Temperature-resistant generation of robust entanglement with blue-detuning driving and mechanical gain

Tie Wang; Liang Wang; Yu-Mu Liu; Cheng-Hua Bai; Dong-Yang Wang; Hong-Fu Wang; Hong-Fu Wang; Shou Zhang; Shou Zhang

doi:10.1364/OE.27.029581

1. Introduction

Cavity optomechanics, as an interdisciplinary subject of quantum optics and mechanics, mainly investigates the influence of radiation pressure on the mechanical element. Benefiting from nano-technology and manufacturing industry, optical microcavities with high fineness have been realized in variety of experimental setups, including standard optical Fabry-Pérot cavity [1,2], whispering-gallery-mode microtoroid resonator [3–5], and “membrane-in-the-middle” setup [6–9], etc. The cavity optomechanical (COM) systems, which are consisted of the optical microcavities, are advanced to the microscopic scale [10–12]. Due to their high sensitivity, many quantum phenomena can be easily achieved on macroscopic objects. Therefore, the COM systems provide an ideal platform to investigate many interesting physical phenomena, such as optomechanically induced transparency [13–19], ground-state cooling of mechanical resonator [20–23], optomechanical entanglement, photon (phonon) block effect, and so on. Thereinto, the optomechanical entanglement, as a unique characteristic of quantum mechanics, provides an essential resource for quantum networks [24], quantum computation, and quantum information processing [25–28]. The latest studies reveal that the optomechanical entanglement plays a significant role in the test of the fundamental principles of quantum mechanics in a single-COM system, such as quantifying the stationary entanglement between a cavity mode and a mechanical mode [29], evaluating the entanglement between two mechanical modes [30], generating robust tripartite entanglement among two longitudinal cavity modes and a mechanical resonator [31].

In the parity-time ($\mathcal {PT}$) symmetric COM system, which is described by the non-Hermitian Hamiltonian, have attracted extensive attention for experimental realization with optical (mechanical) gain. On the one hand, the three-mode $\mathcal {PT}$-symmetric COM system, which consists of an active cavity, a passive cavity, and a mechanical resonator, has been explored [32–36]. Based on the optical gain, the distant entanglement between the mechanical mode and the cavity mode can be generated and enhanced [37]. On the other hand, the $\mathcal {PT}$-symmetric-like single-COM system, which is consisted of a passive cavity and an active mechanical oscillator, has been investigated with coherent manipulation of phonons [38–43]. Based on the mechanical gain, many theoretical findings enable new applications to control the transmission of light beyond what is impossible in a conventional COM system [38,44–48].

To our best knowledge, the stability condition can be satisfied in a single-COM system when the optical cavity is driven by the red-detuning laser, while the stability condition of the system is unsatisfied if the cavity is driven only by the blue-detuning laser. Here we investigate a double-COM system, in which the stable entanglement between two cavity modes driven respectively by the red-detuning laser and the blue-detuning laser can be achieved. In contrast to the red-detuning regime, we find that the entanglement in the blue-detuning regime is extremely robust to temperature in our system. We investigate the destructive influence of the cavity mode driven by the blue-detuning laser on the optomechanical entanglement between the mechanical resonator and the cavity mode driven by the red-detuning laser. The optomechanical entanglement between two cavity modes without direct interaction can be generated in our system, and meanwhile it is robust to temperature. On the other hand, we investigate the effects of temperature on the optomechanical entanglement for different mechanical loss (gain) rates. Further, we introduce the mechanical gain into mechanical resonator to investigate the entanglement properties in the proposed double-COM system. Comparing with the conventional COM system, the proposed COM system with tiny mechanical gain can enhance optomechanical entanglement at high temperature. This would open a new way for the generation of robust entanglement at high temperature and provide an effective method for achieving the amplification of entanglement.

2. System Hamiltonian and dynamical equations

The system we consider is depicted in Fig. 1, which is consisted of two fixed mirrors and a movable total reflection mirror. For convenience, we assume that the two cavities are identical (cavity lengths $L_1=L_2$, loss rates $\kappa _1=\kappa _2=\kappa =\pi c/LF$, resonance frequencies $\omega _{a1}=\omega _{a2}$) and are driven by two pump lasers with amplitudes $E_{L1}$ and $E_{L2}$ and frequencies $\omega _{L1}$ and $\omega _{L2}$, respectively. The Hamiltonian of the system is written as

(1)$$\begin{aligned} H=&\hbar\omega_{a1}a^\dagger_1a_1+\hbar\omega_{a2}a^\dagger_2a_2+\dfrac{\hbar\omega_m}{2}(p^2+q^2)-\hbar g(a^\dagger_1a_1-a^\dagger_2a_2)q \nonumber \\ &+i\hbar(E_{L1}a^\dagger_1 e^{{-}i\omega_{L1}t}+E_{L2}a^\dagger_2e^{{-}i\omega_{L2}t}-\textrm{H.c.}), \end{aligned}$$

where the first two terms are the free Hamiltonians of cavity mode $i$ with annihilation (creation) operator $a_i$ $(a^\dagger _i)$ ($i$=1, 2). The third term accounts for the free Hamiltonian of mechanical resonator, with $q$ and $p$ being the dimensionless position and momentum operators of mechanical resonator with resonance frequency $\omega _m$ and mass $m$, respectively. The fourth term is the interaction Hamiltonian between cavity mode $i$ and mechanical resonator with optomechanical coupling strength $g_i=(\omega _{ai}/L)\sqrt {\hbar /m\omega _m}$ and $g_1=g_2=g$. The last term denotes the interactions between cavity modes and pump lasers with amplitude $E_{Li}=\sqrt {2P_{Li}\kappa _i/\hbar \omega _m}$ (here $P_{Li}$ is the power of pump laser $i$). In the rotating frame at the pump laser frequencies $\omega _{L1}$ and $\omega _{L2}$, the Hamiltonian of the system is given by

