x}[\omega] = \frac{C_{xx}}{4} \big|\kappa_{\mathrm{ext}}\chi_{\mathrm{c}}[\omega_{\mathrm{n}}]  - 1 \big|^2 \times 2 \pi \Big[\delta(\omega - \omega_{\mathrm{n}})  + \delta(\omega + \omega_{\mathrm{n}}) \Big] \ .$$ The absolute value describes the promptly reflected signal, the cavity filtered signal, and their interference. The second term in $S[\omega]$ is the spectrum of $d_{\mathrm{out},z}(t)$, which is proportional to the position spectrum of the mechanical oscillator. We find $$\label{eq:ToySzz}
  S_{z,z}[\omega] = \kappa_{\mathrm{ext}}^2 \alpha^4 C_{xx} |\chi_{\mathrm{c}}[\omega_{\mathrm{n}}]|^4 |\chi_{\mathrm{m}}[\omega_{\mathrm{n}}]|^2 \times  2 \pi \Big[\delta(\omega - \omega_{\mathrm{n}})  + \delta(\omega + \omega_{\mathrm{n}}) \Big] \ .$$ This is proportional to the absolute square of the mechanical susceptibility, which has a Lorentzian dependence on $\omega_{\mathrm{n}}$, as one would expect from a damped and driven harmonic oscillator. Note also that $S_{z,z}[\omega]$ is symmetric in $\omega$ as is required of a spectrum of a real, classical variable [@Clerk2010Introduction].

The last term in $S[\omega]$ results from optomechanical correlations between the modulation $\delta x$ and the oscillator position $z$:

$$\begin{aligned}
  \label{eq:ToySxz}
  S_{\delta x,z}[\omega] & \equiv & \int_{-\infty}^\infty {d}\tau \, {e}^{{i}\omega \tau} \langle d^\ast_{\mathrm{out},z}(t + \tau) d_{\mathrm{out},\delta x}(t)  + d^\ast_{\mathrm{out},\delta x}(t + \tau) d_{\mathrm{out},z}(t) \rangle_\mathrm{time}  \notag\\
  & = & \kappa_{\mathrm{ext}}\alpha^2 C_{xx} |\chi_{\mathrm{c}}[\omega_{\mathrm{n}}]|^2 \Big[ \left( \kappa_{\mathrm{ext}}|\chi_{\mathrm{c}}[\omega_{\mathrm{n}}]| \cos \vartheta_{\mathrm{n}}- \cos 2 \vartheta_{\mathrm{n}}\right) \mathrm{Re} \, \chi_{\mathrm{m}}[\omega_{\mathrm{n}}]  \notag \\
  & &- \left( \kappa_{\mathrm{ext}}|\chi_{\mathrm{c}}[\omega_{\mathrm{n}}]| \sin \vartheta_{\mathrm{n}}- \sin 2 