land on a sampler, and a small amount is sent to the cold cavity. The remainder lands on a “reference” photo diode (RPD) to monitor the heterodyne phase. Light leaving the cavity is collected by another “signal” photodiode (SPD) to monitor the membrane’s motion. The signal laser is locked to the cavity with the Pound-Drever-Hall (PDH) method using the 15 MHz sidebands. The frequency and amplitude of the cooling laser are fine-tuned with an additional AOM (not shown). **b** Mechanical ringdown measurement, showing the membrane’s amplitude after a drive piezo is turned off. **c** Cavity ringdown measurement, showing power leaving the cavity after the drive laser is turned off. The solid lines in **b** and **c** show exponential fits to the data. **d** Power spectral density of the heterodyne sidebands from the membrane’s Brownian motion at 400 mK. The frequency is plotted relative to [$\omega_{\text{if}}/2\pi$]{}= 80 MHz, and the lower sideband (red) has been folded on top of the upper sideband (blue) for comparison. []{data-label="fig1"}](figure1.pdf){width="\textwidth"}

Figure \[fig1\]a shows a schematic of our cryogenic optomechanical system. A 1.5 mm $\times$ 1.5 mm $\times$ 50 nm stoichiometric [Si$_3$N$_4$]{} membrane resides at the center of a (nominally) $3.39$ cm long optical cavity. The 