(2)$$\begin{aligned} H=&\hbar\Delta_{a1}a^\dagger_1a_1+\hbar\Delta_{a2}a^\dagger_2a_2+\dfrac{\hbar\omega_m}{2}(p^2+q^2)-\hbar g(a^\dagger_1a_1-a^\dagger_2a_2)q\nonumber\\ &+i\hbar(E_{L1}a^\dagger_1+E_{L2}a^\dagger_2-\textrm{H.c.}), \end{aligned}$$

where $\Delta _{ai}=\omega _{ai}-\omega _{Li}$ is the detuning between cavity mode $i$ and input laser $i$. The quantum Langevin equations (QLEs) are given by

(3)$$\begin{aligned} \dot{q}&=\omega_mp,\nonumber \\ \dot{p}&=-\omega_mq+g(a_1^\dagger a_1-a_2^\dagger a_2)-\gamma_m p+\xi(t), \nonumber \\ \dot{a_1}&=-(i\Delta_{a1}+\kappa_1)a_1+iga_1q+E_{L1}+\sqrt{2\kappa_1}a^{in}_1(t),\nonumber \\ \dot{a_2}&=-(i\Delta_{a2}+\kappa_2)a_2-iga_2q+E_{L2}+\sqrt{2\kappa_2}a^{in}_2(t), \end{aligned}$$

where $\gamma _m$ is the mechanical loss (gain) rate when $\gamma _m>0$ ($\gamma _m<0$), $a^{in}_i(t)$ is quantum noise operator of cavity mode $i$ with zero mean value, which satisfies correlation relation $\left \langle\delta a^{in}_i(t)\delta a^{in\dagger }_i(t')\right \rangle=\left (N_n+1\right )\delta \left (t-t'\right )$, and $N_n=[$exp$(\hbar \omega _{ai}/k_BT)-1]^{-1}$ is equilibrium mean thermal photon number of cavity mode $i$, with $k_B$ being the Boltzmann constant and $T$ being the temperature of mechanical resonator. We can approximately obtain $N_n\approx 0$ when $\omega _{ai}$ is large enough. And $\xi (t)$ is the Brownian noise operator, which depicts the heat of mechanical resonator by the thermal noise with zero mean value at temperature $T$, with $\left \langle\xi (t)\xi (t')\right \rangle=\dfrac {\gamma _m}{2\pi \omega _m}\displaystyle {\int d\omega }\left [1+\coth \left (\dfrac {\hbar \omega }{2k_BT}\right )\right ]\omega \textrm{exp}[-i\omega (t-t')]$. The standard linearization method can be used and each Heisenberg operator can be rewritten as $O=O_0+\delta O$, where $O_0$ is the steady-state value and $\delta O$ refers to the quantum fluctuation with zero mean value around the steady-state mean value. In the following, we substitute the above expressions into Eq. (3) and the steady-state solutions of the system dynamics can thus be obtained as

(4)$$\begin{aligned} q_0&=\dfrac{g}{\omega_m}(|a_{10}|^2-|a_{20}|^2), \nonumber \\ p_0&=0,\nonumber \\ a_{10}&=\dfrac{E_{L1}}{i\Delta_1+\kappa_1}, \nonumber \\ a_{20}&=\dfrac{E_{L2}}{i\Delta_2+\kappa_2}, \end{aligned}$$

where $\Delta _i=\Delta _{ai}\mp g_iq_0$ denotes the effective detuning between cavity mode $i$ and input laser $i$. In this case, by neglecting the nonlinear terms, we get the linearized QLEs

(5)$$\begin{aligned} \delta\dot{q}&=\omega_m\delta p,\nonumber \\ \delta\dot{p}&=-\omega_m\delta q-\gamma_m\delta p+\sqrt{2}\textrm{Re}[G_1]\delta X_1+\sqrt{2}\textrm{Im}[G_1]\delta Y_1 \nonumber \\ & \quad -\sqrt{2}\textrm{Re}[G_2]\delta X_2-\sqrt{2}\textrm{Im}[G_2]\delta Y_2+\xi(t), \nonumber \\ \delta\dot{X}_{1}&=-\kappa_1\delta X_1+\Delta_1\delta Y_1-\sqrt{2}\textrm{Im}[G_1]\delta q+\sqrt{2\kappa_1}\delta X_1^{in}, \nonumber \\ \delta\dot{Y}_{1}&=-\Delta_1\delta X_1-\kappa_1\delta Y_1+\sqrt{2}\textrm{Re}[G_1]\delta q+\sqrt{2\kappa_1}\delta Y^{in}_1, \nonumber \\ \delta\dot{X}_{2}&=-\kappa_2\delta X_2+\Delta_2\delta Y_2-\sqrt{2}\textrm{Im}[G_2]\delta q+\sqrt{2\kappa}\delta X_2^{in}, \nonumber \\ \delta\dot{Y}_{2}&=-\Delta_2\delta X_2-\kappa_2\delta Y_2+\sqrt{2}\textrm{Re}[G_2]\delta q+\sqrt{2\kappa_2}\delta Y^{in}_2, \end{aligned}$$

where $G_i=ga_{i0}$ is effective optomechanical coupling strength between cavity mode $i$ and mechanical resonator. We define the quadratures of cavity mode $i$ $\delta X_i=(\delta a_i+\delta a_i^\dagger )/\sqrt {2}$ and $\delta Y_i=(\delta a_i-\delta a_i^\dagger )/i\sqrt {2}$, and Hermitian input noise operators $\delta X_i^{in}=(\delta a_i^{in}+\delta a_i^{in\dagger })/\sqrt {2}$ and $\delta Y_i^{in}=(\delta a_i^{in}-\delta a_i^{in\dagger })/i\sqrt {2}$. We can control the effective optomechanical coupling strength directly by changing the laser power.

