltonian for $\beta$ can be written as :
$$H_{\beta} = \frac{\Lambda e^{i(\theta+\pi/2)}g^{*2}}{\kappa^{2}-4\Lambda^{2}}\beta^{\dagger2}+\textrm{H.c.}.$$
The unitary evolution operator described by $H_{\beta }$ can actually be given by a squeeze operator $S(\xi )$, corresponding to mechanical parametric amplification, namely [71]
$$S(\xi) = \textrm{exp}[\frac{1}{2}(\xi^{*}\beta^{2}-\xi\beta^{\dagger2})],$$
where $\xi =re^{i\Phi }$, with $r=\frac {2\Lambda |g|^{2}}{\kappa ^{2}-4\Lambda ^{2}}$ and $\Phi =\theta -2\phi _{0}+\pi /2$ being the squeezing strength and orientation, respectively. As a result, the larger the squeezing of the Bogoliubov mode $\beta$ is, the stronger the mechanical entanglement will be. However, we note that the effective damping of $\beta$ is also amplified by the increase of the OPA gain, giving rise to $\Gamma _m=\frac {|g|^{2}}{\kappa }\frac {1}{1-4\Lambda ^2/\kappa ^2}+\frac {\gamma _m}{2}$. Without OPA pumping, i.e. $\Lambda =0$, the effective damping is given by $\frac {|g|^{2}}{\kappa }+\frac {\gamma _m}{2}$. In comparison, the squeezing strength $r=\frac {2\Lambda }{\kappa }(\frac {|g|^{2}}{\kappa }\frac {1}{1-4\Lambda ^2/\kappa ^2})$ for Bogoliubov mode $\beta$ is improved by an additional factor proportional to the OPA gain for $\Lambda /\kappa \rightarrow 0.5$, namely $r/\Gamma _m \rightarrow 2\Lambda /\kappa$. Thus, the optimal squeezing of the Bogoliubov mode $\beta$ is then achieved by balancing the two processes, see further discussion later. Moreover, it should be noted that the stability condition requires the OPA gain to satisfy $\Lambda /\kappa\;<\;0.5$ [62], which thus gives the upper bound for the degree of mechanical entanglement. On the other hand, the effective mechanical damping $\Gamma _{m}$ as well as the coupling strength $g$ are also amplified by the increasing of driving strength $E$, implying that the excessive increase of $E$ may break the adiabatic approximation and bring side effects to entanglement generation. Therefore, the parameters chosen for generating mechanical entanglement should fulfill all these conditions.

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

$$\begin{aligned} M(t) & = \left[\begin{array}{cccccccc} 0 & \omega_{m} & 0 & 0 & 0 & 0 & 0 & 0\\ -\omega_{m} & -\gamma_{m} & 0 & 0 & G_{x}(t) & G_{y}(t) & 0 & 0\\ 0 & 0 & 0 & \omega_{m} & 0 & 0 & 0 & 0\\ 0 & 0 & -\omega_{m} & -\gamma_{m} & 0 & 0 & G_{x}(t) & G_{y}(t)\\ -G_{y}(t) & 0 & 0 & 0 & R_{1} & R_{2} & 0 & \lambda\\ G_{x}(t) & 0 & 0 & 0 & R_{3} & R_{4} & -\lambda & 0\\ 0 & 0 & -G_{y}(t) & 0 & 0 & \lambda & R_{1} & R_{2}\\ 0 & 0 & G_{x}(t) & 0 & -\lambda & 0 & R_{3} & R_{4} \end{array}\right], \end{aligned}$$
with $n(t)=[0,\xi _{1}(t),0,\xi _{2}(t),\sqrt {2\kappa }\delta x_{1}^{in}(t),\sqrt {2\kappa }\delta y_{1}^{in}(t),\sqrt {2\kappa }\delta x_{2}^{in}(t),\sqrt {2\kappa }\delta y_{2}^{in}(t)]^{T}$. The abbreviation notations are defined as $R_{1}=-\kappa +2\Lambda \cos \bar {\theta },$ $R_{2}=\Delta -2\Lambda \sin \bar {\theta },$ $R_{3}=-\Delta -2\Lambda \sin \bar {\theta },$ and $R_{4}=-\kappa -2\Lambda \cos \bar {\theta }$ with $\bar {\theta }=\Omega t-\theta$. $G_{x}(t)$ and $G_{y}(t)$ are, respectively, the real and imaginary parts of the effective coupling $G(t)=\sqrt {2}\bar {g}\langle a(t)\rangle$. To ensure that the system works in the stability regime, all the eigenvalues of $M(t)$ at any time should have negative real parts, corresponding to the Routh-Hurwitz criterion [72]. For all discussions in the present work, the stability condition for all situations are carefully verified. When the system is stable and the linearization is valid, the system will converge to a unique Gaussian state. Then we can describe the asymptotic state of the quantum fluctuations by a $8\times 8$ covariance matrix (CM) $V(t)$ with $V_{ij}(t)=\langle u_{i}(t)u_{j}(t)+u_{j}(t)u_{i}(t)\rangle /2$ and the diagonal diffusion matrix $D=\textrm {diag}[0,\gamma _{m}(2n_{m}+1),0,\gamma _{m}(2n_{m}+1),\kappa ,\kappa ,\kappa ,\kappa ]$, leading to $\dot {V}(t)=M(t)V(t)+V(t)M(t)^{T}+D$.

