This article is about optomechanical damping rate

Motion equation

$$\begin{aligned}
\dot{\alpha}&=(i\Delta-\frac{\kappa}{2})\alpha+iGx\alpha+\sqrt{\kappa_{ex}}\alpha_{in}\\
m_{eff}\ddot{x}&=-m_{eff}\Omega_m^2x-m_{eff}\Gamma_m\dot{x}+\hbar G|\alpha|^2
\end{aligned}$$

Frequency space

We choice $\alpha=\bar{\alpha}+\delta\alpha$ to linearize, and then we get
$$\begin{aligned}
-i\omega\delta\alpha[\omega]&=(i\Delta-\frac{\kappa}{2})\delta\alpha[\omega]+iG\bar{\alpha}x[\omega]\\
-m_{eff}\omega^2x[\omega]&=-m_{eff}\Omega_m^2x[\omega]+i\omega m_{eff}\Gamma_m x[\omega]\\
&+\hbar G(\bar{\alpha}^{\dagger}\delta\alpha[\omega]+\bar{\alpha}\delta\alpha[-\omega]^{\dagger})
\end{aligned}$$

From equation one, we get
$$\begin{aligned}
\delta\alpha[\omega]&=iG\bar{\alpha}\chi_{opt}[\omega]x[\omega]\\
\delta\alpha[-\omega]^{\dagger}&=-iG\bar{\alpha}^{\dagger}\chi_{opt}[-\omega]^{\dagger}x[-\omega]^{\dagger}
\end{aligned}$$

Insert them back to equation two, we get the modified mechanical susceptibility
$$\begin{aligned}
\chi^{-1}_{m,eff}(\omega)&=m_{eff}((\Omega_m^2-\omega^2)-i\omega\Gamma_m)\\
&+\hbar G^2|\bar{\alpha}|^2(\frac{1}{(\Delta+\omega)+i\kappa/2}+\frac{1}{(\Delta-\omega)-i\kappa/2})\\
&=\chi_m^{-1}(\omega)+\Sigma(\omega)
\end{aligned}$$
By using
$$\hbar G^2|\bar{\alpha}|^2=\hbar g_0^2\frac{2m_{eff}\Omega_m}{\hbar}n_{cav}=2m_{eff}\Omega_mg^2$$
we get
$$\Sigma(\omega)=2m_{eff}\Omega_mg^2(\frac{1}{(\Delta+\omega)+i\kappa/2}+\frac{1}{(\Delta-\omega)-i\kappa/2})$$
From the equation above we can get the effective mechanical damping rate
$$\begin{aligned}
\Gamma_{eff}&=\Gamma_m+\Gamma_{opt}\\
&=\Gamma_m\left(1+\frac{\Omega_m}{\omega}\left(\frac{1}{(\frac{\Delta+\omega}{g})^2\frac{\Gamma_m}{\kappa}+\frac{1}{C}}-\frac{1}{(\frac{\Delta-\omega}{g})^2\frac{\Gamma_m}{\kappa}+\frac{1}{C}}\right)\right)
\end{aligned}$$
Now in weak laser drive, then $g\ll\kappa$, we choice $\omega=\Omega_m$, then we get
$$\Gamma_{eff}=\Gamma_m\left(1+\left(\frac{1}{(\frac{\Delta+\Omega_m}{g})^2\frac{\Gamma_m}{\kappa}+\frac{1}{C}}-\frac{1}{(\frac{\Delta-\Omega_m}{g})^2\frac{\Gamma_m}{\kappa}+\frac{1}{C}}\right)\right)$$
In the blue detuning(red detuning), $\Delta=\pm\Omega_m$, then
$$\Gamma_{eff}|_{\Delta=\pm\Omega_m}=\Gamma_m\left(1\mp C\left(1-\frac{1}{\frac{16\Omega_m^2}{\kappa^2}+1}\right)\right)$$
In the resolved-sideband regime, $\kappa\ll\Omega_m$, we can get two conclusions below

1. In red detuning, $\Gamma_{eff}=\Gamma_m(1+C)$, the mechanic mode line width will increase
2. In blue detuning, $\Gamma_{eff}=\Gamma_m(1-C)$, the mechanic mode line width will decrease. Especially, when C is greater than 1, then we will get an instability.