Phase-Controlled Entanglement in a Four-Mode Optomechanical System

Abstract

We present a scheme for realizing phase-controlled entanglement in a microwave optomechanical system comprising two microwave cavities and two mechanical oscillators. Under specific driving conditions, we show that this optomechanical interface can be exploited to generate simultaneously the stationary cavity–cavity entanglement, mechanical–mechanical entanglement, and cavity–mechanical entanglement. Due to the closed loop interaction, we find that the entanglement can be controlled flexibly by tuning the phase difference between the optomechanical coupling strengths. The dependence of the entanglement on the amplitudes of the optomechanical coupling strengths is also explored in detail. Moreover, the bipartite entanglements are robust against temperature, and it is shown that the mechanical oscillators are cooled to the ground state in the parameter regimes for observing entanglement.

1. Introduction

Entanglement, one essential feature of quantum mechanics, is an indispensable resource for quantum information processing. This phenomenon has been demonstrated in various systems, such as superconducting qubits [1], atomic ensembles [2,3], individual atoms [4] and ions [5], and electron spins [6]. In the past few decades, mechanical oscillators with high resonance frequencies and quality factors have been extensively explored. They can be coupled to a variety of quantum systems, including superconducting qubits [7], quantum dots [8], a single-spin qubit [9], Bose–Einstein condensate [10], and to photons at nearly arbitrary wavelength. Recently, entanglement between mechanical oscillator and atomic spin ensemble [11] as well as the phonon-mediated entanglement between two superconducting qubits [12] have been demonstrated. Therefore, hybrid mechanical systems provide a promising platform for building quantum networks [13].

Cavity optomechanical systems, where the mechanical oscillator interacts with the electromagnetic cavity via radiation pressure, can also serve as an excellent candidate for the interface that connects different components in quantum networks [14,15,16,17,18,19,20,21,22]. Recent milestones in this field include cooling of the mechanical oscillator to the quantum ground state [23,24,25] and observation of quantum effects such as optomechanical squeezing of light and mechanical motion [26,27,28,29]. Moreover, a myriad of schemes have been proposed to generate the entanglement based on optomechanical systems [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]. Recent optomechanical experiments have successfully demonstrated the entanglement between photons and a mechanical oscillator [49], between two mechanical oscillators [50,51,52,53], and between photons at both optical [54] and microwave [55] frequencies. Very recently, significant attention has been paid to the nonreciprocal transmission in multimode optomechanical systems with closed loop interaction [56,57,58,59,60,61,62,63,64,65,66,67,68,69,70], where the phase difference between the optomechanical coupling strengths can be exploited to control the direction of light propagation. In addition, it has been shown that the light-vibration entanglement can be controlled by the phase difference in three-mode optomechanical systems [71,72].

Here, we study the phase-controlled entanglement in a four-mode optomechanical system, which consists of two cavity modes coupled via two mechanical oscillators [64,65,66,67,68]. To generate the entanglement quantified by logarithmic negativity, the cavities are driven by four tones, including two blue tones and two red tones. By tuning the phase difference and the optomechanical coupling strengths, we show that this hybrid system provides a platform to generate simultaneously the cavity–cavity entanglement, mechanical–mechanical entanglement, and cavity–mechanical entanglement. Furthermore, we find that the stationary entanglement is robust against the environmental temperature and both the mechanical oscillators can be cooled to the ground state. Our scheme could be realized based on the current experiments [64,65,66,67].

