## 3 The Effect of the Mechanical Nonlinearity on the EIT  
In this section, we discuss the influence of the nonlinearity strength \(\alpha\) of the movable mirror on the intensity \(|t_{p}|^{2}\) of the output probe field at frequency \(\omega_{p}\). The parameters we use are from the experimental work focusing on slowing, advancing, and switching of the weak probe field [18]. The frequency of the coupling field is \(\omega_{c}=2\pi\times 6.07\) GHz. The cavity decay rate is \(\kappa=2\pi\times 742\) kHz. The effective mass of the movable mirror is \(m=7\) pg, the mechanical frequency \(\omega_{m}=2\pi\times 1.45\) MHz, and the mechanical damping rate is \(\gamma_{m}=2\pi\times 9\) Hz. The optomechanical coupling strength is \(g=2\pi\times 1.26/\sqrt{\frac{\hbar}{2m\omega_{m}}}\) Hz/m. The cavity field is driven by a red-sideband coupling field with \(\Delta=\omega_{m}\). We consider the critically coupled cavity with \(\eta=0.5\). The power of the coupling field is \(\wp_{c}=3\)\(\mu\)W. For the nonlinearity strength \(\alpha=10\times 10^{7}\) N/m\({}^{2}\), the cubic potential energy \(\frac{1}{3}\pi q_{0}^{3}\) is much smaller than the quadratic potential energy \(\frac{1}{2}m\omega_{m}q_{0}^{2}\) since \(|\frac{1}{3}\pi q_{0}^{3}/(\frac{1}{2}m\omega_{m}^{2}q_{0}^{2})|\approx 0.14 \ll 1\). The intensity \(|t_{p}|^{2}\) of the output probe field is plotted against the normalized probe detuning \(\delta/\omega_{m}\) for different values of the nonlinearity strength \(\alpha\) in Figure 2. In the absence of the coupling field (\(\wp_{c}=0\)), it is seen that the intensity curve exhibits an inverted Lorentzian shape, and the value of \(|t_{p}|^{2}\) near \(\delta=\omega_{m}\) is close to zero, thus the input weak probe field is almost completely absorbed by the optomechanical system. In the presence of the coupling field (\(\wp_{c}\
eq 0\)), for the linear mechanical oscillator (\(\alpha=0\)), the intensity curve is symmetric, it has an EIT peak near \(\delta=\omega_{m}\), and the value of \(|t_{p}|^{2}\) near \(\delta=\omega_{m}\) is close to unity, thus the weak probe field is almost totally reflected to the left-hand output port. This is the result of the destructive interference between the input probe field and the first-order anti-Stokes field produced by the interaction of the strong coupling field with the linear mechanical oscillator. For the nonlinear mechanical oscillator (\(\alpha\
eq 0\)), the intensity curve becomes asymmetric, and the transparency peak is shifted to the left, i.e., the output probe frequency \(\omega_{p}\) is lower than the cavity resonance frequency \(\omega_{0}\). Increasing the nonlinearity strength \(\alpha\) will increase the shift of the transparency peak. The dependence of the transparency peak position of the output probe field on the mechanical nonlinearity strength \(\alpha\) is shown in Figure 3. From Figure 3, it is seen that the mechanical nonlinearity strength \(\alpha\) can be easily probed via measuring the transparency peak position of the output probe field. Moreover, in Figure 2, it is seen that the mechanical nonlinearity strength \(\alpha\) can broaden the full width at half maximum (FWHM) of the transparency peak. In the absence of the mechanical nonlinearity (\(\alpha=0\)), the FWHM of the transparency peak is about 0.1 \(\omega_{m}\). In the presence of the mechanical nonlinearity with \(\alpha=10\times 10^{7}\) N/m\({}^{2}\), the FWHM of the transparency peak is about 0.1761 \(\omega_{m}\), which is less than the shift of about 0.257 \(\omega_{m}\) of the transparency peak position. Therefore, when the linewidth of the transparency peak is considered, it is still possible to estimate the mechanical nonlinearity strength \(\alpha\) by detecting the transparency peak position of the output probe field.