subject to Brownian noise from collisions with gas molecules in the cavity. For the photon shot noise, we assume independent lasers and uncorrelated zero temperature noise for which $ \langle {\hat a}_{in}^\dagger(t') {\hat a}_{in}(t)\rangle =0$, while $ \langle {\hat a}^{(i)}_{in}(t') {\hat a}^{(j)\dagger }_{in}(t)\rangle
=\delta(t-t')\delta_{ij}$. For the gas collisions, we take $ \langle {\hat b}_{in}(t') {\hat
b}_{in}^\dagger(t)\rangle =(n_B+1)\delta(t-t')$ and $ \langle {\hat b}_{in}^\dagger(t') {\hat b}_{in}(t)\rangle=
n_B\delta(t-t')$ where the number of surrounding bath phonons $n_B \approx \frac{k_BT}{\hbar \omega_M}$.

The above equations can be integrated in frequency space to obtain analytical expressions for the displacement noise spectra for the arbitrary mode case. We can evaluate the displacement spectrum $S_{xx}(\omega) \equiv \langle |x(\omega)|^2\rangle_{QM}= \frac{1}{2\pi} \int e^{-i\omega \tau} \langle
x(t+\tau) x(t)\rangle d\tau$. We obtain:

$$\begin{aligned}
\langle |x(\omega)|^2\rangle_{QM} |M(\omega)|^2 & = & {\Gamma_M} \left[
|\chi_M(\omega)|^2 n_B + |\chi_M(-\omega)|^2 (n_B+1)\right]\nonumber \\
                                               & + & \frac{\kappa}{2}
|\mu(\omega)|^2 \sum_{j=1,2} g_j^2 |\chi_{jo}(-\omega)|^2,
\label{qnoise}\end{aligned}$$

where the $\chi(\omega)$ represent optical and mechanical susceptibilities: $$\begin{aligned}
\chi_{jo}(\omega)=\left[-i(\omega+\Delta_j^x)+\frac{\kappa}{2}\right]^{-1};  \
 \chi_M(\omega) =\left[-i(\omega-\omega_M)+\frac{\Gamma_M}{2}\right]^{-1}.\end{aligned}$$ with $ \mu(\omega)=\chi_M(\omega)-\chi_M^*(-\omega)$ and $ \eta_j(\omega)=
\chi_{jo}(\omega)-\chi_{jo}^*(-\omega)$; then also $M(\omega)= 1+ \mu(\omega)\sum_j g_j^2 |\eta_{j}(\omega)|^2 $.

![(Colour online) Comparison between quantum and semiclassical displacement spectra for gas pressures of 1mBar and at near vacuum pressure in a strong cooling region. At high vacuum, the ground 