Fig. 1. Schematic illustration of a COM system, which consists of two optical cavities and a mechanical resonator and is driven by two strong driving lasers.

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3. Logarithmic negativity

To investigate the entanglement properties of the double-COM system, we rewrite Eq. (4) as

(6)$$\dot{u}(t)=Au(t)+n(t),$$

where $u^T(t)=(\delta q,\delta p, \delta X_1, \delta Y_1,\delta X_2, \delta Y_2)$, $n^T(t)=(0, \xi (t), \sqrt {2\kappa _1}\delta X^{in}_1, \sqrt {2\kappa _1}\delta Y^{in}_1,\sqrt {2\kappa _2}\delta X^{in}_2$, $\sqrt {2\kappa _2}\delta Y^{in}_2)$. The superscript $T$ is the transpose operation. Then we obtain the drift matrix $A$ as

(7)$$ A=\left(\begin{array}{cccccc} 0 & \omega_m & 0 & 0 & 0 & 0\\ -\omega_m & -\gamma_m & \sqrt{2}\textrm{Re}[G_1] & \sqrt{2}\textrm{Im}[G_1] & -\sqrt{2}\textrm{Re}[G_2] & -\sqrt{2}\textrm{Im}[G_2]\\ -\sqrt{2}\textrm{Im}[G_1] & 0 & -\kappa_1 & \Delta_1 & 0 & 0 \\ \sqrt{2}\textrm{Re}[G_1] & 0 & -\Delta_1 & -\kappa_1 & 0 & 0\\ \sqrt{2}\textrm{Im}[G_2] & 0 & 0 & 0 & -\kappa_2 & \Delta_2\\ -\sqrt{2}\textrm{Re}[G_2] & 0 & 0 & 0 & -\Delta_2 & -\kappa_2\\ \end{array}\right). $$

The system is stable when all the eigenvalues of the matrix $A$ have negative real parts. In terms of Eq. (7), the 6$\times$6 covariance matrix (CM) $V_{ij}$ can be written as

(8)$$V_{ij}=\underset{k,\;l}{\sum}\int^\infty_{0}dt\int^\infty_{0}dt'M_{ik}(t)M_{jl}(t')\Phi_{kl}(t-t'),$$

where $M(t)$=exp($At$) and $\Phi _{kl}$=$\left \langle[n_k(t)n_l(t')+n_k(t')n_k(t)]\right \rangle/2$ is the matrix element of stationary noise correlation functions. When the real parts of all the eigenvalues of drift matrix $A$ are negative, we have $M(\infty )\rightarrow 0$. In the limit of large mechanical quality factor $\left (Q_m=\omega _m/\gamma _m\rightarrow \infty \right )$, there is relevant nonzero correlation function $\left \langle\xi (t)\xi (t')+\xi (t')\xi (t)\right \rangle/2\approx \gamma _m(2n_m+1)\delta (t-t')$, in which $n_m=1/[$exp$(\hbar \omega _m/k_BT)-1]^{-1}$ is the mean thermal phonon number of mechanical resonator. For the Markovian process, we have $\Phi (T-T')=D_{kl}\delta (t-t')$, where $D=\textrm{Diag}[0,\gamma _m(2n_m+1),\kappa _1,\kappa _1,\kappa _2,\kappa _2]$ is a diagonal diffusion matrix, and $V$ in Eq. (8) becomes $V=\int ^{\infty }_{0}dtM(t)DM(t)^T$. The Lyapunov equation can be obtained as

(9)$$AV+VA^T={-}D,$$

where the steady-state covariance matrix is defined as

(10)$$ V=\left[\begin{array}{ccc} V_m & V_{ma_1} & V_{ma_2}\\ V_{ma_1}^T & V_{a_1} & V_{a_1a_2}\\ V_{ma_2}^T & V_{a_1a_2}^T & V_{a_2}\\ \end{array}\right], $$

here each diagonal block matrix ($V_m$, $V_{a1}$ and $V_{a_2}$) represents the variance of each relevant subsystem (mechanical resonator, cavity modes 1 and 2) in the double-COM system and each off-diagonal block matrix represents the covariance between two relevant subsystems. To investigate the entanglement between each two subsystems, we calculate the steady-state covariance sub-matrix as

(11)$$ V_s=\left[\begin{array}{cc} B & C \\ C^T & E\\ \end{array}\right], $$

where $B$ and $E$ represent respectively the variance block matrices of two relevant subsystems, and $C$ represents the covariance block matrix between two subsystems. Using the steady-state covariance sub-matrix $V_s$, the value of entanglement is defined as

(12)$$E_N=\textrm{max}[0,-\textrm{ln}(2\eta^-)],$$

where

(13)$$\begin{aligned} \eta^- &\equiv2^{{-}1/2}\left[\Sigma V_s-\sqrt{\Sigma V_s^2-4\textrm{det} V_s}\right]^{1/2}, \nonumber \\ \Sigma V_s &\equiv\textrm{det}B+\textrm{det}E-2\textrm{det}C. \end{aligned}$$

4. Entanglement properties in double-cavity optomechanical system

In this section, we focus on investigating the entanglement properties of the double-COM system. We choose the experimentally realizable parameters as following [29]: $L_1=L_2$=1 mm, $\lambda$=810 nm, $\omega _{c1}=\omega _{c2}$=2$\pi c/\lambda$, $\omega _m/2\pi$=10 MHz, $m$=5 ng, $\gamma _m/2\pi$=100 Hz, $\mathcal {F}_1=\mathcal {F}_2=1.07\times 10^4$, $P_{L1}=P_{L2}=40$ mW.

