Laser Noise {#section-model}
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In this section, we present and solve the equations of motion for the optical cavity and the mechanical oscillator. Since the local oscillator beam is far off any cavity resonance frequency, we can neglect it here. We will let ${\hat{a}}_{\mathrm{s}}$ be the bosonic annihilation operator of the cavity mode addressed by the lock/signal beam, whereas ${\hat{a}}_{\mathrm{p}}$ is the annihilation operator for the cavity mode addressed by the cooling (pump) beam. The position operator of the mechanical oscillator is $\hat{x} = x_0 + x_\mathrm{zpf} \left(\hat{c} + \hat{c}^\dagger\right)$, where $\hat{c}$ is the phonon annihilation operator, $x_0 = \langle \hat{x} \rangle$ and $x_\mathrm{zpf}$ is the size of the zero point fluctuations. The Hamiltonian is $$\label{eq:Hamiltonian}
  H = \sum_{j = {\mathrm{s}},{\mathrm{p}}} \hbar \left(\omega_j + g_j \hat{x} \right) {\hat{a}}^\dagger_j {\hat{a}}_j + \hbar \omega_{\mathrm{m}}\hat{c}^\dagger \hat{c} + H_\mathrm{drive} + H_\mathrm{diss}$$ The interaction term describes the modulation of the cavity resonance frequencies by the motion of the mechanical oscillator, $H_\mathrm{drive}$ describes the laser drive and $H_\mathrm{diss}$ describes the coupling to both the electromagnetic and mechanical environment. This coupling to external degrees of freedom is conveniently described by input-output theory [@Collett1984Squeezing; @Clerk2010Introduction], which gives rise to the equations of motion $$\begin{aligned}
  \label{eq:EOM}
  \dot{{\hat{a}}}_j  & = 