Dynamic dissipative synchronized cooling of two mechanical resonators in strong coupling optomechanics

Abstract

Ground-state cooling of multiple mechanical resonators is an important goal in the study of quantum optomechanics. Here, we propose a dynamic dissipative synchronous cooling method for the simultaneous cooling of the two mechanical resonators in the strong optomechanical coupling regime. The synchronized modulation of the cavity dissipation can significantly accelerate the cooling process, with both of the resonators reaching the ground state. We derive the analytical cooling limits, which agree with the numerical simulations. The present scheme opens a new prospect for the research of multiple-mode ground-state cooling of mechanical resonators.

Appendices

Appendix A

1.1 The quantum Langevin equations

In the frame rotating at the input laser frequency $\omega_{in}^{{}}$, the Hamiltonian is written as Eq. (2), where $\Delta_{1}^{{}} = \omega_{in} - \omega_{1}$ and $\Delta_{2}^{{}} = \omega_{in} - \omega_{2}$ are the detuning. The quantum Langevin equations are given by

$$ \begin{gathered} \dot{a}_{1}^{{}} = \left( {i\Delta_{1}^{{}} - \frac{{\kappa_{1}^{{}} }}{2}} \right)a_{1}^{{}} - ig_{1}^{{}} a_{1}^{{}} \left( {b_{1}^{\dag } + b_{1}^{{}} } \right) - iJ_{1}^{{}} a_{2}^{{}} - i\Omega_{1}^{{}} - \sqrt {\kappa_{1}^{{}} } a_{in,1}^{{}} , \hfill \\ \dot{a}_{2}^{{}} = \left( {i\Delta_{2}^{{}} - \frac{{\kappa_{2}^{{}} }}{2}} \right)a_{2}^{{}} - ig_{2}^{{}} a_{2}^{{}} \left( {b_{2}^{\dag } + b_{2}^{{}} } \right) - iJ_{1}^{ * } a_{1}^{{}} - i\Omega_{2}^{{}} - \sqrt {\kappa_{2}^{{}} } a_{in,2}^{{}} , \hfill \\ \dot{b}_{1}^{{}} = \left( { - i\omega_{m1}^{{}} - \frac{{\gamma_{1}^{{}} }}{2}} \right)b_{1}^{{}} - ig_{1}^{{}} a_{1}^{\dag } a_{1}^{{}} - iJ_{2}^{{}} b_{2}^{{}} - \sqrt {\gamma_{1}^{{}} } b_{in,1}^{{}} , \hfill \\ \dot{b}_{2}^{{}} = \left( { - i\omega_{m2}^{{}} - \frac{{\gamma_{2}^{{}} }}{2}} \right)b_{2}^{{}} - ig_{2}^{{}} a_{2}^{\dag } a_{2}^{{}} - iJ_{2}^{\dag } b_{1}^{{}} - \sqrt {\gamma_{2}^{{}} } b_{in,2}^{{}} , \hfill \\ \end{gathered} $$

(12)

$\kappa_{l = 1,2}^{{}}$ and $\gamma_{l = 1,2}^{{}}$ are the decay rates of the $l$ cavity mode and the $l$ mechanical mode, respectively. The $a_{in,1}^{{}} ,a_{in,2}^{{}}$ and $b_{in,1}^{{}} ,b_{in,2}^{{}}$ are the corresponding noise operators, in which these operators have zero mean values and the following correlation functions:

$$ \begin{gathered} \left\langle {a_{in,1}^{{}} \left( t \right)a_{in,1}^{\dag } \left( {t{\prime }} \right)} \right\rangle = \left\langle {a_{in,2}^{{}} \left( t \right)a_{in,2}^{\dag } \left( {t{\prime }} \right)} \right\rangle = \delta \left( {t - t{\prime }} \right), \hfill \\ \left\langle {a_{in,1}^{\dag } \left( t \right)a_{in,1}^{{}} \left( {t{\prime }} \right)} \right\rangle = \left\langle {a_{in,2}^{\dag } \left( t \right)a_{in,2}^{{}} \left( {t{\prime }} \right)} \right\rangle = 0, \hfill \\ \left\langle {b_{in,1}^{{}} \left( t \right)b_{in,1}^{\dag } \left( {t{\prime }} \right)} \right\rangle = \left( {n_{th1} + 1} \right)\delta \left( {t - t{\prime }} \right), \hfill \\ \left\langle {b_{in,1}^{\dag } \left( t \right)b_{in,1}^{{}} \left( {t{\prime }} \right)} \right\rangle = n_{th1} \delta \left( {t - t{\prime }} \right), \hfill \\ \left\langle {b_{in,2}^{{}} \left( t \right)b_{in,2}^{\dag } \left( {t{\prime }} \right)} \right\rangle = \left( {n_{th2} + 1} \right)\delta \left( {t - t{\prime }} \right), \hfill \\ \left\langle {b_{in,2}^{\dag } \left( t \right)b_{in,2}^{{}} \left( {t{\prime }} \right)} \right\rangle = n_{th2} \delta \left( {t - t{\prime }} \right). \hfill \\ \end{gathered} $$

(13)

Here, $n_{thl = th1,th2}$ are the thermal phonon number which is given by $n_{thl = th1,th2} = \left[ {\exp \left( {\frac{{\hbar \omega_{ml = m1,m2} }}{{k_{B} T}}} \right) - 1} \right]^{ - 1}$ where $T$ is the environmental temperature and $k_{B}$ is Boltzmann constant.

For strong driving, the Hamiltonian can be linearized, with $a_{1} \to \alpha_{1} + a_{{_{1} }}^{{\prime }}$,$a_{2} \to \alpha_{2} + a_{{_{2} }}^{{\prime }}$,$b_{1} \to \beta_{1} + b_{{_{1} }}^{{\prime }}$,$b_{2} \to \beta_{2} + b_{{_{2} }}^{{\prime }}$. Here, $a_{1}^{{\prime }}$,$a_{2}^{{\prime }}$ and $b_{1}^{{\prime }}$,$b_{2}^{{\prime }}$ describe the quantum fluctuations around the mean values $\alpha_{1} \equiv \left\langle {a_{1}^{{}} } \right\rangle ,\alpha_{2} \equiv \left\langle {a_{2}^{{}} } \right\rangle$ and $\beta_{1} \equiv \left\langle {b_{1}^{{}} } \right\rangle ,\beta_{2} \equiv \left\langle {b_{2}^{{}} } \right\rangle$, respectively. For the sake of simplicity, $a_{1}^{{\prime }}$,$a_{2}^{{\prime }}$ and $b_{1}^{{\prime }}$,$b_{2}^{{\prime }}$ are still written as $a_{1}^{{}}$,$a_{2}^{{}}$ and $b_{1}^{{}}$,$b_{2}^{{}}$ behind. The quantum Langevin equations are rewritten as