4.1 Stability analysis

We now discuss the physical mechanism behind the stability of the double-COM system, as shown in Fig. 2, in which $a_i$ is the annihilation operator of the cavity mode $i$ and $b$ is the annihilation operator of the mechanical resonator. The four arrows of phonon energy transfer respectively represent four processes: backaction heating $a^\dagger b^\dagger$ ($A$), swap cooling $a^\dagger b$ ($B$), backaction cooling $ab$ ($C$), and swap heating $ab^\dagger$ ($D$). We note that the single-COM system in the red-detuning regime is stable. It can be explained by the fact that the red-detuning regime of the system mainly corresponds to the swap cooling process $a^\dagger b$, while the swap heating process $ab^\dagger$ can be ignored under the condition of weak coupling. This process realizes a kind of conversion from phonons to photons, which leads to the decrease of the mean phonon number, thus the system tends to be stable. The quantum correlation between cavity mode and mechanical resonator is produced by the interaction between them. However, the single-COM system in the blue-detuning regime is unstable due to the backaction heating process $a^\dagger b^\dagger$. The phonons are generated by the excess energy when the blue-detuning laser is used to excite the cavity mode. The increasing mean phonon number leads to the instability of the single-COM system. The mean phonon number can be decreased by the red-detuning laser, which ensures the stability of the COM system, we thus need to introduce a cavity mode driven by the red-detuning laser.

Fig. 2. Schematic illustration of the energy transfer in the double-COM system.

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To investigate the stability and entanglement properties of system, we numerically simulate the influence of the two laser powers on the mean phonon number (log$_{10}\langle b^\dagger b\rangle$) at $T$=50 K in Fig. 3(a) when both the cavity modes are driven by the red-detuning laser ($\Delta _1=\Delta _2=\omega _m$). When $P_{L1}=P_{L2}=0$, the mean phonon number is only related to temperature. With the increase of total laser power ($P_{L1}+P_{L2}$), the mean phonon number is constantly reduced before the total laser power reaches 30 mW. While when the total laser power exceeds 30 mW, the mean phonon number is increased. The emergence of the swap heating process $ab^\dagger$ weakens the cooling effect of the mechanical resonator when the laser power is excessively strong. When $P_{L1}+P_{L2}>52$ mW, the system is unstable. According to the Routh-Hurwitz criterion, the stability conditions of the system are as following

(14)$$\begin{aligned} 24(2\kappa^2+G_1^2+G_2^2)\kappa^4\omega_m^2+2(8\kappa^2+G_1^2+G_2^2)\omega_m^6& \nonumber \\ +16\kappa^8+(8\kappa^2-G_1^2-G_2^2)(6\kappa^2+G_1^2+G_2^2)\omega_m^4&>0, \nonumber \\ \kappa^3\omega_m^6\left[16\kappa^8+8\kappa^4(10\kappa^2-G_1^2-G_2^2)+(8\kappa^2+G_1^2+G_2^2)^2\right]&>0, \end{aligned}$$

where the mechanical damping rate is defined as $\gamma _m\rightarrow 0$ in terms of the selected parameters. In the first inequality in Eq. (14), the first three terms are greater than zero. If $8\kappa ^2$ is greater than $G_1^2+G_2^2$, then the last term is positive. In the second inequality, if the decay rates of cavity modes $\kappa >0$ and $10\kappa ^2>G_1^2+G_2^2$, then the inequality is satisfied. Therefore, according to Eq. (14), the stability of the system is related to the sum of squares of two effective optomechanical coupling strengths, i.e., the total power of two pump lasers.

Fig. 3. The mean phonon number ($\textrm{log}_{10}\langle \delta b^\dagger \delta b\rangle$) as a function of two pump powers $P_{L1}$ and $P_{L2}$. (a) $\Delta _1=\Delta _2=\omega _m$. (b) $\Delta _1=-\Delta _2=-\omega _m$. The other parameters are chosen as $L_1=L_2$=1 nm, $\lambda$=810 nm, $\omega _{c1}=\omega _{c2}$=2$\pi c/\lambda$, $\omega _m/2\pi$=10 MHz, $m$=5 ng, $\gamma _m/2\pi$=100 Hz, $\mathcal {F}_1=\mathcal {F}_2=1.07\times 10^4$, $P_{L1}=P_{L2}=40$ mW, and $T$=50 K.

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It is generally known that the single-COM system in the blue-detuning regime is unstable, which can be validated by the Routh-Hewitz criterion. For verifying the unstability, we set $\Delta _1=-\omega _m$, and the inequalities are as below

(15)$$\begin{aligned} (\kappa^2+\omega_m^2)(\kappa^3+3\kappa\omega_m^2)&>0, \nonumber \\ -4\kappa^2\omega_m^2(\kappa^2+\omega_m^2)(G_1^2+2\kappa^2+4\omega_m^2)&>0, \end{aligned}$$

where the second inequality in Eq. (15) is unsatisfied, so the system is unstable. Now we consider a double-COM system, which is consisted of the cavity mode 1 driven by the blue-detuning laser and the cavity mode 2 driven by the red-detuning laser ($\Delta _1=-\Delta _2=-\omega _m$). As shown in Fig. 3(b), we numerically simulate the influence of these two pump powers on the mean phonon number. When $P_{L1}>P_{L2}$, the double-COM system is unstable. In this case, the backaction heating process $a^\dagger b^\dagger$ corresponding to the blue-detuning regime is stronger than the swap cooling process $a^\dagger b$ corresponding to red-detuning regime, which leads to the increasing of the mean phonon number. When $P_{L2}-P_{L1}>50$ mW, the system is unstable, which is caused by excessive effective optomechanical coupling strength $G_2$. When $P_{L1}=0$ mW, the cooling effect of the mechanical oscillator is the most obvious. With the increase of pump power $P_{L1}$, the mean phonon number increases gradually. In addition, we also find that the stable region can be shifted by adjusting the pump powers $P_{L1}$ and $P_{L2}$. In order to analyze the double-COM system in more detail, according to the Routh-Hewitz criterion, we obtain the stability conditions of the system as following

