Fig. \[fig1\]a, the two lasers are locked by picking off a small portion of both beams and generating an error signal based on the frequency of their beat note. We have locked the free-running lasers $\sim 9$ GHz apart with an RMS deviation of $\sim 10$ Hz. When the signal laser is simultaneously locked to the membrane cavity, however, this performance degrades to an RMS deviation of $\sim 1$ kHz; this is because the membrane cavity is quite sensitive to environmental noise such as acoustic vibrations in the room, which injects additional noise into the signal laser’s frequency (this first-generation cryogenic apparatus did not include significant vibration isolation). When the two lasers are locked to each other and the signal laser is locked to the membrane cavity, the cooling laser can then be fine-tuned relative to its cavity mode using an additional AOM (not shown).

![Response of membrane to the cooling laser. **a** Typical heterodyne spectra (red and blue sidebands folded on top of each other) for increasing values of cooling laser power $P_\text{p}$ at $\Delta_\text{p}/2\pi = -250$ kHz. Solid lines are fits to Fano lineshapes (simultaneously fitting the width and frequency of both sidebands). **b** Membrane frequency and damping 