$$ \begin{gathered} \dot{a}_{1}^{{}} = \left( {i\Delta_{1}^{{\prime }} - \frac{{\kappa_{1}^{{}} }}{2}} \right)a_{1}^{{}} - ig_{1}^{{}} \alpha_{1}^{{}} \left( {b_{1}^{\dag } + b_{1}^{{}} } \right) - ig_{1}^{{}} a_{1}^{{}} \left( {b_{1}^{\dag } + b_{1}^{{}} } \right) - iJ_{1}^{{}} a_{2}^{{}} - \sqrt {\kappa_{1}^{{}} } a_{in,1}^{{}} , \hfill \\ \dot{a}_{2}^{{}} = \left( {i\Delta_{2}^{{\prime }} - \frac{{\kappa_{2}^{{}} }}{2}} \right)a_{2}^{{}} - ig_{2}^{{}} \alpha_{2}^{{}} \left( {b_{2}^{\dag } + b_{2}^{{}} } \right) - ig_{2}^{{}} a_{2}^{{}} \left( {b_{2}^{\dag } + b_{2}^{{}} } \right) - iJ_{1}^{ * } a_{1}^{{}} - \sqrt {\kappa_{2}^{{}} } a_{in,2}^{{}} , \hfill \\ \dot{b}_{1}^{{}} = \left( { - i\omega_{m1}^{{}} - \frac{{\gamma_{1}^{{}} }}{2}} \right)b_{1}^{{}} - ig_{1}^{{}} \left( {\alpha_{1}^{ * } a_{1}^{{}} + \alpha_{1}^{{}} a_{1}^{\dag } } \right) - ig_{1}^{{}} a_{1}^{\dag } a_{1}^{{}} - iJ_{2}^{{}} b_{2}^{{}} - \sqrt {\gamma_{1}^{{}} } b_{in,1}^{{}} , \hfill \\ \dot{b}_{2}^{{}} = \left( { - i\omega_{m2}^{{}} - \frac{{\gamma_{2}^{{}} }}{2}} \right)b_{2}^{{}} - ig_{2}^{{}} \left( {\alpha_{2}^{ * } a_{2}^{{}} + \alpha_{2}^{{}} a_{2}^{\dag } } \right) - ig_{2}^{{}} a_{2}^{\dag } a_{2}^{{}} - iJ_{2}^{\dag } b_{1}^{{}} - \sqrt {\gamma_{2}^{{}} } b_{in,2}^{{}} , \hfill \\ \end{gathered} $$

(14)

where $\Delta_{1}^{{\prime }} = \Delta_{1}^{{}} - g_{1} \left( {\beta_{1}^{ * } + \beta_{1}^{{}} } \right)$ and $\Delta_{2}^{{\prime }} = \Delta_{2}^{{}} - g_{2} \left( {\beta_{2}^{ * } + \beta_{2}^{{}} } \right)$ are the optomechanical coupling modified detuning, respectively. Under strong driving conditions, the nonlinear terms $ig_{l}^{{}} a_{l}^{{}} \left( {b_{l}^{\dag } + b_{l}^{{}} } \right)\left| \begin{gathered} \hfill \\ l = 1,2 \hfill \\ \end{gathered} \right.$ and $ig_{l}^{{}} a_{l}^{\dag } a_{l}^{{}} \left| \begin{gathered} \hfill \\ l = 1,2 \hfill \\ \end{gathered} \right.$ in the above equations are neglected. Then, the quantum Langevin equations become linearized, and the linearized system Hamiltonian can be extracted as

$$ \begin{aligned} H_{L} &= - \Delta_{1}^{{\prime }} a_{1}^{\dag } a_{1}^{{}} - \Delta_{2}^{{\prime }} a_{2}^{\dag } a_{2}^{{}} + \omega_{m1} b_{1}^{\dag } b_{1}^{{}} + \omega_{m2} b_{2}^{\dag } b_{2}^{{}} \\ &\quad+ \left( {G_{1}^{{}} a_{1}^{\dag } + G_{1}^{ * } a_{1}^{{}} } \right)\left( {b_{1}^{\dag } + b_{1}^{{}} } \right) + \left( {G_{2}^{{}} a_{2}^{\dag } + G_{2}^{ * } a_{2}^{{}} } \right)\left( {b_{2}^{\dag } + b_{2}^{{}} } \right) \\ &\quad+ \left( {J_{1}^{{}} a_{1}^{\dag } a_{2}^{{}} + J_{1}^{ * } a_{1}^{{}} a_{2}^{\dag } } \right) + \left( {J_{2}^{{}} b_{1}^{\dag } b_{2}^{{}} + J_{2}^{ * } b_{1}^{{}} b_{2}^{\dag } } \right), \\ \end{aligned} $$

(15)

where $G_{1} = g_{1} \alpha_{1}$ and $G_{2} = g_{2} \alpha_{2}$ describe the linear optomechanical coupling strength, respectively.

