& -\left(\frac{\kappa_j}{2} + {i}\omega_j \right) {\hat{a}}_j - {i}g_j \hat{x} {\hat{a}}_j + \sqrt{\kappa_{j,{\mathrm{ext}}}} \, {\hat{a}}_{j,{\mathrm{in}}} + \sqrt{\kappa_{j,\mathrm{int}}} \, {\hat{\xi}}_{j} \quad , \quad j = {\mathrm{s}},{\mathrm{p}}\\
  \dot{\hat{c}} & = & - \left(\frac{\gamma}{2} + {i}\omega_{\mathrm{m}}\right) \hat{c} - {i}\sum_j g_j {\hat{a}}^\dagger_j {\hat{a}}_j + \sqrt{\gamma} \, {\hat{\eta}}\ .\end{aligned}$$ Here, $\kappa_{j,\mathrm{ext}}$ is the decay rate of mode $j$ through the mirror which couples the cavity to the external laser drive, whereas $\kappa_{j,\mathrm{int}}$ describes other types of optical decay. The total linewidth of cavity mode $j$ is $\kappa_j = \kappa_{j,\mathrm{ext}} + \kappa_{j,\mathrm{int}}$. The input modes ${\hat{\xi}}_{j}$ describe optical vacuum noise and fulfill $\langle {\hat{\xi}}_{j}(t) {\hat{\xi}}^\dagger_{j'}(t') \rangle = \delta(t-t') \delta_{j,j'}$ and $\langle {\hat{\xi}}^\dagger_{j}(t) {\hat{\xi}}_{j'}(t') \rangle = 0$. The coupling to the laser drive is described by the input mode $$\label{eq:Input}
  {\hat{a}}_{j,{\mathrm{in}}}(t) = {e}^{-{i}\Omega_{j} t} \left[K_j + \frac{1}{2} \left(\delta x_j(t) + {i}\, \delta y_j(t) \right) \right] + {\hat{\xi}}_{j,{\mathrm{in}}}$$ where $K_j = \sqrt{P_j/\hbar \Omega_j}$, with $\Omega_{\mathrm{s}}$ ($\Omega_{\mathrm{p}}$) being the drive frequency and $P_{\mathrm{s}}$ ($P_{\mathrm{p}}$) the power of the lock (cooling) beam. We have introduced the classical variables $\delta x_j$ and $\delta y_j$ which describe technical laser amplitude and phase noise, respectively. Since we will only be concerned with the noise close to the mechanical frequency $\omega_{\mathrm{m}}$, we can assume a 