(16)$$\begin{aligned} 16\kappa^5+2\kappa\omega_m^2(G_2^2-G_1^2+8\kappa^2)&>0, \nonumber \\ \kappa\omega_m^2(G_2^2-G_1^2)[16\kappa^8+8\kappa^4(G_1^2-G_2^2+10\kappa^2)\omega_m^2+& \nonumber \\ (G_2^2-G_1^2+8\kappa^2)^2\omega_m^4]&>0, \end{aligned}$$

where first inequality can be satisfied when $\kappa >0$ and $G_2^2-G_1^2>-8\kappa ^2\left (\frac {\kappa ^2}{\omega _m^2}+1\right )$. If $\kappa >0$, $G_2^2-G_1^2>0$, and $G_2^2-G_1^2<10\kappa ^2$, then the second inequality is well satisfied. When both two inequalities are satisfied, the double-COM system is stable. Therefore, the stability of the system is related to the difference of the squares of two effective coupling strengths, i.e., the difference between the powers of two pump lasers.

4.2 Stationary optomechanical entanglement

To get the stationary optomechancial entanglement, we find that the stable region in the blue-detuning regime can be produced by introducing a cavity mode driven by the red-detuning laser ($\Delta _2/\omega _m$=1) when $P_{L2}=40$ mW. As shown in Fig. 4, we investigate the influence of detuning parameter $\Delta _1/\omega _m$ and pump power $P_{L1}$ on entanglements between cavity mode 1 and mechanical resonator ($E_N(a_1b)$), cavity mode 2 and mechanical resonator ($E_N(a_2b)$), and two cavity modes ($E_N(a_1a_2)$), respectively. We first analyze the unstable regions of the double-COM system, as shown in the aqua area. The unstable regions in Fig. 4 are divided into the red-detuning region ($\Delta _1/\omega _m>0$) and the blue-detuning region ($\Delta _1/\omega _m<0$). When $P_{L1}=0$, the system is stable in the range of arbitrary detuning parameter $\Delta _1/\omega _m$. The enhancement of $P_{L1}$ leads to the unstable region of the double-COM system first appearing at the red sideband ($\Delta _1/\omega _m=1$) and the range of unstable region gradually becomes enlarger at $\Delta _1/\omega _m>0$. The instability of the system is induced by the excess total laser power $P_{L1}+P_{L2}$. In the following, we consider the unstable region in the blue-detuning regime when $\Delta _1<0$ and $\Delta _2=\omega _m$. The unstable region of the double-COM system first appears at blue sideband ($\Delta _1=-\omega _m$) when $P_{L1}=P_{L2}$. With the increase of pump power $P_{L1}$, the unstable region in the blue-detuning regime gradually becomes enlarger.

Fig. 4. Plots show the logarithmic negativities of the three bipartite subsystems, (a) $E_N(a_1b)$, (b) $E_N(a_2b)$, and (c) $E_N(a_1a_2)$, as functions of the normal detuning $\Delta _1/\omega _m$ and pump phase $P_{L1}$. In addition, $P_{L2}=40$ mW, $T=0.4$ K, and the other parameters are chosen as the same as given in Fig. 3.

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To clearly see the entanglement between each two subsystems in the stable region, the influence of the detuning parameter $\Delta _1/\omega _m$ and the pump power $P_{L1}$ on the entanglement is shown in Fig. 4. Figure 4(a) shows that the entanglement $E_N(a_1b)$ mainly appears around $\Delta _1/\omega _m=-1$, in which its maximum value is close to 0.3. Then we prove that the entanglement $E_N(a_1b)$ can appear in the blue-detuning regime. Figure 4(b) shows the influence of detuning parameter $\Delta _1/\omega _m$ and pump power $P_{L1}$ on the entanglement $E_N(a_2b)$. When $P_{L1}$=0 mW, the maximum value of entanglement $E_N(a_2b)$ is close to 0.13. With the increase of pump power $P_{L1}$, the entanglement becomes weaker when $\Delta _1<0$. It is noteworthy that at $\Delta _1/\omega _m=-1$, the blue-detuning laser in cavity mode 1 has the destructive effect on the entanglement $E_N(a_2b)$. While when $\Delta _1/\omega _m>0$, the influence of cavity mode 1 on the entanglement is weak. The influence of detuning parameter $\Delta _1/\omega _m$ and pump power $P_{L1}$ on entanglement $E_N(a_1a_2)$ is plotted in Fig. 4(c). When $\Delta _1/\omega _m>0$, there is hardly the entanglement $E_N(a_1a_2)$, this is because there is no energy exchange between the two cavity modes. When $\Delta _1/\omega _m<0$, there is weak entanglement $E_N(a_1a_2)$. The blue-detuning laser realizes energy transfer from cavity mode 1 to mechanical resonator, while the red-detuning laser realizes energy transfer from mechanical resonator to cavity mode 2. The energy exchange between the two cavity modes can be indirectly achieved.