2. Model and Theory

We consider the optomechanical system schematically shown in Figure 1a, where two cavity modes are, respectively, coupled to two mechanical oscillators via radiation pressure. The Hamiltonian of the system is given by

$$\begin{array}{ccc}\hfill H/\hslash =& & \sum _{i=1}^2{\omega }_i{a}_i^{†}{a}_i+\sum _{i=1}^2{\mathrm{\Omega }}_i{b}_i^{†}{b}_i+\sum _{i=1}^2\sum _{j=1}^2{g}_{0,ij}{a}_i^{†}{a}_i(b_j^{†}+b_j)\hfill \\ & & +\sum _{i=1}^2\sum _{j=1}^2{\epsilon }_{ij}[a_i{e}^{i({\omega }_{d,ij}t+{\varphi }_{ij})}+a_i^{†}{e}^{-i({\omega }_{d,ij}t+{\varphi }_{ij})}],\hfill \end{array}$$

(1)

where the first term describes the energy of the cavity modes with creation (annihilation) operators $a\left(a^{†}\right)$ and resonance frequency ${\omega }_i$; the second term represents the mechanical modes with creation (annihilation) operators $b\left(b^{†}\right)$ and resonance frequency ${\mathrm{\Omega }}_i$; the third term denotes the optomechanical coupling between the cavity modes $a_i$ and the mechanical modes $b_j$ with $g_{0,ij}$ being the vacuum optomechanical coupling strength; and the last term corresponds to the interaction between the cavity modes and the driving fields with amplitude ${\epsilon }_{ij}$, frequency ${\omega }_{d,ij}$, and phase ${\varphi }_{ij}$. We linearize the Hamiltonian by expanding cavity modes as the sum of their steady-state field amplitudes and the small fluctuations, i.e., $a_i\to a_i+{\sum }_{j=1}^2{\alpha }_{ij}{e}^{-i{\omega }_{d,ij}t}$, where ${\alpha }_{ij}$ is the coherent state amplitude produced in cavity i due to the drive with detuning ${\mathrm{\Delta }}_{ij}={\omega }_i-{\omega }_{d,ij}$. Here, we choose ${\mathrm{\Delta }}_{11}={\mathrm{\Omega }}_1+{\delta }_1,{\mathrm{\Delta }}_{12}=-({\mathrm{\Omega }}_2+{\delta }_2),{\mathrm{\Delta }}_{21}=-({\mathrm{\Omega }}_1+{\delta }_1)$, and ${\mathrm{\Delta }}_{22}={\mathrm{\Omega }}_2+{\delta }_2$ with ${\delta }_i$ being the extra detuning from the exact red (blue) sidebands to study the entanglement in this system, as shown in Figure 1b. Moving into the rotating frame with respect to $H_0/\hslash ={\sum }_{i=1}^2{\omega }_i{a}_i^{†}{a}_i+({\mathrm{\Omega }}_i+{\delta }_i)b_i^{†}{b}_i$ and neglecting the counter-rotating and off-resonant terms under the condition of ${\mathrm{\Omega }}_i,|{\mathrm{\Omega }}_1-{\mathrm{\Omega }}_2|>\{{\kappa }_i,G_{11},G_{22},J_{12},J_{21}\}$, the linearized Hamiltonian can be derived as (see details in Appendix A)

$$\begin{array}{c}\hfill H/\hslash =-\sum _{i=1}^2{\delta }_i{b}_i^{†}{b}_i+(G_{11}{e}^{i{\varphi }_{11}}{a}_1^{†}{b}_1+J_{12}{e}^{i{\varphi }_{12}}{a}_1^{†}{b}_2^{†}+J_{21}{e}^{i{\varphi }_{21}}{a}_2^{†}{b}_1^{†}+G_{22}{e}^{i{\varphi }_{22}}{a}_2^{†}{b}_2+\mathrm{H}.\mathrm{c}),\end{array}$$

