are the classical fluctuations around a mean amplitude $\bar{a}$. In addition to neglecting vacuum noise, we also neglect laser phase noise and thermal noise of the mechanical bath. Finally, we consider the case where the cavity is driven on resonance, i.e. $\Delta = 0$. The equations of motion are then $$\begin{aligned}
  \label{eq:ToyEOM}
  \dot{d} & = & - \frac{\kappa}{2} d - {i}\alpha z + \frac{\sqrt{\kappa_{\mathrm{ext}}}}{2} \delta x(t) \\
  \dot{c} & = & - \left(\frac{\gamma}{2} + {i}\omega_{\mathrm{m}}\right) c - {i}\alpha \left(d + d^\ast \right)\end{aligned}$$ with $\alpha$ real. Instead of considering white amplitude noise, we imagine that the amplitude of the drive is modulated at a frequency $\omega_{\mathrm{n}}$, such that $\delta x(t) = 2 \sqrt{C_{xx}} \cos \omega_{\mathrm{n}}t$. The optical force on the oscillator is then proportional to $$\label{eq:ToyForce}
   d(t) + d^\ast(t) = 2 \sqrt{\kappa_{\mathrm{ext}}C_{xx} }   \, |\chi[\omega_{\mathrm{n}}]| \cos (\omega_{\mathrm{n}}t - \vartheta_{\mathrm{n}})$$ where the phase $\vartheta_{\mathrm{n}}$ is defined by $\chi_{\mathrm{c}}[\omega_{\mathrm{n}}] = |\chi_{\mathrm{c}}[\omega_{\mathrm{n}}]| {e}^{{i}\vartheta_{\mathrm{n}}}$. The dimensionless oscillator position becomes $$\label{eq:ToyzOpt}
  z(t) = 2 \sqrt{\kappa_{\mathrm{ext}}C_{xx} } \, \alpha \, |\chi_{\mathrm{c}}[\omega_{\mathrm{n}}]| \Big[ \cos (\omega_{\mathrm{n}}t - \vartheta_{\mathrm{n}}) \mathrm{Im} \, \chi_{\mathrm{m}}[\omega_{\mathrm{n}}]  - \sin (\omega_{\mathrm{n}}t - \vartheta_n) \mathrm{Re} \, \chi_{\mathrm{m}}[\omega_{\mathrm{n}}] \Big]$$ when assuming $\omega_{\mathrm{n}}$ is positive and close to $\omega_{\mathrm{m}}$, and $\omega_{\mathrm{m}}/\gamma \gg 1$. The real part of the mechanical 