Appendix B

2.1 The differential equations

$$ \begin{aligned} \frac{d}{dt}\left\langle {a_{1}^{{_{\dag } }} a_{1} } \right\rangle &= - \kappa_{1} \left\langle {a_{1}^{{_{\dag } }} a_{1} } \right\rangle - i\left( G_{1} \left\langle {a_{1}^{{_{\dag } }} b_{1}^{{_{\dag } }} } \right\rangle + G_{1} \left\langle {a_{1}^{{_{\dag } }} b_{1} } \right\rangle - G_{1}^{ * } \left\langle {a_{1} b_{1}^{{_{\dag } }} } \right\rangle - G_{1}^{ * } \left\langle {a_{1} b_{1} } \right\rangle\right.\\&\quad\left. + J_{1} \left\langle {a_{1}^{{_{\dag } }} a_{2} }\right\rangle - J_{1}^{ * } \left\langle {a_{1} a_{2}^{{_{\dag } }}} \right\rangle \right) \end{aligned} $$

$$ \begin{aligned} \frac{d}{dt}\left\langle {a_{1}^{{_{\dag } }} a_{2} } \right\rangle &= \frac{{ - \kappa_{1} - \kappa_{2} }}{2}\left\langle {a_{1}^{{_{\dag } }} a_{2} } \right\rangle - i\left[ \left({\Delta_{1}^{^{\prime}} - \Delta_{2}^{^{\prime}} }\right)\left\langle {a_{1}^{{_{\dag } }} a_{2} } \right\rangle -G_{1}^{ * } \left\langle {a_{2} b_{1}^{{_{\dag } }} } \right\rangle - G_{1}^{ * } \left\langle {a_{2} b_{1} } \right\rangle\right.\nonumber\\&\quad\left. + G_{2} \left\langle {a_{1}^{{_{\dag } }} b_{2}^{{_{\dag }}} } \right\rangle + G_{2} \left\langle {a_{1}^{{_{\dag } }} b_{2} }\right\rangle - J_{1}^{ * } \left\langle {a_{2} a_{2}^{{_{\dag } }}} \right\rangle + J_{1}^{ * } \left\langle {a_{1} a_{1}^{{_{\dag }}} } \right\rangle \right] \end{aligned} $$

$$ \begin{aligned} \frac{d}{dt}\left\langle {a_{1}^{{_{\dag } }} b_{1} }\right\rangle &= \frac{{ - \kappa_{1} - \gamma_{1} }}{2}\left\langle {a_{1}^{{_{\dag } }} b_{1} } \right\rangle - i\left[ \left({\Delta_{1}^{^{\prime}} + \omega_{m1} } \right)\left\langle {a_{1}^{{_{\dag } }} b_{1} } \right\rangle -G_{1}^{ * } \left\langle {b_{1} b_{1} } \right\rangle \right.\\&\quad \left. + G_{1} \left\langle {a_{1}^{{_{\dag } }}a_{1}^{{_{\dag } }} } \right\rangle - G_{1}^{ * } \left\langle {b_{1}^{{_{\dag } }} b_{1} } \right\rangle + G_{1}^{* } \left\langle {a_{1}^{{_{\dag } }} a_{1} } \right\rangle -J_{1}^{ * } \left\langle {a_{2}^{{_{\dag } }} b_{1} } \right\rangle + J_{2} \left\langle {a_{1}^{{_{\dag } }} b_{2} } \right\rangle \right]\end{aligned} $$