(2)

where $G_{11}=g_{0,11}\left|{\alpha }_{11}\right|$, $G_{22}=g_{0,22}\left|{\alpha }_{22}\right|$, $J_{12}=g_{0,12}\left|{\alpha }_{12}\right|$, and $J_{21}=g_{0,21}\left|{\alpha }_{21}\right|$ are the field-enhanced optomechanical coupling strengths. Here, we denote $G_{ij}\left(J_{ij}\right)$ as the red (blue) sidebands of cavity $a_i$ on the mechanical mode $b_j$. Due to the loop interaction in this optomechanical system, only the phase difference between the optomechanical coupling strengths has physical effects. By redefining the operators $a_1\to a_1{e}^{i{\varphi }_{11}},b_2\to b_2{e}^{i({\varphi }_{12}-{\varphi }_{11})},a_2\to a_2{e}^{i{\varphi }_{21}}$, the Hamiltonian Equation (2) becomes

$$\begin{array}{cc}\hfill H/\hslash =& -{\delta }_1{b}_1^{†}{b}_1-{\delta }_2{b}_2^{†}{b}_2+G_{11}(a_1^{†}{b}_1+a_1{b}_1^{†})+J_{12}(a_1^{†}{b}_2^{†}+a_1{b}_2)+J_{21}(a_2^{†}{b}_1^{†}+a_2{b}_1)\hfill \\ & +G_{22}(a_2^{†}{b}_2{e}^{-i\varphi }+a_2{b}_2^{†}{e}^{i\varphi }),\hfill \end{array}$$

(3)

with $\varphi ={\varphi }_{11}+{\varphi }_{21}-{\varphi }_{12}-{\varphi }_{22}$ being the phase difference.

According to the linearized Hamiltonian (3) and Heisenberg equation of motion, the quantum Langevin equations (QLEs) can be derived by adding the damping and input noise terms phenomenologically [64,65,67,68], which yields

$$\begin{array}{ccc}& & \dot{a_1}=-\frac{{\kappa }_1}{2}{a}_1-i{G}_{11}{b}_1-i{J}_{12}{b}_2^{†}+\sqrt{{\kappa }_1}{a}_{1,\mathrm{in}},\hfill \\ & & \dot{a_2}=-\frac{{\kappa }_2}{2}{a}_2-i{J}_{21}{b}_1^{†}-i{G}_{22}{b}_2{e}^{-i\varphi }+\sqrt{{\kappa }_2}{a}_{2,\mathrm{in}},\hfill \\ & & \dot{b_1}=-\left(\frac{{\gamma }_1}{2}-i{\delta }_1\right)b_1-i{G}_{11}{a}_1-i{J}_{21}{a}_2^{†}+\sqrt{{\gamma }_1}{b}_{1,\mathrm{in}},\hfill \\ & & \dot{b_2}=-\left(\frac{{\gamma }_2}{2}-i{\delta }_2\right)b_2-i{J}_{12}{a}_1^{†}-i{G}_{22}{a}_2{e}^{i\varphi }+\sqrt{{\gamma }_2}{b}_{2,\mathrm{in}},\hfill \end{array}$$

(4)

where ${\kappa }_i(i=1,2)$ is the decay rate of cavity mode $a_i$ with zero-mean noise operator $a_{i,\mathrm{in}}$, and ${\gamma }_i$ is the damping rate of mechanical mode $b_i$ with zero-mean operator $b_{i,\mathrm{in}}$. Defining the quadratures $U_i=(a_i+a_i^{†})/\sqrt{2}$, $V_i=(a_i-a_i^{†})/i\sqrt{2}$, $X_i=(b_i+b_i^{†})/\sqrt{2}$, $P_i=(b_i-b_i^{†})/i\sqrt{2}$, and the corresponding input noise operators $U_{i,\mathrm{in}}=(a_{i,\mathrm{in}}+a_{i,\mathrm{in}}^{†})/\sqrt{2}$, $V_{i,\mathrm{in}}=(a_{i,\mathrm{in}}-a_{i,\mathrm{in}}^{†})/i\sqrt{2}$, $X_{i,\mathrm{in}}=(b_{i,\mathrm{in}}+b_{i,\mathrm{in}}^{†})/\sqrt{2}$, $P_{i,\mathrm{in}}=(b_{i,\mathrm{in}}-b_{i,\mathrm{in}}^{†})/i\sqrt{2}$, the linearized QLEs can be written in the following compact matrix form:

$$\dot{\mu }\left(t\right)=A\mu \left(t\right)+n\left(t\right),$$

(5)

