## 3 Feedback Cooling of the Movable Mirror
We use the experimentally accessible parameters given in [34]: the wavelength of the external laser \(\lambda=\frac{2\pi c}{\omega_{l}}=1554.35\) nm, the laser power \(\wp=0.01\) mW, the decay rate of the cavity \(\kappa=2\pi\times 423\) MHz, the single-photon optomechanical coupling strength \(g_{0}=2\pi\times 869\times\sqrt{2}\) kHz, the resonant frequency of the movable mirror \(\omega_{m}=2\pi\times 5.25\) GHz, the quality factor of the movable mirror \(Q_{m}=3.8\times 10^{5}\), the mechanical damping rate \(\gamma_{m}=\omega_{m}/Q_{m}=2\pi\times 13.816\) kHz. Due to \(\kappa/\omega_{m}\simeq 0.08\), the system is deep in the resolved sideband regime. The environment is initially at the temperature \(T=300\) K, thus the corresponding thermal phonon number of the movable mirror is \(n_{m}^{th}=\{\exp{[\hbar\omega_{m}/(K_{B}T)]}-1\}^{-1}=1189.61\). The Figure 2 shows that the effective mean phonon number \(n_{eff}\) of the movable mirror versus the normalized effective cavity detuning \(\Delta/\omega_{m}\) for different reflection coefficients \(r=0\), \(0.2\), \(0.3\), \(0.46\). In the absence of the feedback \((r=0)\), the effective mean phonon number \(n_{eff}\) has the minimum value \(11.97\) at \(\Delta/\omega_{m}=0.994\). In the presence of the feedback \((r\
eq 0)\), with increasing the reflection coefficient \(r\) of the beam splitter, the minimum value of \(n_{eff}\) is decreased. For \(r=0.46\), the minimum value of \(n_{eff}\) is \(2.40\) at \(\Delta/\omega_{m}=1\), thus the cooling of the movable mirror can be improved by a factor of about \(5.0\) with respect to the case without the feedback. Therefore, the addition of the feedback to the optomechanical system is able to improve the cooling of the movable mirror. Normal Mode Splitting in the Spectra of the Movable Mirror and the Output Field in the Presence of the Feedback  
In this section, we show the influence of the feedback on the position spectrum of the movable mirror and the spectral density of the output field. From Equation (8), the fluctuation \(\delta c(\omega)\) of the cavity field can be obtained. Then the fluctuation \(\delta c_{out}(\omega)\) of the output field can be found according to the input-output relation \(\delta c_{out}(\omega)=\sqrt{2\kappa}\delta c(\omega)-tc_{in}(\omega)\), which is given by  
\[\delta c_{out}(\omega) = B_{1}(\omega)c_{in}(\omega)+B_{2}(\omega)c_{in}^{\dagger}(- \omega)+B_{3}(\omega)\xi(\omega), \tag{24}\]  
where  
\[ \begin{array}{ll}B_{1}(\omega)=&\frac{2\kappa(1-r)t}{d(\omega)}\big{\{}( \omega_{m}^{2}-\omega^{2}-i\gamma_{m}\omega)[\kappa_{eff}-i(\Delta+\omega)]+ \frac{i}{2}|G|^{2}\omega_{m}\big{\}}-t,\\ B_{2}(\omega)=&\frac{\kappa(1-r)t}{d(\omega)}iG^{2}\omega_{m},\\ B_{3}(\omega)=&\frac{\sqrt{\kappa}iG\omega_{m}}{d(\omega)}[\kappa_{eff}-i( \Delta+\omega)].\end{array}  \tag{25}\]  
In Equation (24), the first two terms arise from the input vacuum noise, the last term originates from the thermal noise of the movable mirror. The spectral density of the output field is defined as  
\[\langle\delta c_{out}^{\dagger}(-\Omega)\delta c_{out}(\omega)\rangle = 2\pi S_{cout}(\omega)\delta(\omega+\Omega). \tag{26}\]  
Figure 2: The effective mean phonon number \(n_{eff}\) of the movable mirror versus the normalized effective cavity detuning \(\Delta/\omega_{m}\) for different reflection coefficients \(r=0\) (black solid), \(0.2\) (blue dashed), \(0.3\) (green dot-dashed), \(0.46\) (red dotted). Using Equations (18) and (26), we calculate the output spectrum as  
\[S_{cout}(\omega) = B_{2}(\omega)B_{2}^{*}(\omega)+B_{3}(\omega)B_{3}^{*}(\omega)\frac {\gamma_{m}}{\omega_{m}}\omega[-1+\coth(\frac{\hbar\omega}{2k_{B}T})]. \tag{27}\]  
It is seen that the output spectrum \(S_{cout}(\omega)\) is the sum of two terms associated respectively with the input vacuum noise and the thermal noise of a movable mirror. It is known that the normal mode splitting is observable when the optomechanical system enters the strong coupling regime \(|G|>\kappa_{eff}\)[25; 27].