$$ \begin{aligned}\frac{d}{dt}\left\langle {a_{1}^{{_{\dag } }} b_{2} } \right\rangle &= \frac{{ - \kappa_{1} - \gamma_{2} }}{2}\left\langle {a_{1}^{{_{\dag } }} b_{2} } \right\rangle - i\left[ \left({\Delta_{1}^{^{\prime}} + \omega_{m2} } \right)\left\langle {a_{1}^{{_{\dag } }} b_{2} } \right\rangle - G_{1}^{ * }\left\langle {b_{1}^{{_{\dag } }} b_{2} } \right\rangle - G_{1}^{ *} \left\langle {b_{1} b_{2} } \right\rangle\right.\\&\quad \left. + G_{2} \left\langle {a_{1}^{{_{\dag } }}a_{2}^{{_{\dag } }} } \right\rangle + G_{2}^{ * } \left\langle {a_{1}^{{_{\dag } }} a_{2} } \right\rangle - J_{1}^{* } \left\langle {a_{2}^{{_{\dag } }} b_{2} } \right\rangle +J_{2}^{ * } \left\langle {a_{1}^{{_{\dag } }} b_{1} } \right\rangle \right]\end{aligned} $$

$$ \frac{d}{dt}\left\langle {a_{1} a_{1} } \right\rangle = - \kappa_{1} \left\langle {a_{1} a_{1} } \right\rangle - i\left( { - 2\Delta_{1}^{^{\prime}} \left\langle {a_{1} a_{1} } \right\rangle + 2G_{1} \left\langle {a_{1} b_{1}^{{_{\dag } }} } \right\rangle + 2G_{1} \left\langle {a_{1} b_{1} } \right\rangle + 2J_{1} \left\langle {a_{1} a_{2} } \right\rangle } \right) $$

$$ \begin{aligned}\frac{d}{dt}\left\langle {a_{1} a_{2} } \right\rangle &= \frac{{ -\kappa_{1} - \kappa_{2} }}{2}\left\langle {a_{1} a_{2} }\right\rangle - i\left[ \left( { - \Delta_{1}^{^{\prime}} -\Delta_{2}^{^{\prime}} } \right)\left\langle {a_{1} a_{2} }\right\rangle + G_{1} \left\langle {a_{2} b_{1}^{{_{\dag } }} }\right\rangle + G_{1} \left\langle {a_{2} b_{1} }\right\rangle\right.\nonumber\\&\quad \left. + G_{2} \left\langle {a_{1} b_{2}^{{_{\dag } }} }\right\rangle + G_{2} \left\langle {a_{1} b_{2} } \right\rangle +J_{1} \left\langle {a_{2} a_{2} } \right\rangle + J_{1}^{ * }\left\langle {a_{1} a_{1} } \right\rangle \right]\end{aligned} $$

$$ \begin{aligned} \frac{d}{dt}\left\langle {a_{1} b_{1} } \right\rangle &= \frac{{ - \kappa_{1} - \gamma_{1} }}{2}\left\langle {a_{1} b_{1}} \right\rangle - i\left[ \left( {\omega_{m1} -\Delta_{1}^{^{\prime}} } \right)\left\langle {a_{1} b_{1} }\right\rangle + G_{1} \left\langle {b_{1} b_{1} } \right\rangle +G_{1} \left\langle {b_{1}^{{_{\dag } }} b_{1} } \right\rangle \right.\\&\quad \left. + G_{1} \left\langle {a_{1}^{{_{\dag } }} a_{1} }\right\rangle + G_{1}^{ * } \left\langle {a_{1} a_{1} }\right\rangle + G_{1} + J_{1} \left\langle {a_{2} b_{1} }\right\rangle + J_{2} \left\langle {a_{1} b_{2} } \right\rangle \right]\end{aligned} $$