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

$$E_{N}=\max[0,-\ln2\eta],$$
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.

 figure: Fig. 4.

Fig. 4. Mechanical entanglement $E_{N}$ versus the evolution time $t$ for different parametric phases $\theta =0$ (blue), $\pi /2$ (green), and $\pi$ (black). Other parameters are the same as those in Fig. 2.

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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.

 figure: Fig. 5.

Fig. 5. Maximum mechanical entanglement $E_{N,max}$ versus the detuning of OPA $\Omega /\omega _{m}$. The vertical black dashed line shows the optimal value of $\Omega /\omega _{m}$ at $\Omega _{\textrm {opt}}/\omega _{m}=1.994$. Other parameters are same as in Fig. 2.

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 figure: Fig. 6.

Fig. 6. $E_{N,max}$ versus the driving laser $E/\omega _{m}$ and cavity decay $\kappa /\omega _{m}$. Here the ratio of the parametric gain to the cavity decay $\Lambda /\kappa =0.3$ and $\Omega =\Omega _\textrm {opt}$ are chosen. Other parameters are same as in Fig. 4.

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 figure: Fig. 7.

Fig. 7. Maximum mechanical entanglement $E_{N,max}$ versus the ratio of parametric gain to the cavity decay $\Lambda /\kappa$ for different cavity decay $\kappa /\omega _{m}=0.2$ (blue), $0.4$ (orange), $0.6$ (green) and $0.8$ (black) when $E/\omega _{m}=1.5\times 10^{5}$. For each curve the parametric gains $\Lambda /\kappa$ is rescaled by corresponding $\kappa$. Other parameters are the same as in Fig. 6.

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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.

 figure: Fig. 8.

Fig. 8. (a) Without OPAs the steady-state mechanical entanglement $E_{N}$ versus the cavity decay $\kappa /\omega _{m}$ and thermal phonon number $n_{m}$, with $\lambda /\omega _{m}=20$ and $E/\omega _{m}=1\times 10^{7}$. (b) $E_N$ versus time $t$ with $\lambda /\omega _{m}=2$, $E/\omega _{m}=1.5\times 10^{5}$, $\kappa /\omega _m=0.2$ and the rescaled parametric gain $\Lambda /\kappa =0.49$ for different thermal phonon number $n_m=0$ (blue) and $n_m=0.2$ (black). The other parameters are the same as in Fig. 6.

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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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68. A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82(2), 1155–1208 (2010). [CrossRef]  

69. J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452(7183), 72–75 (2008). [CrossRef]  

70. A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A 86(1), 013820 (2012). [CrossRef]  

71. C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University, 2004).

72. E. X. DeJesus and C. Kaufman, “Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35(12), 5288–5290 (1987). [CrossRef]  

73. G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65(3), 032314 (2002). [CrossRef]  

74. G. Adesso, A. Serafini, and F. Illuminati, “Extremal entanglement and mixedness in continuous variable systems,” Phys. Rev. A 70(2), 022318 (2004). [CrossRef]  

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2019 (7)

C. H. Bai, D. Y. Wang, S. Zhang, S. Liu, and H. F. Wang, “Modulation-based atom-mirror entanglement and mechanical squeezing in an unresolved-sideband optomechanical system,” Ann. Phys. 531(7), 1800271 (2019).
[Crossref]

H. T. Tan, W. W. Deng, and L. Sun, “Directional steering as a sufficient and necessary condition for gaussian entanglement swapping: Application to distant optomechanical oscillators,” Phys. Rev. A 99(4), 043834 (2019).
[Crossref]

J. Y. Yang, D. Y. Wang, C. H. Bai, S. Y. Guan, X. Y. Gao, A. D. Zhu, and H. F. Wang, “Ground-state cooling of mechanical oscillator via quadratic optomechanical coupling with two coupled optical cavities,” Opt. Express 27(16), 22855–22867 (2019).
[Crossref]

J. H. Gan, Y. C. Liu, C. Lu, X. Wang, M. K. Tey, and L. You, “Intracavity-squeezed optomechanical cooling,” Laser Photonics Rev. 13(11), 1900120 (2019).
[Crossref]

G. Huang, W. Deng, H. T. Tan, and G. Cheng, “Generation of squeezed states and single-phonon states via homodyne detection and photon subtraction on the filtered output of an optomechanical cavity,” Phys. Rev. A 99(4), 043819 (2019).
[Crossref]

C. H. Bai, D. Y. Wang, S. Zhang, and H. F. Wang, “Qubit-assisted squeezing of mirror motion in a dissipative cavity optomechanical system,” Sci. China: Phys., Mech. Astron. 62(7), 970311 (2019).
[Crossref]

H. T. Tan and H. Zhan, “Strong mechanical squeezing and optomechanical steering via continuous monitoring in optomechanical systems,” Phys. Rev. A 100(2), 023843 (2019).
[Crossref]

2018 (11)

C. S. Hu, Z. B. Yang, H. Wu, Y. Li, and S. B. Zheng, “Twofold mechanical squeezing in a cavity optomechanical system,” Phys. Rev. A 98(2), 023807 (2018).
[Crossref]

X. W. Xu, H. Q. Shi, A. X. Chen, and Y. X. Liu, “Cross-correlation between photons and phonons in quadratically coupled optomechanical systems,” Phys. Rev. A 98(1), 013821 (2018).
[Crossref]

