membrane is mounted on a three-axis cryogenic actuator allowing us to tilt the membrane about two axes and displace it along the cavity axis. The cavity, membrane, and a small set of guiding optics are cooled to approximately 400 mK in a $^3$He cryostat. Free-space laser light is coupled to the cavity via one of the cryostat’s clear-shot tubes.

The most reliable way to measure the membrane’s mechanical quality at 400 mK is to perform a mechanical ringdown by driving the membrane at its resonant frequency ($\omega_\text{m}= 2\pi \times 261.15$ kHz) to large amplitude with a nearby piezo, shutting off the drive, and monitoring the decay of the membrane’s vibrations. We monitor the membrane’s motion interferometrically using a laser of wavelength 935 nm, which is far enough from the design wavelength of our cavity mirror coatings (1064 nm) that the cavity finesse is $\sim 1$; this ensures the measurement exerts no significant back action upon the membrane. Figure \[fig1\]b shows a typical mechanical ringdown measurement. To ensure the membrane motion is in the linear regime, we let it ring down until its frequency stabilizes before fitting the data to an exponential curve (the inferred time constant is then insensitive to 