velocity for a room temperature thermal distribution.

It can be shown that the perturbation theory argument above can be adapted to obtain equilibrium phonon numbers for a given cooling rate $\Gamma_{opt}$:

$$\begin{aligned}
\langle{n}\rangle_{PT}=\frac{\frac{k_BT_B}{\hbar\omega_M} {\Gamma_M} + 
\left[S_1(\omega_M )+
S_2(\omega_M )\right]}
{\Gamma_M + |\Gamma_{opt}|},
\label{PT}\end{aligned}$$

where $T_B\simeq 300$K. Alternatively, the final equilibrium temperatures $T_{eq} = \frac{\Gamma_M T_{B}
+ |\Gamma| T_{vac}}{\Gamma_M+|\Gamma|}$, where $T_{vac}$ is the equilibrium oscillator temperature which would have been obtained in a perfect vacuum.

Quantum and semiclassical noise spectra
---------------------------------------

Although we investigate only a two optical mode system, generalization to more optical modes is straightforward. We consider a set of equations of motion, for $j=1,...N$: $$\begin{aligned}
 { \dot { \hat b} } & = &  -( i\omega_M (\Delta_1^x,...\Delta_j^x)+
\frac{\Gamma_M}{2} ) {\hat b} +
i\sum_j g_j(\Delta_1^x,...\Delta_j^x) ({\hat a}_j+{\hat a}_j^\dagger)   +
\sqrt{\Gamma_M} {\hat b}_{in}    \nonumber \\
{\dot {\hat a}_j} & = & (i \Delta_j^x-\frac{\kappa}{2}) {\hat a}_j + i 
g_j(\Delta_1^x,...\Delta_j^x)\alpha_j({\hat b}+{\hat b}^\dagger)
   + \sqrt{\kappa}{\hat a}_{in}^{(j)}, \nonumber\\
\label{QM}\end{aligned}$$

where the optomechanical strengths $g_j(\Delta_1^x,...\Delta_j^x)= -kA X_{ZPF}
\sin 2(kx_0-\phi_j)$ depend on the detunings (as does the mechanical frequency $\omega_M$). In the two mode case we consider, we take $\phi_1=0$ and $\phi_2=\pi/4$.

The optical modes are subject to photon shot noise, while the mechanical modes are 