Y. H. Chen, Z. C. Shi, J. Song, and Y. Xia, “Invariant-based inverse engineering for fluctuation transfer between membranes in an optomechanical cavity system,” Phys. Rev. A 97(2), 023841 (2018).
[Crossref]

J. Li, S. Gröblacher, S. Y. Zhu, and G. S. Agarwal, “Generation and detection of non-gaussian phonon-added coherent states in optomechanical systems,” Phys. Rev. A 98(1), 011801 (2018).
[Crossref]

X. Y. Lü, L. L. Zheng, G. L. Zhu, and Y. Wu, “Single-photon-triggered quantum phase transition,” Phys. Rev. Appl. 9(6), 064006 (2018).
[Crossref]

G. L. Zhu, X. Y. Lü, L. L. Wan, T. S. Yin, Q. Bin, and Y. Wu, “Controllable nonlinearity in a dual-coupling optomechanical system under a weak-coupling regime,” Phys. Rev. A 97(3), 033830 (2018).
[Crossref]

D. G. Lai, F. Zou, B. P. Hou, Y. F. Xiao, and J. Q. Liao, “Simultaneous cooling of coupled mechanical resonators in cavity optomechanics,” Phys. Rev. A 98(2), 023860 (2018).
[Crossref]

S. Huang and A. X. Chen, “Improving the cooling of a mechanical oscillator in a dissipative optomechanical system with an optical parametric amplifier,” Phys. Rev. A 98(6), 063818 (2018).
[Crossref]

C. G. Liao, R. X. Chen, H. Xie, and X. M. Lin, “Reservoir-engineered entanglement in a hybrid modulated three-mode optomechanical system,” Phys. Rev. A 97(4), 042314 (2018).
[Crossref]

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556(7702), 473–477 (2018).
[Crossref]

J. Li, S. Y. Zhu, and G. S. Agarwal, “Magnon-photon-phonon entanglement in cavity magnomechanics,” Phys. Rev. Lett. 121(20), 203601 (2018).
[Crossref]

2017 (2)

R. G. Yang, N. Li, J. Zhang, J. Li, and T. C. Zhang, “Enhanced entanglement of two optical modes in optomechanical systems via an optical parametric amplifier,” J. Phys. B: At., Mol. Opt. Phys. 50(8), 085502 (2017).
[Crossref]

Y. Li, Y. Y. Huang, X. Z. Zhang, and L. Tian, “Optical directional amplification in a three-mode optomechanical system,” Opt. Express 25(16), 18907–18916 (2017).
[Crossref]

2016 (3)

G. S. Agarwal and S. Huang, “Strong mechanical squeezing and its detection,” Phys. Rev. A 93(4), 043844 (2016).
[Crossref]

X. W. Xu, Y. Li, A. X. Chen, and Y. X. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93(2), 023827 (2016).
[Crossref]

Z. J. Deng, X. B. Yan, Y. D. Wang, and C. W. Wu, “Optimizing the output-photon entanglement in multimode optomechanical systems,” Phys. Rev. A 93(3), 033842 (2016).
[Crossref]

2015 (4)

X. Y. Lü, J. Q. Liao, L. Tian, and F. Nori, “Steady-state mechanical squeezing in an optomechanical system via duffing nonlinearity,” Phys. Rev. A 91(1), 013834 (2015).
[Crossref]

K. Qu and G. S. Agarwal, “Generating quadrature squeezed light with dissipative optomechanical coupling,” Phys. Rev. A 91(6), 063815 (2015).
[Crossref]

X. W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91(5), 053854 (2015).
[Crossref]

X. Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, “Squeezed optomechanics with phase-matched amplification and dissipation,” Phys. Rev. Lett. 114(9), 093602 (2015).
[Crossref]

2014 (5)

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity–time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

R. X. Chen, L. T. Shen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Enhancement of entanglement in distant mechanical vibrations via modulation in a coupled optomechanical system,” Phys. Rev. A 89(2), 023843 (2014).
[Crossref]

J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89(1), 014302 (2014).
[Crossref]

L. Ying, Y. C. Lai, and C. Grebogi, “Quantum manifestation of a synchronization transition in optomechanical systems,” Phys. Rev. A 90(5), 053810 (2014).
[Crossref]

2013 (6)

D. D. B. Rao and K. Mølmer, “Dark entangled steady states of interacting rydberg atoms,” Phys. Rev. Lett. 111(3), 033606 (2013).
[Crossref]

H. T. Tan, G. X. Li, and P. Meystre, “Dissipation-driven two-mode mechanical squeezed states in optomechanical systems,” Phys. Rev. A 87(3), 033829 (2013).
[Crossref]

W. J. Gu and G. X. Li, “Squeezing of the mirror motion via periodic modulations in a dissipative optomechanical system,” Opt. Express 21(17), 20423–20440 (2013).
[Crossref]

T. P. Purdy, P. L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X 3(3), 031012 (2013).
[Crossref]

A. Kronwald, F. Marquardt, and A. A. Clerk, “Arbitrarily large steady-state bosonic squeezing via dissipation,” Phys. Rev. A 88(6), 063833 (2013).
[Crossref]

Y. D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110(25), 253601 (2013).
[Crossref]

2012 (7)

A. Xuereb, M. Barbieri, and M. Paternostro, “Multipartite optomechanical entanglement from competing nonlinearities,” Phys. Rev. A 86(1), 013809 (2012).
[Crossref]