4.3 Robust optomechanical entanglement

To investigate the influence of the temperature on entanglement, we numerically simulate the entanglement between each two subsystems when $\Delta _2/\omega _m=1$. In the case of $P_{L1}=20$ mW and $P_{L2}=25$ mW, we plot the influence of detuning parameter $\Delta _1/\omega _m$ and environment temperature $T$ on the entanglements $E_N(a_1b)$ and $E_N(a_2b)$, as shown in Figs. 5(a) and 5(b). One can see from Fig. 5(a) that the entanglement $E_N(a_1b)$ can appear at $T<30$ K in the case of $\Delta _1/\omega _m>0$. While when $\Delta _1/\omega _m<0$, the entanglement $E_N(a_1b)$ has remarkable resistance to environment temperature. Therefore, the entanglement that is extremely robust to temperature can be achieved in the blue-detuning regime, and the optimal value of entanglement appears at $\Delta _1/\omega _m=-1$. In Fig. 5(b), the entanglement $E_N(a_2b)$ only exists at $T<20$ K. When $-2<\Delta _1/\omega _m<0$, due to the destructive influence of the blue-detuning laser in cavity mode 1 on the entanglement, there is no the entanglement existing in this area, i.e., $E_N(a_2b)=0$, which is consistent with Fig. 4(b). The reason is that the blue-detuning laser leads to the increase of the mean phonon number, which destroys the entanglement $E_N(a_2b)$. Taking into account this, we can control indirectly the generation and disappearance of entanglement $E_N(a_2b)$ by the blue-detuning laser. When $P_{L1}=20$ mW and $P_{L2}=40$ mW, the influence of temperature $T$ and detuning parameter $\Delta _1/\omega _m$ on entanglement $E_N(a_1a_2)$ is shown in Fig. 5(c). In this regard the energy exchange between two cavity modes leads to a weak entanglement $E_N(a_1a_2)$ around $\Delta _1/\omega _m=-1$. What is important in this case is that the entanglement is strongly robust to temperature.

Fig. 5. Plots show the logarithmic negativities of three bipartite subsystems, (a) $E_N(a_1b)$, (b) $E_N(a_2b)$, and (c) $E_N(a_1a_2)$, as functions of normalized detuning $\Delta _1/\omega _m$ and environment temperature $T$. (a) $P_{L1}=20$ mW and $P_{L2}=25$ mW. (b) $P_{L1}=20$ mW and $P_{L2}=25$ mW. (c) $P_{L1}=20$ mW and $P_{L2}=40$ mW. The other parameters are chosen as the same as given in Fig. 3.

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5. The influence of the mechanical gain on optomechanical entanglement

In this section, we mainly investigate the amplification and robustness of entanglement when mechanical gain and environment temperature are taken into account. The relevant parameters of the double-COM system with mechanical gain are set as: $L_1=L_2$=1 mm, $\lambda$=810 nm, $\omega _{c1}=\omega _{c2}$=2$\pi c/\lambda$, $\omega _m/2\pi$=10 MHz, $m$=5 ng, and $\mathcal {F}_1=\mathcal {F}_2=1.07\times 10^4$.

5.1 The red-detuning regime

To see how the mechanical gain and temperature affect the entanglement $E_N(a_2b)$, with the choice of $P_1=0$ mW and $P_2=50$ mW, we plot $E_N(a_2b)$ as a function of the normalized detuning $\Delta _2/\omega _m$ and the temperature $T$, as shown in Fig. 6. Figure 6(a) shows the entanglement $E_N(a_2b)$ versus normalized detuning $\Delta _2/\omega _m$ for different mechanical loss (gain) rates when $T=$1 K. For a fixed value of the normalized detuning $\Delta _2/\omega _m$, one can see that $E_N(a_2b)$ increases (decreases) with the increase of mechanical gain (loss) rate $\gamma _m$. While for a fixed value of mechanical loss (gain) rate, $E_N(a_2b)$ first increases and then decreases with the increase of the normalized detuning $\Delta _2/\omega _m$. The height difference between any two lines becomes larger with the increase of the normalized detuning $\Delta _2/\omega _m$. On the other hand, it should be worth noting that the amplification of entanglement can be successfully achieved by taking mechanical gain into account in the red-detuning regime.

Fig. 6. (a) The entanglement $E_N(a_2b)$ as a function of the normalized detuning $\Delta _2/\omega _m$ when $T=1$ K. (b) The entanglement $E_N(a_2b)$ as a function of temperature $T$ with $\Delta _2=\omega _m$. The other parameters are set as $L_1=L_2$=1 nm, $\lambda$=810 nm, $\omega _{c1}=\omega _{c2}$=2$\pi c/\lambda$, $\omega _m/2\pi$=10 MHz, $m$=5 ng, $\mathcal {F}_1=\mathcal {F}_2=1.07\times 10^4$, $P_{1}=0$ mW, and $P_{2}=30$ mW.

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With the choice of $\Delta _2/\omega _m$=1, we calculate the entanglement $E_N(a_2b)$ versus the temperature $T$ for different mechanical loss (gain) rates, as shown in Fig. 6(b). One can see that $E_N(a_2b)$ decreases rapidly with the increase of temperature $T$ for a large mechanical loss rate. When the mechanical loss rate decreases, we find that the declining trend of the entanglement slows down and the robustness of entanglement on temperature becomes stronger. Especially, when the mechanical loss rate is close to zero ($\gamma _m/2\pi =0$ Hz), the effect of temperature on the entanglement disappears such that the optomechanical entanglement is almost independent of temperature. For the tiny mechanical gain rate ($\gamma _m/2\pi <0$), we find that the amplified entanglement appears at a suitable temperature. By adjusting the mechanical gain rate, therefore, the considerable entanglement can be achieved in a wide temperature range. For a fixed value of temperature $T$, it is seen that $E_N(a_2b)$ increases with the increase of mechanical gain rate in a reasonable range of temperature. Those can be observed from $\gamma _m(2n_m+1)$ in the diffusion matrix. By adjusting mechanical damping rate, we can change the effect of temperature on the entanglement.