where $\mu \left(t\right)={(U_1\left(t\right),V_1\left(t\right),U_2\left(t\right),V_2\left(t\right),X_1\left(t\right),P_1\left(t\right),X_2\left(t\right),P_2\left(t\right))}^T$ is the vector of the fluctuation operators, $n\left(t\right)={(\sqrt{{\kappa }_1}{U}_{1,\mathrm{in}}\left(t\right),\sqrt{{\kappa }_1}{V}_{1,\mathrm{in}}\left(t\right),\sqrt{{\kappa }_2}{U}_{2,\mathrm{in}}\left(t\right),\sqrt{{\kappa }_2}{V}_{2,\mathrm{in}}\left(t\right),\sqrt{{\gamma }_1}{X}_{1,\mathrm{in}}\left(t\right),\sqrt{{\gamma }_1}{P}_{1,\mathrm{in}}\left(t\right),\sqrt{{\gamma }_2}{X}_{2,\mathrm{in}}\left(t\right),\sqrt{{\gamma }_2}{P}_{2,\mathrm{in}}\left(t\right))}^T$ is the vector of the corresponding noises, and the coefficient matrix

$$A=\left(\begin{array}{cccccccc}-{\kappa }_1/2& 0& 0& 0& 0& G_{11}& 0& -J_{12}\\ 0& -{\kappa }_1/2& 0& 0& -G_{11}& 0& -J_{12}& 0\\ 0& 0& -{\kappa }_2/2& 0& 0& -J_{21}& -G_{22}sin\varphi & G_{22}cos\varphi \\ 0& 0& 0& -{\kappa }_2/2& -J_{21}& 0& -G_{22}cos\varphi & -G_{22}sin\varphi \\ 0& G_{11}& 0& -J_{21}& -{\gamma }_1/2& -{\delta }_1& 0& 0\\ -G_{11}& 0& -J_{21}& 0& {\delta }_1& -{\gamma }_1/2& 0& 0\\ 0& -J_{12}& G_{22}sin\varphi & G_{22}cos\varphi & 0& 0& -{\gamma }_2/2& -{\delta }_2\\ -J_{12}& 0& -G_{22}cos\varphi & G_{22}sin\varphi & 0& 0& {\delta }_2& -{\gamma }_2/2\end{array}\right).$$

(6)

The formal solution of Equation (5) is

$$\mu \left(t\right)=M\left(t\right)\mu \left(0\right)+{\int }_0^{t}dsM\left(s\right)n(t-s),$$

(7)

where

$$M\left(t\right)=exp\left\{At\right\}.$$

(8)

The system is stable when all of the eigenvalues of A have negative real parts, which can be derived from the Routh–Hurwitz criterion [73]. When the stability conditions are fulfilled, one can obtain $M(\infty )=0$ in the steady state.

The steady state of the system is a zero-mean Gaussian state due to the linearized dynamics and the Gaussian nature of the input quantum noises, which is fully characterized by an $8\times 8$ correlation matrix (CM) V, with its components

$$V_{ij}=\left[\langle {\mu }_i(\infty ){\mu }_j(\infty )+{\mu }_j(\infty ){\mu }_i(\infty )\rangle \right]/2.$$

(9)

When the system is stable, by substituting Equation (7) into Equation (9), we can obtain

$$V_{ij}=\sum _{k,l}{\int }_0^{\infty }ds{\int }_0^{\infty }d{s}^{\prime }{M}_{ik}\left(s\right)M_{jl}\left(s^{\prime }\right){\Phi }_{kl}(s-s^{\prime }),$$

