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Consider the standard model of optomechanical systems with a single optical cavity mode coupled to a mechanical oscillator, which is canonically modeled as a FP cavity with one fixed mirror and one movable mirror mounted on a spring. By setting $\hbar=1$, the Hamiltonian reads

\begin{equation}\label{eq:Ham} H=-\Delta a^{\dagger}a+\omega_{m}b^{\dagger}b+g a^{\dagger}a(b^{\dagger}+b) +\eta a^{\dagger}+\eta^*a, \end{equation} where $\Delta_d =\omega_m-\omega_c$ is the detuning between the cavity mode and the external deriving laser, $a (b)$ describes the cavity (mechanical) mode, $g$ represents the single-photon coupling strength, and $\mathcal{E}$ denotes the amplitude of the external driving laser. Could any one write me any comments or suggestion the reduced master equation for the mechanical modes after tracing out the cavity modes (or any reference as link)

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Following the elimination of the cavity field, there is a nice work along this line for the non-degenerate multimode leads to the effective master equation for the mechanics.

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