mean phonon number is $\gg1$.

The second purpose of this apparatus is to manipulate the membrane with optical forces, and so we include a second (cooling / pump) laser that addresses a different longitudinal mode of the cavity. If the cooling and signal beams address the same cavity mode, the beating between the two beams leads to a large heterodyne signal that clouds our measurement and a strong mechanical drive at the beat frequency (which usually is close to the mechanical frequency). This can cause the system to be unstable and makes the data difficult to interpret. To overcome this challenge, we lock the cooling and signal lasers such that they address different longitudinal cavity modes roughly 9 GHz apart. The longitudinal modes are chosen to be two free spectral ranges apart so that the dependence of cavity resonance frequency on membrane displacement is approximately the same for the two modes. This way, drift or vibrations in the membrane mount will (to lowest order) not change the relative frequencies of the modes. With the lasers locked in this way, any beating between the cooling and signal lasers occurs at frequencies that are irrelevant to the membrane’s mechanics.

As shown in 