M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. 109(23), 233906 (2012).
[Crossref]

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: Discord and related measures,” Rev. Mod. Phys. 84(4), 1655–1707 (2012).
[Crossref]

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

A. Mari and J. Eisert, “Opto- and electro-mechanical entanglement improved by modulation,” New J. Phys. 14(7), 075014 (2012).
[Crossref]

C. Joshi, J. Larson, M. Jonson, E. Andersson, and P. Öhberg, “Entanglement of distant optomechanical systems,” Phys. Rev. A 85(3), 033805 (2012).
[Crossref]

A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A 86(1), 013820 (2012).
[Crossref]

2011 (4)

K. G. H. Vollbrecht, C. A. Muschik, and J. I. Cirac, “Entanglement distillation by dissipation and continuous quantum repeaters,” Phys. Rev. Lett. 107(12), 120502 (2011).
[Crossref]

J. Q. Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83(3), 033820 (2011).
[Crossref]

L. Zhou, Y. Han, J. Jing, and W. Zhang, “Entanglement of nanomechanical oscillators and two-mode fields induced by atomic coherence,” Phys. Rev. A 83(5), 052117 (2011).
[Crossref]

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475(7356), 359–363 (2011).
[Crossref]

2010 (2)

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464(7289), 697–703 (2010).
[Crossref]

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82(2), 1155–1208 (2010).
[Crossref]

2009 (5)

Z. Q. Yin and Y. J. Han, “Generating epr beams in a cavity optomechanical system,” Phys. Rev. A 79(2), 024301 (2009).
[Crossref]

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79(1), 013821 (2009).
[Crossref]

S. Huang and G. S. Agarwal, “Normal-mode splitting in a coupled system of a nanomechanical oscillator and a parametric amplifier cavity,” Phys. Rev. A 80(3), 033807 (2009).
[Crossref]

A. Mari and J. Eisert, “Gently modulating optomechanical systems,” Phys. Rev. Lett. 103(21), 213603 (2009).
[Crossref]

S. Huang and G. S. Agarwal, “Entangling nanomechanical oscillators in a ring cavity by feeding squeezed light,” New J. Phys. 11(10), 103044 (2009).
[Crossref]

2008 (4)

G. Vacanti, M. Paternostro, G. M. Palma, and V. Vedral, “Optomechanical to mechanical entanglement transformation,” New J. Phys. 10(9), 095014 (2008).
[Crossref]

C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77(5), 050307 (2008).
[Crossref]

C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78(3), 032316 (2008).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452(7183), 72–75 (2008).
[Crossref]

2007 (3)

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99(9), 093902 (2007).
[Crossref]

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99(9), 093901 (2007).
[Crossref]

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref]

2006 (1)

S. Pirandola, D. Vitali, P. Tombesi, and S. Lloyd, “Macroscopic entanglement by entanglement swapping,” Phys. Rev. Lett. 97(15), 150403 (2006).
[Crossref]

2005 (1)

M. Pinard, A. Dantan, D. Vitali, O. Arcizet, T. Briant, and A. Heidmann, “Entangling movable mirrors in a double-cavity system,” Europhys. Lett. 72(5), 747–753 (2005).
[Crossref]

2004 (1)

G. Adesso, A. Serafini, and F. Illuminati, “Extremal entanglement and mixedness in continuous variable systems,” Phys. Rev. A 70(2), 022318 (2004).
[Crossref]

2003 (1)

J. Zhang, K. Peng, and S. L. Braunstein, “Quantum-state transfer from light to macroscopic oscillators,” Phys. Rev. A 68(1), 013808 (2003).
[Crossref]

2002 (1)

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65(3), 032314 (2002).
[Crossref]

2001 (1)

J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73(3), 565–582 (2001).
[Crossref]

1987 (1)

E. X. DeJesus and C. Kaufman, “Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35(12), 5288–5290 (1987).
[Crossref]

1986 (1)

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57(20), 2520–2523 (1986).
[Crossref]

Adesso, G.

G. Adesso, A. Serafini, and F. Illuminati, “Extremal entanglement and mixedness in continuous variable systems,” Phys. Rev. A 70(2), 022318 (2004).
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Agarwal, G. S.

J. Li, S. Y. Zhu, and G. S. Agarwal, “Magnon-photon-phonon entanglement in cavity magnomechanics,” Phys. Rev. Lett. 121(20), 203601 (2018).
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J. Li, S. Gröblacher, S. Y. Zhu, and G. S. Agarwal, “Generation and detection of non-gaussian phonon-added coherent states in optomechanical systems,” Phys. Rev. A 98(1), 011801 (2018).
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G. S. Agarwal and S. Huang, “Strong mechanical squeezing and its detection,” Phys. Rev. A 93(4), 043844 (2016).
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K. Qu and G. S. Agarwal, “Generating quadrature squeezed light with dissipative optomechanical coupling,” Phys. Rev. A 91(6), 063815 (2015).
[Crossref]

S. Huang and G. S. Agarwal, “Entangling nanomechanical oscillators in a ring cavity by feeding squeezed light,” New J. Phys. 11(10), 103044 (2009).
[Crossref]

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79(1), 013821 (2009).
[Crossref]

S. Huang and G. S. Agarwal, “Normal-mode splitting in a coupled system of a nanomechanical oscillator and a parametric amplifier cavity,” Phys. Rev. A 80(3), 033807 (2009).
[Crossref]

Allman, M. S.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475(7356), 359–363 (2011).
[Crossref]

Andersson, E.

