## 2 Model  
We consider a Fabry-Perot optical cavity with a moving-end mirror, as depicted in Figure 1. The movable mirror is a cubic anharmonic oscillator. An intense coupling field with frequency \(\omega_{c}\) and a weak probe field with frequency \(\omega_{p}\) are sent into the optical cavity through the left fixed mirror. We assume that a single-mode cavity field having a resonant frequency of \(\omega_{0}\) is generated inside the optical cavity. The moving mirror with effective mass \(m\) and resonance frequency \(\omega_{m}\) makes one-dimensional motion along the cavity axis due to the radiation pressure force exerted by the  
Figure 1: Schematic diagram of the studied system. A coupling field with frequency \(\omega_{c}\) and a probe field with frequency \(\omega_{p}\) enter the cavity through the left fixed mirror. The right movable mirror is a cubic anharmonic oscillator. photons within the cavity. By moving to a rotating frame at the coupling frequency \(\omega_{c}\), the Hamiltonian of the system reads  
\[H = \hbar(\omega_{0}-\omega_{c})c^{\dagger}c+\hbar gc^{\dagger}cq+H_{m} +i\hbar\sqrt{\eta\kappa}\varepsilon_{c}(c^{\dagger}-c)+i\hbar\sqrt{\eta\kappa}( \varepsilon_{p}c^{\dagger}e^{-i\delta t}-\varepsilon_{p}^{*}ce^{i\delta t}),\] \[H_{m} = \tfrac{1}{2}\left(\tfrac{p^{2}}{m}+m\omega_{m}^{2}q^{2}\right)+ \tfrac{1}{3}\alpha q^{3}, \tag{1}\]  
where \(c\) and \(c^{\dagger}\) denote the annihilation and creation operators of the cavity mode, satisfying the commutation relation \([c,c^{\dagger}]=1\), \(g\) is the optomechanical coupling constant, \(q\) and \(p\) are the displacement and momentum operators of the movable mirror, obeying the relation \([q,p]=i\hbar\). The \(\delta=\omega_{p}-\omega_{c}\) is the probe tuning from the coupling field. The amplitudes \(\varepsilon_{c}\) and \(\varepsilon_{p}\) of the coupling and probe fields are determined by their respective powers \(\wp_{c}\) and \(\wp_{p}\) by \(\varepsilon_{c}=\sqrt{\wp_{c}/(\hbar\omega_{c})}\) and \(\varepsilon_{p}=\sqrt{\wp_{p}/(\hbar\omega_{p})}\). The \(\eta\) is the cavity coupling parameter defined by \(\eta=\kappa_{e}/(\kappa_{0}+\kappa_{e})=\kappa_{e}/\kappa\), where \(\kappa\) is the total cavity decay rate, \(\kappa_{0}\) and \(\kappa_{e}\) are the photon loss rates due to the intrinsic dissipation of the cavity and the transmission of the left fixed mirror, respectively. In Equation (1), the first term gives the energy of the cavity field, the second term describes the coupling between the optical cavity field and the movable mirror, the third term is the energy of the movable mirror which is modeled as a cubic anharmonic oscillator, the four terms give the interaction between the coupling field and the cavity field, the last term describes the interaction between the probe field and the cavity field. The \(H_{m}\) has two terms, the first term corresponds to the energy of the simple harmonic oscillator, the second term proportional to the third order of the mechanical displacement \(q\) is the potential energy generated by the mechanical nonlinearity, and the parameter \(\alpha\) represents the nonlinearity strength of the mechanical oscillator. The cubic potential \(\tfrac{1}{3}\alpha q^{3}\) is assumed to be much smaller than the quadratic potential \(\tfrac{1}{2}m\omega_{m}^{2}q^{2}\), thus the cubic potential \(\tfrac{1}{3}\alpha q^{3}\) is a small perturbation. A cubic anharmonic oscillator is a typical example of the simplest perturbation to the harmonic oscillator in quantum mechanics [35].