$$ \begin{aligned}\frac{d}{dt}\left\langle {a_{1} b_{2} } \right\rangle &= \frac{{ -\kappa_{1} - \gamma_{2} }}{2}\left\langle {a_{1} b_{2} }\right\rangle - i\left[ \left( {\omega_{m2} - \Delta_{1}^{^{\prime}}} \right)\left\langle {a_{1} b_{2} } \right\rangle + G_{1}\left\langle {b_{1}^{{_{\dag } }} b_{2} }\right\rangle + G_{1} \left\langle {b_{1} b_{2} } \right\rangle \right.\\&\quad \left. + G_{2} \left\langle {a_{1} a_{2}^{{_{\dag } }} }\right\rangle + G_{2}^{ * } \left\langle {a_{1} a_{2} }\right\rangle + J_{1} \left\langle {a_{2} b_{2} } \right\rangle +J_{2}^{ * } \left\langle {a_{1} b_{1} } \right\rangle \right]\end{aligned} $$

$$\frac{d}{dt}\left\langle {a_{2}^{{_{\dag } }} a_{2} } \right\rangle = - \kappa_{2} \left\langle {a_{2}^{{_{\dag } }} a_{2} } \right\rangle - i\left( {G_{2} \left\langle {a_{2}^{{_{\dag } }} b_{2}^{{_{\dag } }} } \right\rangle + G_{2} \left\langle {a_{2}^{{_{\dag } }} b_{2} } \right\rangle - G_{2}^{ * } \left\langle {a_{2} b_{2}^{{_{\dag } }} } \right\rangle - G_{2}^{ * } \left\langle {a_{2} b_{2} } \right\rangle - J_{1} \left\langle {a_{1}^{{_{\dag } }} a_{2} } \right\rangle + J_{1}^{ * } \left\langle {a_{1} a_{2}^{{_{\dag } }} } \right\rangle } \right) $$

$$ \begin{aligned} \frac{d}{dt}\left\langle {a_{2}^{{_{\dag } }} b_{1} }\right\rangle & = \frac{{ - \kappa_{2} - \gamma_{1}}}{2}\left\langle {a_{2}^{{_{\dag } }} b_{1} } \right\rangle -i\left[ \left( {\Delta_{2}^{^{\prime}} + \omega_{m1} }\right)\left\langle {a_{2}^{{_{\dag } }} b_{1} } \right\rangle +G_{1} \left\langle {a_{1}^{{_{\dag } }} a_{2}^{{_{\dag } }} }\right\rangle + G_{1}^{ * } \left\langle {a_{1} a_{2}^{{_{\dag } }}} \right\rangle\right.\\&\quad \left. - G_{2}^{* } \left\langle {b_{1} b_{2}^{{_{\dag } }} } \right\rangle -G_{2}^{ * } \left\langle {b_{1} b_{2} } \right\rangle - J_{1}\left\langle {a_{1}^{{_{\dag } }} b_{1} } \right\rangle + J_{2}\left\langle {a_{2}^{{_{\dag } }} b_{2} } \right\rangle \right]\end{aligned} $$

$$ \begin{aligned}\frac{d}{dt}\left\langle {a_{2}^{{_{\dag } }} b_{2} } \right\rangle &= \frac{{ - \kappa_{2} - \gamma_{2} }}{2}\left\langle {a_{2}^{{_{\dag } }} b_{2} } \right\rangle - i\left[ \left({\Delta_{2}^{^{\prime}} + \omega_{m2} } \right)\left\langle {a_{2}^{{_{\dag } }} b_{2} } \right\rangle + G_{2} \left\langle {a_{2}^{{_{\dag } }} a_{2}^{{_{\dag } }} } \right\rangle - G_{2}^{ *} \left\langle {b_{2} b_{2} } \right\rangle\right.\\&\quad \left. - G_{2}^{ * } \left\langle {b_{2}^{{_{\dag } }} b_{2}} \right\rangle + G_{2}^{ * } \left\langle {a_{2}^{{_{\dag } }}a_{2} } \right\rangle - J_{1} \left\langle {a_{1}^{{_{\dag } }}b_{2} } \right\rangle + J_{2}^{ * } \left\langle {a_{2}^{{_{\dag }}} b_{1} } \right\rangle \right]\end{aligned} $$

