the result of the unitary transformation with $U_I(t)=\exp\{-i\int_0^t d\tau (H_r+\omega_c\hat{a}^\dagger\hat{a}+\omega_m\hat{b}^\dagger\hat{b})\}$. Similar to the practices in [@supp1], we make a rotating-wave approximation to neglect the rapidly oscillating terms in Eq. (\[couple\]), together with an approximation of smooth system-reservoir couplings to reduce the associated coupling intensities to constants, i.e, $\kappa(\omega)\rightarrow \sqrt{\kappa/\pi}$ and $\gamma_m(\omega)\rightarrow \sqrt{\gamma_m/ \pi}$. Then the system-reservoir coupling will be simplified to $$\begin{aligned}
H_{sr}(t)=i\sqrt{2\kappa }\{\hat{a}^{\dagger }\hat{\xi}_{c}(t)-\hat{a}\hat{\xi}^{\dagger}_{c}(t)\}+i\sqrt{2\gamma_m }\{\hat{b}^{\dagger }
\hat{\xi}_{m}(t)-\hat{b}\hat{\xi}^{\dagger }_{m}(t)\},
\label{couple1}\end{aligned}$$ where $\hat{\xi}_{c,m}(t)=\frac{1}{\sqrt{2\pi}}\int d\omega \hat{\xi}_{c,m}(\omega)e^{-i(\omega-\omega_{c,m})t}$. For the relatively slow interaction processes described by Eq. (\[couple1\]), the quantum noise represented by $\hat{\xi}_{c,m}(t)$ can be approximated as a white one satisfying $[\hat{\xi}_{c,m}(t),\hat{\xi}^\dagger_{c,m}(t')]=\delta(t-t')$ [@supp1].

Meanwhile, under the above-mentioned rotation by $U_I(t)$, the external drive Hamiltonian and the optomechanical coupling Hamiltonian will become $H_e(t)=iE(\hat{a}^{\dagger}e^{i\Delta t}-\hat{a}e^{-i\Delta t})$, where $\Delta=\omega_c-\omega_l$, and $H_{om}(t)=-g\hat{a}^\dagger\hat{a}(\hat{b}e^{-i\omega_m t}+\hat{b}^\dagger e^{i\omega_m t})$, respectively. The action $U(t)={\cal T}\exp\{-i\int_0^t d\tau H(\tau)\}$ of the total Hamiltonian $H(t)=H_e(t)+H_{om}(t)+H_{sr}(t)$, which is called the formal solution to Quantum Stochastic Schrödinger equations in [@supp1], will lead to the following nonlinear dynamical equations of the system modes: $$\begin{aligned}
\dot{\hat{a}}&=&-\kappa\hat{a}+ig(\hat{b}e^{-i\omega_m t}+\hat{b}^\dagger e^{i\omega_m t})\hat{a}+E e^{i\Delta t}+\sqrt{2\kappa}\hat{\xi}_c(t),\nonumber\\
\dot{\hat{b}}&=&-\gamma_m\hat{b}+ige^{i\omega_m t}\hat{a}^\dagger\hat{a}+\sqrt{2\gamma_m}\hat{\xi}_m(t).
\label{e1}\end{aligned}$$ Among the literature, the dynamics of an OMS is also described in another rotating frame only with respect to the drive frequency $\omega_l$ (see, e.g. 