5.2 The blue-detuning regime

The entanglement $E_N(a_1b)$ as a function of the normalized detuning $\Delta _1/\omega _m$ and temperature $T$ is shown in Fig. 7 in the case of $P_1=30$ mW, $P_2=50$ mW, and $\Delta _2/\omega _m=1$. Figure 7(a) shows $E_N(a_1b)$ versus the normalized detuning $\Delta _1/\omega _m$ for different mechanical loss (gain) rates when the temperature is $T=$10 K. For a fixed value of normalized detuning $\Delta _1/\omega _m$, it is seen that $E_N(a_1b)$ increases (decreases) with the increase of mechanical gain (loss) rate $\gamma _m$. For a fixed value of mechanical loss rate, $E_N(a_1b)$ first increases and then decreases with the increase of the normalized detuning $\Delta _1/\omega _m$. While for a fixed value of the mechanical gain rate, $E_N(a_1b)$ first increases, then decreases, and then increases again with the increase of the normalized detuning $\Delta _1/\omega _m$. The height difference between any two lines becomes larger with the increase of the normalized detuning $\Delta _1/\omega _m$. Therefore, the amplification of entanglement can be achieved when mechanical gain is taken into account in the blue-detuning regime.

Fig. 7. (a) The entanglement $E_N(a_1b)$ as a function of the normalized detuning $\Delta _1/\omega _m$ when $T=10$ K. (b) The entanglement $E_N(a_1b)$ as a function of temperature $T$ in the case of $\Delta _1=-\omega _m$. In (a) and (b), $P_{1}=30$ mW, $P_{2}=50$ mW, and $\Delta _2=\omega _m$. The other parameters are chosen as the same as given in Fig. 6.

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When $\Delta _1/\omega _m$=$-$1, the entanglement $E_N(a_1b)$ versus the temperature $T$ for different mechanical loss (gain) rates is shown in Fig. 7(b). One can see that $E_N(a_1b)$ decreases rapidly with the increase of temperature for a large mechanical loss rate. When the mechanical loss rate decreases, we find that the declining trend of the entanglement slows down and the robustness of entanglement on temperature becomes stronger. Especially, when the mechanical loss rate is close to zero ($\gamma _m/2\pi =0$ Hz), the destructive effect of temperature on entanglement disappears. For a tiny mechanical gain, the entanglement can be significantly amplified. Therefore, the considerable entanglement can be obtained in a wide temperature range by adjusting the mechanical gain rate. It is worth noting that the influence of the mechanical gain on entanglement in the blue-detuning regime is similar to that in the red-detuning regime. With the increase of temperature, the influence of the mechanical loss (gain) rate on entanglement becomes larger.

6. Conclusions

In conclusions, we have shown that the robust optomehanical entanglement can be achieved in a double-COM system with blue-detuning laser and mechanical gain. In the stable regime of the system, we find that the optomechanical entanglement is extremely robust to temperature can be generated in the blue-detuning regime. The blue-detuning laser in cavity mode 1 can control the entanglement between cavity mode 2 and mechanical resonator indirectly. Moreover, we find that the optomechanical entanglement between two subsystems without direct coupling can be achieved. Furthermore, we investigate the effects of mechanical gain on the entanglement, and find that the entanglement between mechanical resonator and each cavity mode is hardly affected by the temperature when the mechanical loss rate is close to zero. And the entanglement amplification at high temperature can be achieved by adjusting the mechanical gain appropriately. This might open a new way for the generation of robust entanglement in high temperature and provides an effective method for achieving the amplification of entanglement.

Funding

National Natural Science Foundation of China (11465020, 61465013, 61822114); Project of Jilin Science and Technology Development for Leading Talent of Science and Technology Innovation in Middle and Young and Team Project (20160519022JH).

Disclosures

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

References

1. S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444(7115), 67–70 (2006). [CrossRef]

2. S. GröBlacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009). [CrossRef]

3. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003). [CrossRef]

4. H. Jing, S. K. Özdemir, X. Y. Lü, J. Zhang, L. Yang, and F. Nori, “$\mathcal {PT}$-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014). [CrossRef]

5. X. W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90(3), 033809 (2014). [CrossRef]

6. J. D. Thompson, B. M. Zwickl, A. M. Jayich, M. Florian, 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]

7. M. J. Hartmann and M. B. Plenio, “Steady state entanglement in the mechanical vibrations of two dielectric membranes,” Phys. Rev. Lett. 101(20), 200503 (2008). [CrossRef]

8. 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]

9. M. Bhattacharya and P. Meystre, “Multiple membrane cavity optomechanics,” Phys. Rev. A 78(4), 041801 (2008). [CrossRef]

10. J. D. Teufel, J. W. Harlow, C. A. Regal, and K. W. Lehnert, “Dynamical backaction of microwave fields on a nanomechanical oscillator,” Phys. Rev. Lett. 101(19), 197203 (2008). [CrossRef]

11. T. Rocheleau, T. Ndukum, C. Macklin, J. B. Hertzberg, A. A. Clerk, and K. C. Schwab, “Preparation and detection of a mechanical resonator near the ground state of motion,” Nature 463(7277), 72–75 (2010). [CrossRef]

12. N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Rev. Mod. Phys. 86(2), 419–478 (2014). [CrossRef]

13. P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90(4), 043825 (2014). [CrossRef]

14. L. Liu, G. Zhang, X. Wang, L. Liu, G. Zhang, X. Wang, L. Liu, G. Zhang, and X. Wang, “Tunable double optomechanically induced transparency in a dual-species bose-einstein condensate,” Laser Phys. Lett. 14(10), 105201 (2017). [CrossRef]

15. M. Fleischhauer, A. Imamoglu, and J. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005). [CrossRef]

16. G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010). [CrossRef]

17. B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92(3), 033829 (2015). [CrossRef]

18. T. Wang, M. H. Zheng, C. H. Bai, D. Y. Wang, A. D. Zhu, H. F. Wang, and S. Zhang, “Normal-mode splitting and optomechanically induced absorption, amplification, and transparency in a hybrid optomechanical system,” Ann. Phys. 530(10), 1800228 (2018). [CrossRef]