(10)

where ${\Phi }_{kl}(s-s^{\prime })=\left(\langle n_k\left(s\right)n_k\left(s^{\prime }\right)+n_l\left(s^{\prime }\right)n_k\left(s\right)\rangle \right)/2=D_{kl}\delta (s-s^{\prime })$. According to the correlation functions that the noise operators satisfy,

$$\begin{array}{ccc}& & \langle a_{i,\mathrm{in}}\left(s\right)a_{i,\mathrm{in}}^{†}\left(s^{\prime }\right)\rangle =\delta (s-s^{\prime }),\langle a_{i,\mathrm{in}}^{†}\left(s^{\prime }\right)a_{i,\mathrm{in}}\left(s\right)\rangle =0,\hfill \\ & & \langle b_{i,\mathrm{in}}\left(s\right)b_{i,\mathrm{in}}^{†}\left(s^{\prime }\right)\rangle =(n_{m,i}+1)\delta (s-s^{\prime }),\langle b_{i,\mathrm{in}}^{†}\left(s^{\prime }\right)b_{i,\mathrm{in}}\left(s\right)\rangle =n_{m,i}\delta (s-s^{\prime }),\hfill \end{array}$$

(11)

where $n_{m,i}={\{exp[\hslash {\mathrm{\Omega }}_i/\left(k_{\mathrm{B}}T\right)]-1\}}^{-1}$ denotes the thermal phonon number of the mechanical mode $b_i$ with $k_B$ being the Boltzmann constant and T being the environment temperature, we can obtain $D=\mathrm{Diag}[{\kappa }_1/2,{\kappa }_1/2,{\kappa }_2/2,{\kappa }_2/2,(2n_{m,1}+1){\gamma }_1/2,(2n_{m,1}+1){\gamma }_1/2,(2n_{m,2}+1){\gamma }_2/2,(2n_{m,2}+1){\gamma }_2/2]$, and Equation (10) becomes $V={\int }_0^{\infty }dsM\left(s\right)DM{\left(s\right)}^T$. The time evolution of the correlation matrix V satisfies $\dot{V}\left(t\right)=AV\left(t\right)+V\left(t\right)A^T+D$ [36]. In the steady state, the correlation matrix V fulfills the Lyapunov equation [30]:

$$AV+V{A}^T=-D,$$

(12)

which is a linear equation for V and can be straightforwardly solved for given parameters in matrix A and matrix D; but the general analytical expression is too cumbersome to be reported here. The bipartite entanglement between different modes can be quantified by the logarithmic negativity [30,31,74]

$$E_N=max[0,-ln2{\eta }^{-}],$$

(13)

where ${\eta }^{-}\equiv 2^{-1/2}{\{\Sigma \left(V_{\mathrm{bp}}\right)-{[\Sigma {\left(V_{\mathrm{bp}}\right)}^2-4det{V}_{\mathrm{bp}}]}^{1/2}\}}^{1/2}$, with $\Sigma \left(V_{\mathrm{bp}}\right)\equiv detB+det{B}^{\prime }-2detC$. The reduced CM $V_{\mathrm{bp}}$ contains the entries of V, and it can be obtained by selecting the rows and columns of the interesting mode in V. By writing the reduced CM $V_{\mathrm{bp}}$ in a $2\times 2$ block form, we have

$$V_{\mathrm{bp}}\equiv \left(\begin{array}{cc}B& C\\ C^T& B^{\prime }\end{array}\right).$$

(14)

Note that matrix A in Equation (6) contains the phase difference $\varphi$; hence, the correlation matrix V and the logarithmic negativity $E_N$ depend on $\varphi$. We will study how to control the entanglement quantified by Equation (13) in this optomechanical system by tuning the phase difference $\varphi$.

Furthermore, based on the correlation matrix V, the final average phonon numbers in the mechanical modes $b_1$ and $b_2$ can be calculated by

$$\begin{array}{ccc}& & \langle b_1^{†}{b}_1\rangle =n_1^f=\frac{V_{5,5}+V_{6,6}-1}{2},\hfill \\ & & \langle b_2^{†}{b}_2\rangle =n_2^f=\frac{V_{7,7}+V_{8,8}-1}{2},\hfill \end{array}$$

(15)

which affect the quantum entanglement in the hybrid system.

