## 3 Feedback Cooling of the Movable Mirror  
In this section, we show how the feedback scheme affects the cooling of the movable mirror. Fourier transforming of all the fluctuation and noise operators in Equation (8) by \(f(\tau)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}f(\omega)e^{-i\omega\tau}d\omega\) and solving it, we find the position fluctuation of the movable mirror  
\[\delta Q(\omega) = A_{1}(\omega)c_{in}(\omega)+A_{2}(\omega)c_{in}^{\dagger}(- \omega)+A_{3}(\omega)\xi(\omega), \tag{15}\]where  
\[ \begin{array}{ll}A_{1}(\omega)=&\frac{\omega_{m}}{d(\omega)}\sqrt{\kappa}(1-r)tG ^{*}[\kappa_{eff}-i(\Delta+\omega)],\\ A_{2}(\omega)=&\frac{\omega_{m}}{d(\omega)}\sqrt{\kappa}(1-r)tG[\kappa_{eff}+i( \Delta-\omega)],\\ A_{3}(\omega)=&\frac{\omega_{m}}{d(\omega)}[(\kappa_{eff}-i\omega)^{2}+\Delta ^{2}],\\ d(\omega)=&(\omega_{m}^{2}-\omega^{2}-i\gamma_{m}\omega)[(\kappa_{eff}-i \omega)^{2}+\Delta^{2}]-\omega_{m}\Delta|G|^{2}.\end{array}  \tag{16}\]  
In Equation (15), the term dependent on \(\xi(\omega)\) is the contribution of the thermal noise of the movable mirror, the other two terms are the contribution of radiation pressure. In the absence of the optomechanical coupling (\(G=0\)), Equation (15) is reduced to \(\delta Q(\omega)=\frac{\omega_{m}}{\omega_{m}^{2}-\omega^{2}-i\gamma_{m}\omega }\xi(\omega)\), the position fluctuation of the movable mirror is only determined by the thermal noise, hence the movable mirror is undergoing Brownian motion due to its contact with the thermal environment. Furthermore, the position spectrum of the movable mirror is defined by  
\[2\pi S_{Q}(\omega)\delta(\omega+\Omega)=\frac{1}{2}[\langle\delta Q(\omega) \delta Q(\Omega)\rangle+\langle\delta Q(\Omega)\delta Q(\omega)\rangle]. \tag{17}\]  
Applying the Fourier transform to Equations (4) and (5) yields  
\[ \begin{array}{ll}\langle c_{in}(\omega)c_{in}^{\dagger}(-\Omega)\rangle=&2 \pi\delta(\omega+\Omega),\\ \langle\xi(\omega)\xi(\Omega)\rangle=&2\pi\frac{\gamma_{m}}{\omega_{m}}\omega [1+\coth(\frac{\hbar\omega}{2k_{B}T})]\delta(\omega+\Omega).\end{array}  \tag{18}\]  
Combining Equation (17) with Equation (18), we find the position spectrum of the movable mirror  
\[S_{Q}(\omega) = \frac{1}{2}A_{1}(\omega)A_{2}(-\omega)+\frac{1}{2}A_{2}(\omega)A_ {1}(-\omega)+\frac{\gamma_{m}}{\omega_{m}}\omega\coth(\frac{\hbar\omega}{2k_{B }T})A_{3}(\omega)A_{3}(-\omega). \tag{19}\]  
It is noted that the position spectrum \(S_{Q}(\omega)\) of the movable mirror depends on the radiation pressure (the first two terms) and the thermal noise of the movable mirror (the last term). Next we get the momentum fluctuation of the movable mirror \(\delta P(\omega)=-\frac{i\omega}{\omega_{m}}\delta Q(\omega)\) by taking the Fourier transform of the fluctuation operators in Equation (8). Thus the momentum spectrum of the movable mirror is found to be  
\[S_{P}(\omega)=\frac{\omega^{2}}{\omega_{m}^{2}}S_{Q}(\omega). \tag{20}\]  
The position and momentum variances \(\langle\delta Q^{2}\rangle\) and \(\langle\delta P^{2}\rangle\) of the movable mirror are determined by  
\[ \begin{array}{ll}\langle\delta Q^{2}\rangle=&\frac{1}{2\pi}\int_{-\infty}^{+ \infty}d\omega S_{Q}(\omega),\\ \langle\delta P^{2}\rangle=&\frac{1}{2\pi}\int_{-\infty}^{+\infty}d\omega S_{P }(\omega).\end{array}  \tag{21}\]  
The interesting quantity is the effective mean phonon number \(n_{eff}\) of the movable mirror, which can be obtained from the stationary mean energy of the mechanical oscillator given by  
\[\frac{\hbar\omega_{m}}{2}[\langle\delta Q^{2}\rangle+\langle\delta P^{2} \rangle]=\hbar\omega_{m}(n_{eff}+\frac{1}{2}). \tag{22}\]  
We find  
\[n_{eff}=\frac{1}{2}[\langle\delta Q^{2}\rangle+\langle\delta P^{2}\rangle-1]. \tag{23}\]  
When \(n_{eff}=0\), the movable mirror is in its quantum ground state.