19. M. H. Zheng, T. Wang, D. Y. Wang, C. H. Bai, S. Zhang, C. S. An, and H. F. Wang, “Manipulation of multi-transparency windows and fast-slow light transitions in a hybrid cavity optomechanical system,” Sci. China: Phys., Mech. Astron. 62(5), 950311 (2019). [CrossRef]

20. J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6(9), 707–712 (2010). [CrossRef]

21. Y. M. Liu, C. H. Bai, D. Y. Wang, T. Wang, M. H. Zheng, H. F. Wang, A. D. Zhu, and S. Zhang, “Ground-state cooling of rotating mirror in double-laguerre-gaussian-cavity with atomic ensemble,” Opt. Express 26(5), 6143 (2018). [CrossRef]

22. E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482(7383), 63–67 (2012). [CrossRef]

23. D. Y. Wang, C. H. Bai, S. Liu, S. Zhang, and H. F. Wang, “Optomechanical cooling beyond the quantum backaction limit with frequency modulation,” Phys. Rev. A 98(2), 023816 (2018). [CrossRef]

24. R. Simon and M. J. Hartmann, “Quantum information processing with nanomechanical qubits,” Phys. Rev. Lett. 110(12), 120503 (2013). [CrossRef]

25. K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012). [CrossRef]

26. A. H. Safavi-Naeini and O. Painter, “Proposal for an optomechanical traveling wave phonon-photon translator,” New J. Phys. 13(1), 013017 (2011). [CrossRef]

27. 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]

28. 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]

29. M. Aspelmeyer, “Towards experimental optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007). [CrossRef]

30. N. Yazdanpanah and M. K. Tavassoly, “Spectral properties of trapped two-level ions interacting with quantized fields,” Phys. Rev. A 95(6), 063815 (2017). [CrossRef]

31. X. Yang, Y. Ling, X. Shao, and M. Xiao, “Generation of robust tripartite entanglement with a single-cavity optomechanical system,” Phys. Rev. A 95(5), 052303 (2017). [CrossRef]

32. C. Cao, S. C. Mi, Y. P. Gao, L. Y. He, D. Yang, T. J. Wang, R. Zhang, and C. Wang, “Tunable high-order sideband spectra generation using a photonic molecule optomechanical system,” Sci. Rep. 6(1), 22920 (2016). [CrossRef]

33. B. He, Y. Liu, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A 94(3), 031802 (2016). [CrossRef]

34. D. Schönleber, A. Eisfeld, and R. El-Ganainy, “Optomechanical interactions in non-hermitian photonic molecules,” New J. Phys. 18(4), 045014 (2016). [CrossRef]

35. D. Y. Wang, C. H. Bai, S. Liu, S. Zhang, and H. F. Wang, “Distinguishing photon blockade in a $\mathcal {PT}$-symmetric optomechanical system,” Phys. Rev. A 99(4), 043818 (2019). [CrossRef]

36. Z. P. Liu, J. Zhang, Åđ. K. Özdemir, B. Peng, H. Jing, X. Y. Lü, C. W. Li, L. Yang, F. Nori, and Y. X. Liu, “Metrology with $\mathcal {PT}$-symmetric cavities: Enhanced sensitivity near the $\mathcal {PT}$-phase transition,” Phys. Rev. Lett. 117(11), 110802 (2016). [CrossRef]

37. C. Tchodimou, P. Djorwé, and S. G. Nana Engo, “Distant entanglement enhanced in $\mathcal {PT}$-symmetric optomechanics,” Phys. Rev. A 96(3), 033856 (2017). [CrossRef]

38. Y. L. Liu, R. Wu, J. Zhang, Åđ. K. Özdemir, and Y. X. Liu, “Controllable optical response by modifying the gain and loss of a mechanical resonator and cavity mode in an optomechanical system,” Phys. Rev. A 95(1), 013843 (2017). [CrossRef]

39. I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104(8), 083901 (2010). [CrossRef]

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

41. J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9(11), 712–716 (2013). [CrossRef]

42. I. Mahboob, K. Nishiguchi, H. Okamoto, and H. Yamaguchi, “Phonon-cavity electromechanics,” Nat. Phys. 8(5), 387–392 (2012). [CrossRef]

43. I. Mahboob, K. Nishiguchi, A. Fujiwara, and H. Yamaguchi, “Phonon lasing in an electromechanical resonator,” Phys. Rev. Lett. 110(12), 127202 (2013). [CrossRef]

44. A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011). [CrossRef]

45. C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource for maximally entangled photonic states: Application to interferometry,” Phys. Rev. A 66(1), 013804 (2002). [CrossRef]

46. F. Massel, S. U. Cho, J. M. Pirkkalainen, P. J. Hakonen, T. T. Heikkilä, and M. A. Sillanpää, “Multimode circuit optomechanics near the quantum limit,” Nat. Commun. 3(1), 987 (2012). [CrossRef]

47. X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. 9(3), 179–184 (2013). [CrossRef]

48. T. S. Duffy, “Earth science: probing the core’s light elements,” Nature 479(7374), 480–481 (2011). [CrossRef]

Temperature-resistant generation of robust entanglement with blue-detuning driving and mechanical gain

Abstract

1. Introduction

2. System Hamiltonian and dynamical equations

3. Logarithmic negativity

4. Entanglement properties in double-cavity optomechanical system

4.1 Stability analysis

4.2 Stationary optomechanical entanglement

4.3 Robust optomechanical entanglement

5. The influence of the mechanical gain on optomechanical entanglement

5.1 The red-detuning regime

5.2 The blue-detuning regime

6. Conclusions

Funding

Disclosures

References

Cited By

Figures (7)

Equations (16)

Optics Express