$$ \frac{d}{dt}\left\langle {a_{2} a_{2} } \right\rangle = - \kappa_{2} \left\langle {a_{2} a_{2} } \right\rangle - i\left( { - 2\Delta_{2}^{^{\prime}} \left\langle {a_{2} a_{2} } \right\rangle + 2G_{2} \left\langle {a_{2} b_{2}^{{_{\dag } }} } \right\rangle + 2G_{2} \left\langle {a_{2} b_{2} } \right\rangle + 2J_{1}^{ * } \left\langle {a_{1} a_{2} } \right\rangle } \right) $$

$$ \begin{aligned}\frac{d}{dt}\left\langle {a_{2} b_{1} } \right\rangle &= \frac{{ -\kappa_{2} - \gamma_{1} }}{2}\left\langle {a_{2} b_{1} }\right\rangle - i\left[ \left( {\omega_{m1} - \Delta_{2}^{^{\prime}}} \right)\left\langle {a_{2} b_{1} } \right\rangle + G_{1}\left\langle {a_{1}^{{_{\dag } }} a_{2} } \right\rangle + G_{1}^{ *} \left\langle {a_{1} a_{2} }\right\rangle \right.\\&\quad\left. + G_{2} \left\langle {b_{1} b_{2}^{{_{\dag } }} }\right\rangle + G_{2} \left\langle {b_{1} b_{2} } \right\rangle +J_{1}^{ * } \left\langle {a_{1} b_{1} } \right\rangle + J_{2}\left\langle {a_{2} b_{2} } \right\rangle \right]\end{aligned} $$

$$ \begin{aligned}\frac{d}{dt}\left\langle {a_{2} b_{2} } \right\rangle &= \frac{{ -\kappa_{2} - \gamma_{2} }}{2}\left\langle {a_{2} b_{2} }\right\rangle - i\left[ \left( {\omega_{m2} - \Delta_{2}^{^{\prime}}} \right)\left\langle {a_{2} b_{2} } \right\rangle + G_{2}\left\langle {a_{2}^{{_{\dag } }} a_{2} } \right\rangle + G_{2}\left\langle {b_{2}^{{_{\dag } }} b_{2} }\right\rangle + G_{2}\right.\\&\quad \left. + G_{2} \left\langle {b_{2} b_{2} } \right\rangle +G_{2}^{* } \left\langle {a_{2} a_{2} } \right\rangle + J_{1}^{ * }\left\langle {a_{1} b_{2} } \right\rangle + J_{2}^{ * } \left\langle {a_{2} b_{1} } \right\rangle \right]\end{aligned} $$

$$ \frac{d}{dt}\left\langle {b_{1}^{{_{\dag } }} b_{1} } \right\rangle = - \gamma_{1} \left\langle {b_{1}^{{_{\dag } }} b_{1} } \right\rangle + \gamma_{1} \times n_{th1} - i\left( {G_{1} \left\langle {a_{1}^{{_{\dag } }} b_{1}^{{_{\dag } }} } \right\rangle - G_{1} \left\langle {a_{1}^{{_{\dag } }} b_{1} } \right\rangle + G_{1}^{ * } \left\langle {a_{1} b_{1}^{{_{\dag } }} } \right\rangle - G_{1}^{ * } \left\langle {a_{1} b_{1} } \right\rangle + J_{2} \left\langle {b_{1}^{{_{\dag } }} b_{2} } \right\rangle - J_{2}^{ * } \left\langle {b_{1} b_{2}^{{_{\dag } }} } \right\rangle } \right) $$