3. Results and Discussion

In this section, we choose the parameters from a recent experiment to demonstrate the numerical results about the phase-controlled entanglement in this system. The parameters are [65] ${\mathrm{\Omega }}_1/2\pi =4.34$ MHz, ${\mathrm{\Omega }}_2/2\pi =5.64$ MHz, ${\kappa }_1/2\pi =2.42$ MHz, ${\kappa }_2/2\pi =1.98$ MHz, ${\gamma }_1/2\pi =40$ Hz, ${\gamma }_2/2\pi =80$ Hz, and $T=20$ mK. The system works in the resolved sideband regime and the rotating-wave approximation is valid.

In Figure 2, we plot the logarithmic negativity $E_N$ between (a) the two cavity modes as well as (b) the two mechanical modes versus the optomechanical coupling strength $G_{22}/2\pi$ and the phase difference $\varphi /\pi$. At $G_{22}=0$, the optomechanical interactions cannot form a closed loop and the phase-dependent effect is absent. With the increase in the coupling strength $G_{22}$, it is evident that the quantum entanglement depends on the phase difference $\varphi$. Note that cavity mode $a_1$ is driven on the blue (red) sideband of the mechanical mode $b_2$ ($b_1$), while cavity mode $a_2$ is driven on the blue (red) sideband of the mechanical mode $b_1$ ($b_2$). Therefore, the parametric-amplifier interaction $J_{12}(a_1^{†}{b}_2^{†}+a_1{b}_2)$ entangles $a_1$ and $b_2$, and the interaction $J_{21}(a_2^{†}{b}_1^{†}+a_2{b}_1)$ entangles $a_2$ and $b_1$. Moreover, the beam-splitter interaction $G_{11}(a_1^{†}{b}_1+a_1{b}_1^{†})$ swaps $a_1$ and $b_1$, and the interaction $G_{22}(a_2^{†}{b}_2{e}^{-i\varphi }+a_2{b}_2^{†}{e}^{i\varphi })$ swaps $a_2$ and $b_2$. Consequently, the two cavity modes can be entangled via the routes $a_1\leftrightarrow b_1\leftrightarrow a_2$ and $a_1\leftrightarrow b_2\leftrightarrow a_2$, and the two mechanical modes can also be entangled via the routes $b_1\leftrightarrow a_1\leftrightarrow b_2$ and $b_1\leftrightarrow a_2\leftrightarrow b_2$, where the phase difference $\varphi$ plays a vital role. At $\varphi =\pi$, we can see that both the cavity–cavity and mechanical–mechanical entanglement firstly decrease from an initial value to zero when $G_{22}$ increases. With further increasing $G_{22}$ above a critical value, it is shown that the quantum entanglement starts to increase again. This phenomenon is closely related to the quantum interference between the eigenmodes of the system [34].

We proceed to study the effect of the entangling-interaction strength $J_{12}$ on the entanglement in Figure 3, assuming $J_{21}=J_{12}$ for simplicity. Figure 3a and Figure 3c plot, respectively, the dynamics of the cavity–cavity entanglement and mechanical–mechanical entanglement for $\varphi =0,\pi /2$, and $\pi$. It is shown that stationary entanglement can be achieved after $t\approx 2$ μs, which depends on the phase difference $\varphi$. Moreover, Figure 3b and Figure 3d plot the stationary cavity–cavity entanglement and mechanical–mechanical entanglement as functions of the entangling-interaction strength $J_{12}/2\pi$ for different values of phase difference $\varphi$. At $\varphi =0$, both the cavity–cavity entanglement and the mechanical–mechanical entanglement are enhanced monotonically with the increase in the coupling strength $J_{12}$ in the parameter regimes. When the phase difference $\varphi$ is tuned to be $\pi /2$ or $\pi$, the stationary logarithmic negativity $E_N$ becomes larger and reaches the maximum with the increase of $J_{12}$. With further increasing $J_{12}$, the entanglement will decrease and be zero above a critical value of $J_{12}$. On one hand, it is straightforward that the entanglement can be enhanced by increasing the coupling strengths $J_{12}\left(J_{21}\right)$, which are strengthened by the blue-detuned driving fields. On the other hand, increasing $J_{12}\left(J_{21}\right)$ can suppress the cavity-cooling effect for given $G_{12}\left(G_{21}\right)$ and increase the effective temperature of the cavity and mechanical modes, which will reduce the entanglement. In addition, when the coupling strengths $J_{12}$ and $J_{21}$ are too strong, the system will become unstable. Therefore, the maximum entanglement is achieved by balancing these two competing effects [35]. Throughout this work, we have checked numerically that the chosen parameters satisfy the stability condition.

