## 5 Conclusions  
We have investigated that the propagation of a weak probe field in an optomechanical system with a cubic anharmonic oscillator. When the optomechanical cavity is pumped by a strong red-sideband laser, the nonlinearity of the movable mirror causes an asymmetric line shape to appear in the intensity of the output probe field. We show that the nonlinearity strength of the movable mirror can be detected by measuring the shift of the transparency peak of the output probe field. We also show that the mechanical nonlinearity results in the enhancement of the intensity of the second-order upper sideband generation. Thus, compared to the optomechanical system with a mechanical harmonic oscillator, the optomechanical system with a cubic anharmonic oscillator offers more efficient conversion from the incident lights (the coupling light and the probe light) to the second-order upper sideband generation; this is useful in frequency conversion in all-optical communication networks. Future work will extend these studies to the single-photon strong-coupling regime [47; 48], where the nonlinear coupling between the optical cavity field and the movable mirror could not be neglected.  
Finally, we briefly discuss the possibility of the generation of a strong cubic mechanical nonlinearity. It has been reported that a cubic anharmonic oscillator can be realized by coupling a linear mechanical oscillator to a three-level auxiliary via the linear interaction \(\mu xV\)[25], where \(\mu\) is the interaction strength, \(x\) is the dimensionless displacement operator of the mechanical oscillator and is related to the mechanical displacement operator \(q\) via \(q=\sqrt{\frac{\hbar}{m\omega_{m}}}x\), and \(V\) is the operator of the three-level auxiliary. The auxiliary system has a diagonal Hamiltonian \(\hat{H}_{0}\), whose three energy levels are given by \(E_{j}^{(0)}=j\Delta_{a}\), \(j=0,1,2\), and \(\Delta_{a}\) is the energy difference between adjacent energy levels. By placing the auxiliary in the eigenstate corresponding to the energy level \(E_{1}\), choosing the appropriate operator \(V\), a cubic potential in the form of \(H_{m}^{(3)}=2\hbar\Delta_{a}\left(\frac{\mu}{\Delta_{a}}\right)^{3}x^{3}\) could be created. By choosing \(\Delta_{a}=2\pi\times 5\) GHz, \(\mu=2\pi\times 3.7\) MHz, \(\mu/\Delta_{a}\ll 1\), we have \(H_{m}^{(3)}\approx 2.685\times 10^{-33}x^{3}\). By using \(H_{m}^{(3)}=\frac{1}{3}\alpha q^{3}\), we find that the cubic nonlinearity strength \(\alpha\) is about \(1.2\times 10^{8}\) N/m\({}^{2}\), which is close to the parameter we use in the numerical calculations.  
Figure 4: The intensity \(|u|^{2}\) of the second-order upper sideband generation in the output 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\({}^{2}\) when \(\wp_{p}=0.01\wp_{c}\). The other parameters are the same as those in Figure 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. Here, the red solid line for \(\alpha=10\times 10^{7}\) N/m\({}^{2}\) has been divided by 10.  
**Author Contributions:** Conceptualization, S.H. and A.C.; methodology, S.H. and A.C.; software, S.H. and H.H.; formal analysis, S.H. and H.H.; writing--original draft preparation, S.H. and H.H.; writing--review and editing, S.H., H.H., and A.C. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by Science Foundation of Zhejiang Sci-Tech University (Grant Nos. 18062121-Y, 17062071-Y), by the National Natural Science Foundation of China (Grant Nos. 11775190,91636108), and by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LZ20A040002).  
**Conflicts of Interest:** The authors declare no conflict of interest.