C. Joshi, J. Larson, M. Jonson, E. Andersson, and P. Öhberg, “Entanglement of distant optomechanical systems,” Phys. Rev. A 85(3), 033805 (2012).
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Ansmann, M.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464(7289), 697–703 (2010).
[Crossref]

Arcizet, O.

M. Pinard, A. Dantan, D. Vitali, O. Arcizet, T. Briant, and A. Heidmann, “Entangling movable mirrors in a double-cavity system,” Europhys. Lett. 72(5), 747–753 (2005).
[Crossref]

Aspelmeyer, M.

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556(7702), 473–477 (2018).
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M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
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D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
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Bai, C. H.

C. H. Bai, D. Y. Wang, S. Zhang, S. Liu, and H. F. Wang, “Modulation-based atom-mirror entanglement and mechanical squeezing in an unresolved-sideband optomechanical system,” Ann. Phys. 531(7), 1800271 (2019).
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J. Y. Yang, D. Y. Wang, C. H. Bai, S. Y. Guan, X. Y. Gao, A. D. Zhu, and H. F. Wang, “Ground-state cooling of mechanical oscillator via quadratic optomechanical coupling with two coupled optical cavities,” Opt. Express 27(16), 22855–22867 (2019).
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C. H. Bai, D. Y. Wang, S. Zhang, and H. F. Wang, “Qubit-assisted squeezing of mirror motion in a dissipative cavity optomechanical system,” Sci. China: Phys., Mech. Astron. 62(7), 970311 (2019).
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Barbieri, M.

A. Xuereb, M. Barbieri, and M. Paternostro, “Multipartite optomechanical entanglement from competing nonlinearities,” Phys. Rev. A 86(1), 013809 (2012).
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Barnard, A.

M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. 109(23), 233906 (2012).
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Bender, C. M.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity–time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Bialczak, R. C.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464(7289), 697–703 (2010).
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Bin, Q.

G. L. Zhu, X. Y. Lü, L. L. Wan, T. S. Yin, Q. Bin, and Y. Wu, “Controllable nonlinearity in a dual-coupling optomechanical system under a weak-coupling regime,” Phys. Rev. A 97(3), 033830 (2018).
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Böhm, H. R.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref]

Braunstein, S. L.

J. Zhang, K. Peng, and S. L. Braunstein, “Quantum-state transfer from light to macroscopic oscillators,” Phys. Rev. A 68(1), 013808 (2003).
[Crossref]

Briant, T.

M. Pinard, A. Dantan, D. Vitali, O. Arcizet, T. Briant, and A. Heidmann, “Entangling movable mirrors in a double-cavity system,” Europhys. Lett. 72(5), 747–753 (2005).
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Brodutch, A.

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: Discord and related measures,” Rev. Mod. Phys. 84(4), 1655–1707 (2012).
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Brune, M.

J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73(3), 565–582 (2001).
[Crossref]

Cable, H.

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: Discord and related measures,” Rev. Mod. Phys. 84(4), 1655–1707 (2012).
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Chen, A. X.

S. Huang and A. X. Chen, “Improving the cooling of a mechanical oscillator in a dissipative optomechanical system with an optical parametric amplifier,” Phys. Rev. A 98(6), 063818 (2018).
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X. W. Xu, H. Q. Shi, A. X. Chen, and Y. X. Liu, “Cross-correlation between photons and phonons in quadratically coupled optomechanical systems,” Phys. Rev. A 98(1), 013821 (2018).
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X. W. Xu, Y. Li, A. X. Chen, and Y. X. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93(2), 023827 (2016).
[Crossref]

Chen, J. P.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99(9), 093902 (2007).
[Crossref]

Chen, R. X.

C. G. Liao, R. X. Chen, H. Xie, and X. M. Lin, “Reservoir-engineered entanglement in a hybrid modulated three-mode optomechanical system,” Phys. Rev. A 97(4), 042314 (2018).
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R. X. Chen, L. T. Shen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Enhancement of entanglement in distant mechanical vibrations via modulation in a coupled optomechanical system,” Phys. Rev. A 89(2), 023843 (2014).
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Chen, Y. H.

Y. H. Chen, Z. C. Shi, J. Song, and Y. Xia, “Invariant-based inverse engineering for fluctuation transfer between membranes in an optomechanical cavity system,” Phys. Rev. A 97(2), 023841 (2018).
[Crossref]

Chen, Z. B.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012).
[Crossref]

Cheng, G.

G. Huang, W. Deng, H. T. Tan, and G. Cheng, “Generation of squeezed states and single-phonon states via homodyne detection and photon subtraction on the filtered output of an optomechanical cavity,” Phys. Rev. A 99(4), 043819 (2019).
[Crossref]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

Cicak, K.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475(7356), 359–363 (2011).
[Crossref]

Cirac, J. I.

K. G. H. Vollbrecht, C. A. Muschik, and J. I. Cirac, “Entanglement distillation by dissipation and continuous quantum repeaters,” Phys. Rev. Lett. 107(12), 120502 (2011).
[Crossref]

Cleland, A. N.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464(7289), 697–703 (2010).
[Crossref]

Clerk, A. A.

