we calculate the average phonon number $n_{\mathrm{m}}= \langle \hat{c}^\dagger \hat{c} \rangle$. In the weak coupling limit $|\alpha_{\mathrm{s}}|, |\alpha_{\mathrm{p}}| \ll \kappa_{\mathrm{s}}, \kappa_{\mathrm{p}}$, one finds $$\label{eq:19}
  n_{\mathrm{m}}= \frac{\gamma n_{\mathrm{th}}+ \sum_j \gamma_{j} n_{j}}{\tilde{\gamma}} \ .$$ Here, $\tilde{\gamma} = \gamma + \gamma_{\mathrm{s}}+ \gamma_{\mathrm{p}}$ is the effective mechanical linewidth, and the optical contributions to it are given by $$\label{eq:gammaOpt}
  \gamma_{j} =   -4 |\chi_{j,{\mathrm{c}}}[\omega_{\mathrm{m}}]|^2 |\chi_{j,{\mathrm{c}}}[-\omega_{\mathrm{m}}]|^2 \Delta_j |\alpha_j|^2 \kappa_j \, \omega_{\mathrm{m}}\ .$$ Furthermore, we define

$$\begin{aligned}
  \label{eq:23}
  \gamma_{j} n_{j} &=& \frac{|\alpha_j|^2}{4} \Big\{ \kappa_{j,\mathrm{ext}} \Big[|B_{j,+}[\omega_{\mathrm{m}}]|^2 C_{j,{xx}} + |B_{j,-}[\omega_{\mathrm{m}}]|^2 C_{j,{yy}} \notag \\
  & & + 2 \, \mathrm{Im} (B_{j,+}[\omega_{\mathrm{m}}] B^\ast_{j,-}[\omega_{\mathrm{m}}]) C_{j,{xy}} \Big] + \kappa_j |\chi_{j,{\mathrm{c}}}[-\omega_{\mathrm{m}}]|^2 \Big\} \ \end{aligned}$$

with $B_{j,\pm}[\omega] = {e}^{-{i}\phi_j} \chi_{j,{\mathrm{c}}}[\omega] \pm {e}^{{i}\phi_j} \chi^\ast_{j,{\mathrm{c}}}[-\omega]$ and ${e}^{{i}\phi_j} = \alpha_j/|\alpha_j|$. Finally, we also note that the optical spring effect leads to an effective mechanical resonance frequency $\tilde{\omega}_\text{m} = \omega_\text{m} + \delta_{\mathrm{s}}+ \delta_{\mathrm{p}}$, where $$\label{eq:deltaOmegam}
  \delta_j = 2 |\chi_{j,{\mathrm{c}}}[\omega_{\mathrm{m}}]|^2 |\chi_{j,{\mathrm{c}}}[-\omega_{\mathrm{m}}]|^2 \Delta_j |\alpha_j|^2 [(\kappa_j/2)^2 - \omega_{\mathrm{m}}^2 + \Delta_j^2]$$ is the shift due to mode $j$.

Toy example
===========

To illustrate the role of technical noise in the optical sidebands, we consider a simplified example. We treat the optomechanical system classically, and focus on a single optical mode (omitting the index) with amplitude $a(t) = {e}^{-{i}\Omega t} (\bar{a} + d(t))$, where $d(t)$ 