$$ \begin{aligned} \frac{d}{dt}\left\langle {b_{1}^{{_{\dag } }} b_{2} } \right\rangle &= \frac{{ - \gamma_{1} - \gamma_{2} }}{2}\left\langle {b_{1}^{{_{\dag } }} b_{2} } \right\rangle - i\left[ {\left( {\omega_{m2} - \omega_{m1} } \right)\left\langle {b_{1}^{{_{\dag } }} b_{2} } \right\rangle - G_{1} \left\langle {a_{1}^{{_{\dag } }} b_{2} } \right\rangle - G_{1}^{ * } \left\langle {a_{1} b_{2} } \right\rangle} \right. \\ &\quad \left.{ + G_{2} \left\langle {a_{2}^{{_{\dag } }} b_{1}^{{_{\dag } }} } \right\rangle + G_{2}^{ * } \left\langle {a_{2} b_{1}^{{_{\dag } }} } \right\rangle + J_{2}^{ * } \left\langle {b_{1} b_{1}^{{_{\dag } }} } \right\rangle - J_{2}^{ * } \left\langle {b_{2} b_{2}^{{_{\dag } }} } \right\rangle } \right] \end{aligned}$$

$$ \frac{d}{dt}\left\langle {b_{1} b_{1} } \right\rangle = - \gamma_{1} \left\langle {b_{1} b_{1} } \right\rangle - i\left( {2\omega_{m1} \left\langle {b_{1} b_{1} } \right\rangle + 2G_{1} \left\langle {a_{1}^{{_{\dag } }} b_{1} } \right\rangle + 2G_{1}^{ * } \left\langle {a_{1} b_{1} } \right\rangle + 2J_{2} \left\langle {b_{1} b_{2} } \right\rangle } \right) $$

$$ \begin{aligned} \frac{d}{dt}\left\langle {b_{1} b_{2} } \right\rangle &= \frac{{ - \gamma_{1} - \gamma_{2} }}{2}\left\langle {b_{1} b_{2} } \right\rangle - i\left[ {\left( {\omega_{m2} + \omega_{m1} } \right)\left\langle {b_{1} b_{2} } \right\rangle + G_{1} \left\langle {a_{1}^{{_{\dag } }} b_{2} } \right\rangle }\right. \\ &\quad\left. {+ G_{1}^{ * } \left\langle {a_{1} b_{2} } \right\rangle + G_{2} \left\langle {a_{2}^{{_{\dag } }} b_{1} } \right\rangle + G_{2}^{ * } \left\langle {a_{2} b_{1} } \right\rangle + J_{2} \left\langle {b_{2} b_{2} } \right\rangle + J_{2}^{ * } \left\langle {b_{1} b_{1} } \right\rangle } \right]\end{aligned} $$

$$ \begin{aligned}\frac{d}{dt}\left\langle {b_{2}^{{_{\dag } }} b_{2} } \right\rangle &= - \gamma_{2} \left\langle {b_{2}^{{_{\dag } }} b_{2} } \right\rangle + \gamma_{2} \times n_{th2} \\ &\quad- i\left( {G_{2} \left\langle {a_{2}^{{_{\dag } }} b_{2}^{{_{\dag } }} } \right\rangle - G_{2} \left\langle {a_{2}^{{_{\dag } }} b_{2} } \right\rangle + G_{2}^{ * } \left\langle {a_{2} b_{2}^{{_{\dag } }} } \right\rangle - G_{2}^{ * } \left\langle {a_{2} b_{2} } \right\rangle - J_{2} \left\langle {b_{1}^{{_{\dag } }} b_{2} } \right\rangle + J_{2}^{ * } \left\langle {b_{1} b_{2}^{{_{\dag } }} } \right\rangle } \right) \end{aligned}$$

$$ \frac{d}{dt}\left\langle {b_{2} b_{2} } \right\rangle = - \gamma_{2} \left\langle {b_{2} b_{2} } \right\rangle - i\left( {2\omega_{m2} \left\langle {b_{2} b_{2} } \right\rangle + 2G_{2} \left\langle {a_{2}^{{_{\dag } }} b_{2} } \right\rangle + 2G_{2}^{ * } \left\langle {a_{2} b_{2} } \right\rangle + 2J_{2}^{ * } \left\langle {b_{1} b_{2} } \right\rangle } \right) $$