Y. D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110(25), 253601 (2013).
[Crossref]

A. Kronwald, F. Marquardt, and A. A. Clerk, “Arbitrarily large steady-state bosonic squeezing via dissipation,” Phys. Rev. A 88(6), 063833 (2013).
[Crossref]

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82(2), 1155–1208 (2010).
[Crossref]

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99(9), 093902 (2007).
[Crossref]

Dantan, A.

M. Pinard, A. Dantan, D. Vitali, O. Arcizet, T. Briant, and A. Heidmann, “Entangling movable mirrors in a double-cavity system,” Europhys. Lett. 72(5), 747–753 (2005).
[Crossref]

DeJesus, E. X.

E. X. DeJesus and C. Kaufman, “Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35(12), 5288–5290 (1987).
[Crossref]

Deng, W.

G. Huang, W. Deng, H. T. Tan, and G. Cheng, “Generation of squeezed states and single-phonon states via homodyne detection and photon subtraction on the filtered output of an optomechanical cavity,” Phys. Rev. A 99(4), 043819 (2019).
[Crossref]

Deng, W. W.

H. T. Tan, W. W. Deng, and L. Sun, “Directional steering as a sufficient and necessary condition for gaussian entanglement swapping: Application to distant optomechanical oscillators,” Phys. Rev. A 99(4), 043834 (2019).
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Deng, Z. J.

Z. J. Deng, X. B. Yan, Y. D. Wang, and C. W. Wu, “Optimizing the output-photon entanglement in multimode optomechanical systems,” Phys. Rev. A 93(3), 033842 (2016).
[Crossref]

Devoret, M. H.

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82(2), 1155–1208 (2010).
[Crossref]

Donner, T.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475(7356), 359–363 (2011).
[Crossref]

Eisert, J.

A. Mari and J. Eisert, “Opto- and electro-mechanical entanglement improved by modulation,” New J. Phys. 14(7), 075014 (2012).
[Crossref]

A. Mari and J. Eisert, “Gently modulating optomechanical systems,” Phys. Rev. Lett. 103(21), 213603 (2009).
[Crossref]

Fan, S.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity–time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Farace, A.

A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A 86(1), 013820 (2012).
[Crossref]

Ferreira, A.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref]

Gan, J. H.

J. H. Gan, Y. C. Liu, C. Lu, X. Wang, M. K. Tey, and L. You, “Intracavity-squeezed optomechanical cooling,” Laser Photonics Rev. 13(11), 1900120 (2019).
[Crossref]

Gao, X. Y.

Genes, C.

C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77(5), 050307 (2008).
[Crossref]

C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78(3), 032316 (2008).
[Crossref]

Gerry, C.

C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University, 2004).

Gianfreda, M.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity–time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Gigan, S.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref]

Giovannetti, V.

A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A 86(1), 013820 (2012).
[Crossref]

Girvin, S. M.

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82(2), 1155–1208 (2010).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452(7183), 72–75 (2008).
[Crossref]

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99(9), 093902 (2007).
[Crossref]

Grebogi, C.

L. Ying, Y. C. Lai, and C. Grebogi, “Quantum manifestation of a synchronization transition in optomechanical systems,” Phys. Rev. A 90(5), 053810 (2014).
[Crossref]

Gröblacher, S.

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556(7702), 473–477 (2018).
[Crossref]

J. Li, S. Gröblacher, S. Y. Zhu, and G. S. Agarwal, “Generation and detection of non-gaussian phonon-added coherent states in optomechanical systems,” Phys. Rev. A 98(1), 011801 (2018).
[Crossref]

Gu, W. J.

Guan, S. Y.

Guerreiro, A.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref]

Hall, J. L.

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57(20), 2520–2523 (1986).
[Crossref]

Han, Y.

L. Zhou, Y. Han, J. Jing, and W. Zhang, “Entanglement of nanomechanical oscillators and two-mode fields induced by atomic coherence,” Phys. Rev. A 83(5), 052117 (2011).
[Crossref]

Han, Y. J.

Z. Q. Yin and Y. J. Han, “Generating epr beams in a cavity optomechanical system,” Phys. Rev. A 79(2), 024301 (2009).
[Crossref]

Harlow, J. W.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475(7356), 359–363 (2011).
[Crossref]

Haroche, S.

J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73(3), 565–582 (2001).
[Crossref]

Harris, J. G. E.

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452(7183), 72–75 (2008).
[Crossref]

Heidmann, A.

M. Pinard, A. Dantan, D. Vitali, O. Arcizet, T. Briant, and A. Heidmann, “Entangling movable mirrors in a double-cavity system,” Europhys. Lett. 72(5), 747–753 (2005).
[Crossref]

Hofheinz, M.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464(7289), 697–703 (2010).
[Crossref]

Hong, S.