Furthermore, we study the phase dependence of the optomechanical entanglement in Figure 4. The logarithmic negativity $E_N$ between the two cavity modes is plotted as a function of the phase difference $\varphi$ for different values of $J_{12}$ in Figure 4a. At $J_{12}/\left(2\pi \right)=0.1$ MHz, the maximum cavity–cavity entanglement is achieved around $\varphi =\pi$ and the minimum value locates at $\varphi =0\left(2\pi \right)$. However, when the value of $J_{12}$ is increased to be 0.2 MHz and 0.25 MHz, the peak around $\varphi =\pi$ turns into a dip. Especially, the cavity–cavity entanglement in the vicinity of $\varphi =\pi$ is absent for $J_{12}=0.25$ MHz. Therefore, the phase difference $\varphi$ can be exploited as a switch of the cavity–cavity entanglement between zero and nonzero value. Similarly, Figure 4b plots the logarithmic negativity $E_N$ between the two mechanical modes versus the phase difference $\varphi$. It is shown that the mechanical–mechanical entanglement reaches the minimum at $\varphi =\pi$ and the maximum at $\varphi =0\left(2\pi \right)$ for $J_{12}/2\pi =0.1$, 0.2, and 0.25 MHz. Moreover, the entanglement can be strengthened monotonically by increasing the coupling strength $J_{12}$ at $\varphi =0\left(2\pi \right)$. Consequently, the quantum entanglement can be controlled by the phase difference in this optomechanical system.

As mentioned above, the entanglement between the two cavity (mechanical) modes is realized via the entanglement transfer between the cavity and mechanical modes. In Figure 5, we study the optomechanical entanglement between the cavity mode $a_1\left(a_2\right)$ and mechanical mode $b_2\left(b_1\right)$. Figure 5a plots the time evolution of the logarithmic negativity $E_N$ between the cavity mode $a_1$ and mechanical mode $b_2$ with $J_{12}/2\pi =0.15$ MHz for different values of phase difference $\varphi$. It is shown that stationary entanglement can be achieved after $t\approx 2$ μs and it is strengthened when the the phase difference $\varphi$ is increased from 0 to $\pi$. The logarithmic negativity $E_N$ between $a_1$ and $b_2$ versus the entangling-interaction strength $J_{12}$ is plotted in Figure 5b, which shows that stationary entanglement can be enhanced by increasing the strength $J_{12}$ and reaches the maximum at some critical value for $\varphi =0$ and $\pi /2$. In the chosen parameter regime, the stationary entanglement increases monotonically with the increase in $J_{12}$ for $\varphi =\pi$. In addition, Figure 5c plots the time evolution of the logarithmic negativity $E_N$ between $a_2$ and $b_1$ for $\varphi =0,\pi /2$, and $\pi$, where stationary entanglement can also be obtained. The stationary entanglement between $a_2$ and $b_1$ versus $J_{12}$ is plotted in Figure 5d, where the maximum entanglement is achieved around $J_{12}/2\pi =0.15$ MHz for various phase differences. Figure 5 also shows that the stationary entanglement between $a_2$ and $b_1$ is weaker than the entanglement between $a_1$ and $b_2$, which is mainly due to $G_{22}>G_{11}$ under consideration.

