of the main text. Raw spectra are normalized to the noise level with the pump off, to which the only contributions are vacuum noise and measurement chain added noise; the peak height is $\sigma$ in these units. The peak exhibits a linear temperature dependence due to the thermally driven motion of the mechanical oscillator (Fig. \[fig:noise\]), except at the lowest temperatures where the membrane evidently falls out of equilibrium with the dilution refrigerator (open circles, Fig. \[fig:noise\]c and d). The temperature below which the membrane falls out of equilibrium is higher in the optical measurement, indicating that optical heating is a significant factor below $T=100~\mathrm{mK}$. We find that this membrane heating occurs whenever the locking beam is introduced, but is independent of damping power, consistent with the observations in Ref. [@peterson_laser_2016]. The spectra are calibrated by fitting the observed peak heights $\sigma(T)$ to an expected linear dependence obtained from combining the full optomechanical equations of motion [@andrews_bidirectional_2014] with independently measured parameters in Table \[tab:params\] [@purdy_cavity_2012]. The only fit parameter is a proportionality factor that converts uncalibrated thermal peaks to units of photons at the converter output; the normalization of the uncalibrated spectra implies that the single-quadrature background noise (vacuum noise 