## 4 The Enhancement of the Second-Order Upper Sideband Generation by the Mechanical Nonlinearity  
In this section, we discuss the influence of the mechanical nonlinearity \(\alpha\) on the second-order upper sideband generation at frequency \(2\omega_{p}-\omega_{c}\) generated by the four wave-mixing process. In the four wave-mixing process, two photons at frequency \(\omega_{p}\) are absorbed and one photon at frequency \(\omega_{c}\) is emitted.  
The intensity \(|u|^{2}\) of the second-order upper sideband generation in the output field is plotted against the normalized probe detuning \(\delta/\omega_{m}\) for different values of the nonlinearity strength \(\alpha\) when the power of the probe field is \(\wp_{p}=0.01\wp_{c}\) in Figure 4. Without the mechanical nonlinearity (\(\alpha=0\)), a dip is observed near \(\delta=\omega_{m}\) in the intensity \(|u|^{2}\) of the second-order upper sideband generation. The value of \(|u|^{2}\) at the dip near \(\delta=\omega_{m}\) is about zero. For \(\alpha=0\), we have mentioned above that the value of \(|t_{p}|^{2}\) at the transparency peak near \(\delta=\omega_{m}\) in the first-order upper sideband generation in Figure 2 is about unity. Hence, when the input probe field is completely reflected to the left-hand output port, the second-order anti-Stokes field vanishes. This is consistent with the result in [28]. Moreover, as the nonlinearity strength \(\alpha\) is increased, the maximum intensity \(|u|^{2}\) of the second-order upper sideband generation is increased. When \(\alpha=0\), \(10\times 10^{7}\) N/m\({}^{2}\), the maximum values of \(|u|^{2}\) are about 0.0063, 0.1003, respectively. Thus, the intensity \(|u|^{2}\) of the second-order upper sideband generation with the mechanical nonlinearity \(\alpha=10\times 10^{7}\) N/m\({}^{2}\) is enhanced by a factor of about 15.9  
Figure 3: The transparency peak position of the output probe field as a function of the nonlinearity strength \(\alpha\).  
Figure 2: The intensity \(|t_{p}|^{2}\) of the output probe field versus the normalized probe detuning \(\delta/\omega_{m}\) for different values of the nonlinearity strength \(\alpha=0,5\times 10^{7},10\times 10^{7}\) N/m\({}^{2}\). The black dotted, blue dashed, and red solid lines correspond to \(\alpha=0\), \(5\times 10^{7}\) N/m\({}^{2}\), \(10\times 10^{7}\) N/m\({}^{2}\), respectively. The dot-dashed green line is the intensity \(|t_{p}|^{2}\) without the coupling laser.  
compared to that without the mechanical nonlinearity \(\alpha=0\). In addition, the intensity \(|u|^{2}\) of the second-order upper sideband generation can be largely enhanced by further increasing the mechanical nonlinearity strength \(\alpha\). For example, for \(\alpha=11\times 10^{7}\) N/m\({}^{2}\), the cubic potential energy \(\frac{1}{3}\alpha q_{0}^{3}\) is still much smaller than the quadratic potential energy \(\frac{1}{2}m\omega_{m}^{2}q_{0}^{2}\) due to \(\left|\frac{1}{3}\alpha q_{0}^{3}/\left(\frac{1}{2}m\omega_{m}^{2}q_{0}^{2} \right)\right|\approx 0.16\ll 1\), the maximum value of \(|u|^{2}\) is about \(0.4886\). Hence, the intensity \(|u|^{2}\) of the second-order upper sideband generation with the mechanical nonlinearity \(\alpha=11\times 10^{7}\) N/m\({}^{2}\) is enhanced by a factor of about \(77.5\) compared to that without the mechanical nonlinearity \(\alpha=0\). Therefore, the mechanical nonlinearity can enhance the intensity \(|u|^{2}\) of the second-order upper sideband generation. This is because the mechanical nonlinearity enhances the four-wave mixing process in the optical cavity.