as seen from the Hamiltonian $iE(\hat{a}^{\dagger}e^{-i\omega_l t}-\hat{a}e^{i\omega_l t})$, will build up the cavity field. Under the radiation pressure of the cavity field, the size of the cavity will be changed from $L$ to $L+\hat{x}_m$, where $\hat{x}_m=\sqrt{\frac{1}{2m\omega_m}}(\hat{b}+\hat{b}^\dagger)$ is the displacement of the mechanical oscillator with the effective mass $m$ (the notation of $\hbar=1$ is adopted), and is treated as a q-number for a quantum OMS. The consequentially modified cavity oscillation term $n\pi c/(L+\hat{x}_m)\hat{a}^\dagger\hat{a}\approx \omega_c\hat{a}^\dagger\hat{a}-(\omega_c/L)\hat{x}_m\hat{a}^\dagger\hat{a}$ gives the optomechanical coupling Hamiltonian as the second term written as $-g\hat{a}^\dagger\hat{a}(\hat{b}+\hat{b}^\dagger)$, where $g=\sqrt{\frac{1}{2m\omega_m}}\frac{\omega_c}{L}$.

In addition to the above-mentioned elements, the OMS is an open quantum system coupled to the environment known as the cavity and mechanical reservoirs, which are modeled as ensembles of oscillators with the Hamiltonian $$\begin{aligned}
H_r=\int_0^\infty d\omega_1\omega_1\hat{\xi}^\dagger_c\hat{\xi}_c(\omega_1)+\int_0^\infty d\omega_2\omega_2\hat{\xi}^\dagger_m\hat{\xi}_m(\omega_2)\end{aligned}$$ where $[\hat{\xi}_{c,m}(\omega),\hat{\xi}^\dagger_{c,m}(\omega')]=\delta(\omega-\omega')$. The Hamiltonian for the coupling between the system and the reservoirs takes the general form $$\begin{aligned}
H_{s-r}=i\int d\omega_1 \kappa(\omega_1)(\hat{a}-\hat{a}^\dagger)\{\hat{\xi}^\dagger_c(\omega_1)+\hat{\xi}_c(\omega_1)\}+i\int d\omega_2 \gamma_m(\omega_2)(\hat{b}-\hat{b}^\dagger)\{\hat{\xi}^\dagger_m(\omega_2)+\hat{\xi}_m(\omega_2)\}.\end{aligned}$$ In the rotating frame with respect to the cavity frequency $\omega_c$ and the mechanical frequency $\omega_m$, as well as to the frequencies of the reservoir modes, the system-reservoir coupling Hamiltonian will become $$\begin{aligned}
H^{(i)}_{s-r}&=&i\int d\omega_1 \kappa(\omega_1)(\hat{a}e^{-i\omega_c t}-\hat{a}^\dagger e^{i\omega_c t})\{\hat{\xi}^\dagger_c(\omega_1)e^{i\omega_1 t}+\hat{\xi}_c(\omega_1)e^{-i\omega_1 t}\}\nonumber\\
&+&i\int d\omega_2 \gamma_m(\omega_2)(\hat{b}e^{-i\omega_m t}-\hat{b}^\dagger e^{i\omega_m t})\{\hat{\xi}^\dagger_m(\omega_2)e^{i\omega_2 t}+\hat{\xi}_m
(\omega_2)e^{-i\omega_2 t}\},
\label{couple}\end{aligned}$$ as 