- {i}\Delta_j}$$ and $\Delta_j = \Omega_j - \omega_j - g_j x_0$ is the laser detuning from the cavity resonance in the presence of a static membrane. Defining the dimensionless position operator ${\hat{z}}= \hat{c} + \hat{c}^\dagger$, the Fourier transform as $f^{(\dagger)}[\omega] = \int_{-\infty}^{\infty} {d}t \, {e}^{{i}\omega t} f^{(\dagger)}(t)$, and the susceptibilities $$\label{eq:Susceptibilities}
  \chi_{j,{\mathrm{c}}}[\omega] = \frac{1}{\kappa_j/2 - {i}(\omega + \Delta_j)} \quad , \quad \chi_{\mathrm{m}}[\omega] = \frac{1}{\gamma/2 - {i}(\omega - \omega_{\mathrm{m}})} \ ,$$ the solution to the linearized equations can be expressed as

$$\begin{aligned}
  \label{eq:SolutionEOM}
  {\hat{d}}_j[\omega] & = & \chi_{j,{\mathrm{c}}}[\omega] \Big(\zeta_j[\omega] - {i}\alpha_j \hat{z}[\omega] \Big) \\ 
  \hat{z}[\omega] & = & \frac{1}{N[\omega]} \Bigg[\sqrt{\gamma} \left( \chi_{\mathrm{m}}^{-1 \, \ast}[-\omega] \eta[\omega] + \chi_{\mathrm{m}}^{-1}[\omega] \eta^\dagger[\omega] \right) \notag\\
  &&- 2 \omega_{\mathrm{m}}\sum_j \left( \alpha_j^\ast \chi_{j,{\mathrm{c}}}[\omega] \zeta_j[\omega] + \alpha \chi_{j,{\mathrm{c}}}^\ast[-\omega] \zeta^\dagger[\omega] \right) \Bigg] \ . 
$$

We have introduced the effective coupling rates $\alpha_j = g_j x_\mathrm{zpf} \bar{a}_j$, the operators $$\label{eq:zetadef}
  \zeta_j[\omega] = \sqrt{\kappa_{j,{\mathrm{ext}}}} \left[ \frac{1}{2} \left(\delta x_j[\omega] + {i}\delta y_j[\omega] \right) + {\hat{\xi}}_{j,{\mathrm{in}}}[\omega] \right] + \sqrt{\kappa_{j,\mathrm{int}}} \, {\hat{\xi}}_{j}[\omega] \ ,$$ and the function $$\label{eq:Ndef}
  N[\omega] = \chi^{-1}_{\mathrm{m}}[\omega] \chi^{-1 \, \ast}_{\mathrm{m}}[-\omega] - 2 {i}\omega_{\mathrm{m}}\sum_j |\alpha_j|^2 \left( \chi_{j,{\mathrm{c}}}[\omega] - \chi^\ast_{j,{\mathrm{c}}}[-\omega] \right) \ .$$ Eqs.  gives the optical output field $\hat{a}_{j,\mathrm{out}}(t) = \sqrt{\kappa_{j,\mathrm{ext}}} \, {\hat{a}}_j(t) - {\hat{a}}_{j,{\mathrm{in}}}(t)$ from mode $j$.

For later use, 