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556(7702), 473–477 (2018).
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Figures (8)

Fig. 1.
Fig. 1. Schematic diagram of our system to entangle distant mechanical modes. Two optomechanical cavities are driven by classical laser fields, and the coupling between the two distant cavities is realized by an optical fiber. Each cavity contains an OPA pumped as shown, and the amount degree of entanglement between the two mechanical modes can be generated by the especially tuned OPA.
Fig. 2.
Fig. 2. (a) The phase space trajectories of $\langle a(t)\rangle$ from $t=0$ to $t=900\tau$ . The numerical result (blue solid) simulated by Eq. (5) agrees well with the analytical result (green dashed) simulated by Eq. (6). (b) Time evolution of the modulus of the cavity-mode mean values for the time interval [ $895\tau$ , $900\tau$ ]. The synchronization of $|\langle a_1(t)\rangle |$ (thick orange solid) and $|\langle a_2(t)\rangle |$ (thin blue solid) implies that the classical dynamics of the two subsystems can be effectively described by a set of common reduced equations of motion Eq. (5). The analytical solution $|\langle a(t)\rangle |$ (thin green dashed) fits well with the numerical results $|\langle a_1(t)\rangle |$ and $|\langle a_2(t)\rangle |$ , which further proves the validity of Eq. (5). Here the time unit $\tau =2\pi /\Omega$ . The system parameters are [69] (in units of $\omega _{m}$ ): $(E,g,\gamma _{m},\Delta _{0},\lambda )/\omega _{m}=(1.5\times 10^{5},4\times 10^{-6},2\times 10^{-3},3,2)$ , cavity decay $\kappa /\omega _{m}=0.2$ , $n_{m}=0$ , and $n_{a}=0$ . As for OPA $\Lambda /\kappa =0.3$ , $\theta =0$ and $\Omega /\omega _{m}=2$ .
Fig. 3.
Fig. 3. Phase space trajectories of the classical mean values $\langle q(t)\rangle$ and $\langle p(t)\rangle$ for time intervals (a) $[0,20\tau ]$ , (b) $[980\tau ,1000\tau ]$ , [ $1980\tau$ , $2000\tau$ ], and (d) [ $2980\tau$ , $3000\tau$ ]. Again, the blue solid and green dashed lines are obtained from numerical and analytical solutions, respectively, and the parameters are the same as in Fig. 2.
Fig. 4.
Fig. 4. Mechanical entanglement $E_{N}$ versus the evolution time $t$ for different parametric phases $\theta =0$ (blue), $\pi /2$ (green), and $\pi$ (black). Other parameters are the same as those in Fig. 2.
Fig. 5.
Fig. 5. Maximum mechanical entanglement $E_{N,max}$ versus the detuning of OPA $\Omega /\omega _{m}$ . The vertical black dashed line shows the optimal value of $\Omega /\omega _{m}$ at $\Omega _{\textrm {opt}}/\omega _{m}=1.994$ . Other parameters are same as in Fig. 2.
Fig. 6.
Fig. 6. $E_{N,max}$ versus the driving laser $E/\omega _{m}$ and cavity decay $\kappa /\omega _{m}$ . Here the ratio of the parametric gain to the cavity decay $\Lambda /\kappa =0.3$ and $\Omega =\Omega _\textrm {opt}$ are chosen. Other parameters are same as in Fig. 4.
Fig. 7.
Fig. 7. Maximum mechanical entanglement $E_{N,max}$ versus the ratio of parametric gain to the cavity decay $\Lambda /\kappa$ for different cavity decay $\kappa /\omega _{m}=0.2$ (blue), $0.4$ (orange), $0.6$ (green) and $0.8$ (black) when $E/\omega _{m}=1.5\times 10^{5}$ . For each curve the parametric gains $\Lambda /\kappa$ is rescaled by corresponding $\kappa$ . Other parameters are the same as in Fig. 6.
Fig. 8.
Fig. 8. (a) Without OPAs the steady-state mechanical entanglement $E_{N}$ versus the cavity decay $\kappa /\omega _{m}$ and thermal phonon number $n_{m}$ , with $\lambda /\omega _{m}=20$ and $E/\omega _{m}=1\times 10^{7}$ . (b) $E_N$ versus time $t$ with $\lambda /\omega _{m}=2$ , $E/\omega _{m}=1.5\times 10^{5}$ , $\kappa /\omega _m=0.2$ and the rescaled parametric gain $\Lambda /\kappa =0.49$ for different thermal phonon number $n_m=0$ (blue) and $n_m=0.2$ (black). The other parameters are the same as in Fig. 6.

Equations (20)

