susceptibility is a Lorentzian as a function of $\omega_{\mathrm{n}}$, whereas the imaginary part is antisymmetric around the mechanical frequency: $$\label{eq:ToyRealImag}
  \mathrm{Re} \, \chi_{\mathrm{m}}[\omega_{\mathrm{n}}]  = \frac{\gamma/2}{(\gamma/2)^2 + (\omega_{\mathrm{n}}- \omega_{\mathrm{m}})^2} \quad , \quad \mathrm{Im} \, \chi_{\mathrm{m}}[\omega_{\mathrm{n}}]  = \frac{\omega_{\mathrm{n}}- \omega_{\mathrm{m}}}{(\gamma/2)^2 + (\omega_{\mathrm{n}}- \omega_{\mathrm{m}})^2} \ .$$ As one would expect, the mechanical oscillation goes through a phase shift of $\pi$ as the modulation frequency $\omega_{\mathrm{n}}$ is swept through the mechanical resonance, and the oscillation is out of phase with the force at resonance $\omega_{\mathrm{n}}= \omega_{\mathrm{m}}$.

We write the optical output amplitude $d_\mathrm{out}(t) = \sqrt{\kappa_{\mathrm{ext}}} d(t) - \delta x(t)/2$ as a sum of two terms, $$\label{eq:ToydOut}
  d_\mathrm{out}(t) = d_{\mathrm{out},\delta x}(t) + d_{\mathrm{out},z}(t) \ ,$$ where $d_{\mathrm{out},\delta x}(t)$ is the amplitude for the reflected and cavity filtered signal $\delta x(t)$, whereas $d_{\mathrm{out},z}(t)$ is the part that comes from the motion of the mechanical oscillator. We define the output spectrum as $$\label{eq:ToySpectrum}
  S[\omega] = \int_{-\infty}^\infty {d}\tau \, {e}^{{i}\omega \tau}  \langle d_\mathrm{out}^\ast(t + \tau) d_\mathrm{out}(t) \rangle_\mathrm{time} \ ,$$ where $\langle \ \rangle_\mathrm{time}$ denotes averaging over the time $t$.

The spectrum consists of three terms, $S[\omega] = S_{\delta x,\delta x}[\omega] + S_{z,z}[\omega] + S_{\delta x,z}[\omega]$. The first term is the spectrum of $d_{\mathrm{out},\delta x}(t)$, which becomes $$\label{eq:ToySxx}
  S_{\delta x,\delta 