## 2 Model
Combining Equation (3) with Equation (2), and equating coefficients belonging to the same frequency, we find the amplitudes of the components oscillating at the frequency \(\omega_{c}\) of the coupling field  
\[p_{0} = 0,\] \[q_{0} = -\frac{\hbar g|c_{0}|^{2}}{m\omega_{m}^{2}+\alpha q_{0}}, \tag{4}\] \[c_{0} = \frac{\sqrt{\eta\kappa}c}{\frac{\kappa}{2}+i\Delta},\]  
the amplitudes of the first-order sideband generations  
\[c_{1+} = \frac{\sqrt{\eta\kappa}\epsilon_{p}}{d(\delta)}\left\{\left[ \frac{\kappa}{2}-i(\Delta+\delta)\right]f(\delta)+i\hbar g^{2}|c_{0}|^{2} \right\},\] \[c_{1-} = \frac{i\hbar g^{2}c_{0}^{2}\sqrt{\eta\kappa}\epsilon_{p}^{*}}{d ^{*}(\delta)}, \tag{5}\] \[q_{1+} = -\frac{\hbar g(c_{0}^{*}c_{1+}+c_{0}c_{1-}^{*})}{f(\delta)},\]  
and the amplitude of the second-order anti-Stokes field  
\[c_{2+} = \frac{-ig}{d(2\delta)}\Big{\{}q_{1+}c_{1+}\Big{\{}\Big{[}\frac{ \kappa}{2}-i(\Delta+2\delta)\Big{]}f(2\delta)+i\hbar g^{2}|c_{0}|^{2}\Big{\}}- \alpha c_{0}q_{1+}^{2}\Big{[}\frac{\kappa}{2}-i(\Delta+2\delta)\Big{]}\] \[-\hbar gc_{0}\left\{igc_{0}q_{1+}+c_{1+}\Big{[}\frac{\kappa}{2}- i(\Delta+2\delta)\Big{]}\right\}c_{1-}^{*}\Big{\}},\]  
where \(\Delta=\omega_{0}-\omega_{c}+gq_{0}\) is the displacement-dependent effective cavity detuning, \(d(\delta)=[\frac{\kappa}{2}+i(\Delta-\delta)][\frac{\kappa}{2}-i(\Delta+ \delta)]f(\delta)-2\Delta\hbar g^{2}|c_{0}|^{2},f(\delta)=m\omega_{m}^{2}+2 \alpha q_{0}-im\delta(\gamma_{m}-i\delta)\). The \(q_{0}\) and \(p_{0}\) represent the steady-state values of the displacement and momentum of the movable mirror, respectively. It is seen that the steady-state displacement \(q_{0}\) of the mechanical oscillator is related to the mechanical nonlinearity \(\alpha\). Moreover, we note that the amplitudes \(c_{1+}\) and \(c_{2+}\) of the first-order and second-order anti-Stokes fields depend on the mechanical nonlinearity strength \(\alpha\) and the steady-state displacement \(q_{0}\) of the movable mirror. In addition, the amplitude \(c_{2+}\) of the second-order anti-Stokes field is dependent on the amplitudes \(c_{1+}\) and \(c_{1-}\) of the first-order anti-Stokes and Stokes fields and the amplitude \(q_{1+}\) of the mechanical excitation. Furthermore, the output field is connected with the cavity field and the two input fields via the standard input-output relation [15,46], which yields  
\[\varepsilon_{out}(t)=\varepsilon_{c}+\varepsilon_{p}e^{-i\delta t}-\sqrt{\eta \kappa}\langle c\rangle. \tag{7}\]  
The output field \(\varepsilon_{out}(t)\) consists of five components  
\[p_{0} = 0,\]0  
where \(\varepsilon_{out0}\), \(\varepsilon_{out1+}\), \(\varepsilon_{out1-}\), \(\varepsilon_{out2+}\), \(\varepsilon_{out2-}\) are the amplitudes of the five components of the output field, respectively, and their frequencies are \(\omega_{c}\), \(\omega_{p}\), and \(2\omega_{c}-\omega_{p}\), \(2\omega_{p}-\omega_{c}\), \(3\omega_{c}-2\omega_{p}\), respectively. From Equations (7) and (8), we obtain  
\[p_{0} = 0,\]1 \[p_{0} = 0,\]2Next, the output probe field at frequency \(\omega_{p}\) is rescaled by \(\varepsilon_{p}\), which gives  
\[p_{0} = 0,\]3  
Without the coupling field (\(\wp_{c}=0\)), \(t_{p}\) is given by  
\[p_{0} = 0,\]4  
which is independent of the oscillation of the movable mirror. Moreover, the outgoing field from the cavity at the second-order upper sideband is renormalized as  
\[p_{0} = 0,\]5  
Without the coupling field (\(\wp_{c}=0\)), \(u=0\), no second-order upper sideband is generated.