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H = j = 1 2 [ Δ 0 a j a j + ω m 2 ( p j 2 + q j 2 ) g ¯ a j a j q j + i E ( a j a j ) + i Λ ( e i θ e i Ω t a j 2 e i θ e i Ω t a j 2 ) ] + λ ( a 1 a 2 + a 1 a 2 ) ,
q ˙ 1 = ω m p 1 , q ˙ 2 = ω m p 2 , p ˙ 1 = ω m q 1 γ m p 1 + g ¯ a 1 a 1 + ξ 1 ( t ) , p ˙ 2 = ω m q 2 γ m p 2 + g ¯ a 2 a 2 + ξ 2 ( t ) , a ˙ 1 = ( κ + i Δ 0 ) a 1 + i g ¯ a 1 q 1 + E + 2 Λ e i ( Ω t θ ) a 1 i λ a 2 + 2 κ a 1 i n ( t ) , a ˙ 2 = ( κ + i Δ 0 ) a 2 + i g ¯ a 2 q 2 + E + 2 Λ e i ( Ω t θ ) a 2 i λ a 1 + 2 κ a 2 i n ( t ) ,
δ q ˙ 1 = ω m δ p 1 , δ q ˙ 2 = ω m δ p 1 , δ p ˙ 1 = ω m δ q 1 γ m δ p 1 + g ¯ [ a 1 ( t ) δ a + a 1 ( t ) δ a 1 ] + ξ 1 ( t ) , δ p ˙ 2 = ω m δ q 2 γ m δ p 2 + g ¯ [ a 2 ( t ) δ a + a 2 ( t ) δ a 2 ] + ξ 2 ( t ) , δ a ˙ 1 = [ κ + i Δ j ( t ) ] δ a 1 + i g ¯ a 1 ( t ) δ q 1 + 2 Λ e i ( Ω t θ ) δ a 1 i λ δ a 2 + 2 κ a 1 i n ( t ) , δ a ˙ 2 = [ κ + i Δ j ( t ) ] δ a 2 + i g ¯ a 2 ( t ) δ q 2 + 2 Λ e i ( Ω t θ ) δ a 2 i λ δ a 1 + 2 κ a 2 i n ( t ) .
H lin = j = 1 2 { Δ j ( t ) δ a j δ a j + ω m 2 ( δ p j 2 + δ q j 2 ) g ¯ [ a j ( t ) δ a j + a j ( t ) δ a j ] δ q j + i Λ [ e i ( Ω t θ ) δ a j 2 e i ( Ω t θ ) δ a j 2 ] } + λ ( δ a 1 δ a 2 + δ a 1 δ a 2 ) .
q ˙ ( t ) = ω m p ( t ) , p ˙ ( t ) = ω m q ( t ) γ m p ( t ) + g ¯ | a ( t ) | 2 , a ˙ ( t ) = [ κ + i ( Δ 0 + λ ) ] a ( t ) + i g ¯ a ( t ) q ( t ) + E + 2 Λ e i ( Ω t θ ) a ( t ) .
O ( t ) = n = O n e i n Ω t = j = 0 n = O n , j e i n Ω t g ¯ j .
p n , 0 = 0 , q n , 0 = 0 , a 0 , 0 = [ κ i ( Δ 0 + λ Ω ) ] E [ κ i ( Δ 0 + λ Ω ) ] [ κ + i ( Δ 0 + λ ) ] 4 Λ 2 , a 1 , 0 = 2 Λ e i θ E [ κ + i ( Δ 0 + λ Ω ) ] [ κ i ( Δ 0 + λ ) ] 4 Λ 2 , a n , 0 = 0 , n 0 , 1.
q n , j = ω m k = 0 j 1 m = a m , k a n + m , j k 1 ω m 2 ( n Ω ) 2 + i γ m n Ω , p n , j = i n Ω ω m q n , j , a n , j = 2 Λ e i θ a n 1 , j κ + i ( Δ 0 + λ + n Ω ) + i k = 0 j 1 m = a m , k q n m , j k 1 κ + i ( Δ 0 + λ + n Ω ) .
H lin = j = 1 2 { Δ ( t ) δ a j δ a j + ω m δ b j δ b j 1 2 [ G ( t ) δ a j + G ( t ) δ a j ] ( δ b j + δ b j ) + i Λ [ e i ( Ω t θ ) δ a j 2 e i ( Ω t θ ) δ a j 2 ] } + λ ( δ a 1 δ a 2 + δ a 1 δ a 2 ) .
H j lin = Δ ( t ) A j A j + ω m B j B j 1 2 [ G ( t ) A j + G ( t ) A j ] ( B j + B j ) + i Λ [ e i ( Ω t θ ) A j 2 e i ( Ω t θ ) A j 2 ] ,
H j lin = { 1 2 [ g 1 e i ( Ω Δ j ω m ) t B j + g 0 e i ( Δ j ω m ) t B j ] A j 1 2 [ g 0 e i ( Δ j + ω m ) t B j + g 1 e i ( Ω Δ j + ω m ) t B j ] A j + i Λ e i θ e i ( Ω 2 Δ j ) t A j 2 } + H.c. .
H 1 lin 0 , H 2 lin 1 2 { [ g 1 B 2 + g 0 e i δ 0 t B 2 ] A 2 + H.c. } + i Λ ( e i θ e i δ 0 t A 2 2 e i θ e i δ 0 t A 2 2 ) .
H 2 lin [ 1 2 ( g 1 B 2 + g 0 B 2 ) A 2 + H.c. ] + i Λ ( e i θ A 2 2 e i θ A 2 2 ) ,
H 2 lin = g β A 2 g β A 2 + i Λ ( e i θ A 2 2 e i θ A 2 2 ) .
β ˙ = i g A 2 γ m 2 β , A ˙ 2 = κ A 2 + i g β + 2 Λ e i θ A 2 .
β ˙ = i g 1 κ 2 4 Λ 2 ( i κ g β i 2 Λ e i θ g β ) γ m 2 β , = ( κ | g | 2 κ 2 4 Λ 2 + γ m 2 ) β + 2 Λ e i θ g 2 κ 2 4 Λ 2 β .
H β = Λ e i ( θ + π / 2 ) g 2 κ 2 4 Λ 2 β 2 + H.c. .
S ( ξ ) = exp [ 1 2 ( ξ β 2 ξ β 2 ) ] ,
M ( t ) = [ 0 ω m 0 0 0 0 0 0 ω m γ m 0 0 G x ( t ) G y ( t ) 0 0 0 0 0 ω m 0 0 0 0 0 0 ω m γ m 0 0 G x ( t ) G y ( t ) G y ( t ) 0 0 0 R 1 R 2 0 λ G x ( t ) 0 0 0 R 3 R 4 λ 0 0 0 G y ( t ) 0 0 λ R 1 R 2 0 0 G x ( t ) 0 λ 0 R 3 R 4 ] ,
E N = max [ 0 , ln 2 η ] ,

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