white noise model where $$\begin{aligned}
  \label{eq:ClassNoise}
  \langle \delta x_j(t) \delta x_{j'}(t') \rangle & = & C_{j,{xx}} \delta(t-t') \delta_{j,j'} \\
  \langle \delta y_j(t) \delta y_{j'}(t') \rangle & = & C_{j,{yy}} \delta(t-t')  \delta_{j,j'}  \nonumber \\
  \langle \delta x_j(t) \delta y_{j'}(t') \rangle & = & C_{j,{xy}} \delta(t-t') \delta_{j,j'} \nonumber  \end{aligned}$$ The amplitude and phase noise is characterized by the real numbers $C_{j,{xx}}, C_{j,{yy}} \geq 0$ and $C_{j,{xy}}$ that are proportional to laser power. The Cauchy-Bunyakovsky-Schwarz inequality dictates that $C_{j,{xy}}^2 \leq C_{j,{xx}} C_{j,{yy}}$. Note that $C_{j,{xx}} = 1$ or $C_{j,{yy}} = 1$ corresponds to the condition in which the laser’s classical noise is equal to its quantum noise. The operator ${\hat{\xi}}_{j,{\mathrm{in}}}$ describes vacuum noise and obeys the same relations as ${\hat{\xi}}_{j}$. The intrinsic linewidth of the mechanical oscillator is $\gamma$, and ${\hat{\eta}}$ describes thermal noise obeying $\langle {\hat{\eta}}(t) {\hat{\eta}}^\dagger(t') \rangle \approx \langle {\hat{\eta}}^\dagger(t) {\hat{\eta}}(t') \rangle = n_{\mathrm{th}}\delta(t-t')$, where $n_{\mathrm{th}}\approx k_\mathrm{B}T/\hbar \omega_{\mathrm{m}}$ is the phonon number in the absence of laser driving.

For sufficiently strong driving and weak optomechanical coupling, we can linearize the equations of motion by considering small fluctuations around an average cavity amplitude. We write $$\label{eq:MeanFluct}
  {\hat{a}}_j(t) = {e}^{-{i}\Omega_{j} t} \left( \bar{a}_j + {\hat{d}}_j(t) \right)$$ where $$\label{eq:Mean}
  \bar{a}_j = \frac{\sqrt{\kappa_{j,{\mathrm{ext}}}} \, K_j}{\kappa_j/2 