The robustness of the quantum entanglement with respect to the environmental temperature is shown in Figure 6. The value of $E_N$ decreases with the increasing temperature for various phase differences. Figure 6a plots the logarithmic negativity $E_N$ between the cavity modes $a_1$ and $a_2$ versus the temperature T. At $\varphi =0$, the cavity–cavity entanglement persists up to 150 mK. The entanglement vanishes at higher temperature for $\varphi =\pi /2$ and $\varphi =\pi$. The entanglement between the two mechanical modes shown in Figure 6b is stronger than the cavity–cavity entanglement and survives at higher temperature. In addition, the entanglement between the cavity mode $a_1$ and mechanical mode $b_2$ versus the temperature is plotted in Figure 6c, where $E_N\approx 0.11$ can be obtained for $T=400$ mK and $\varphi =\pi$. The logarithmic negativity $E_N$ between the cavity mode $a_2$ and mechanical mode $b_1$ is much smaller than the value in Figure 6c and survives at much lower temperature, which results from $G_{22}>G_{11}$. The value of $E_N$ between $a_2$ and $b_1$ can be increased by reducing $G_{22}$, but the entanglement between the two cavity (mechanical) modes will be absent in some parameter regimes shown in Figure 2.

Finally, the entanglement usually survives at low final phonon numbers. The beam-splitter interaction such as $G_{11}(a_1^{†}{b}_1+a_1{b}_1^{†})$ can not only result in state transfer between $a_1$ and $b_1$ but also cool the mechanical mode $b_1$. In Figure 7, we numerically study the final average phonon numbers $n_1^f$ and $n_2^f$ by choosing the parameters for observing the entanglement mentioned above. Figure 7a plots the phonon numbers $n_1^f$ and $n_2^f$ as a function of the optomechanical coupling strength $J_{12}/2\pi$ with $\varphi =0$, which shows that both $n_1^f$ and $n_2^f$ increase monotonically with the enhancement of $J_{12}$. The parametric-amplifier interaction $J_{12}(a_1^{†}{b}_2^{†}+a_1{b}_2)$ is caused by driving the cavity mode $a_1$ on the blue sideband of mechanical mode $b_2$, where a drive photon is scattered into a photon at frequency ${\omega }_1$ and a phonon at frequency ${\mathrm{\Omega }}_2$, resulting in the monotonic increase in phonon number at larger $J_{12}$. It is noted that quantum phase transition in the two-mode Rabi model has been recently investigated by analyzing the eigenenergy of the system [75], but quantum phase transition of the phonon number of the mechanical mode in optomechanical systems has seldom been reported. In addition, here, we show that both $n_1^f$ and $n_2^f$ are smaller than 1 in the chosen parameter regime. Therefore, simultaneous cooling of the mechanical modes and quantum entanglement can be realized in this optomechanical system. In addition, the phase dependence of phonon numbers is plotted in Figure 7b. The final average phonon numbers of the two mechanical modes reach the maximum at $\varphi =\pi$ and the minimum at $\varphi =0\left(2\pi \right)$, which is contrary to the entanglement between the two mechanical modes shown in Figure 4b.

4. Conclusions

In conclusion, we have explored the phase-controlled entanglement in a multimode optomechanical system, where two cavity modes are, respectively, coupled to two mechanical modes via radiation pressure. Four tones are applied to the cavity modes to generate the entanglement, where each cavity is driven on the red sideband and blue sideband of the two mechanical modes, respectively. The dynamics of the bipartite entanglement is studied, which shows that the stationary entanglement can be generated. We have shown that the entanglement can be effectively modulated by the phase difference between the optomechanical coupling strengths that form a loop interaction. By tuning the optomechanical coupling strengths and phase difference in some parameter regimes, we can realize simultaneously the entanglement between the two cavity mode, between the two mechanical modes, and between the cavity and mechanical modes. We also find that the entanglement is robust against temperature and the mechanical resonators can be cooled to the ground state when the entanglement is generated.