drift by a few Hz on the hour time scale. This adds a constant offset to the frequency plot in Fig. \[fig2\]b. Finally, for this particular experiment we locked the signal beam as close to resonance as possible, but as this tends to drift on the scale of hours, we left $\Delta_\text{s}$ as a fitting parameter. $\Delta_\text{s}$ is responsible for adding a very small constant offset to the damping and spring. All other parameters such as the cavity finesse and input power were measured independently. The simultaneous fit is thus heavily constrained and agrees with the data very well. We also find that the fit is similarly convincing if we simply fix $\Delta_\text{s} = 0$ and $\omega_\text{m}/2\pi = 261.15$ kHz (a typical value of $\omega_\text{m}$).

While the optical spring and damping in Fig. \[fig2\]b are well-modeled by standard theory, the interpretation of the sideband amplitudes and lineshapes in Fig. \[fig2\]a is not obvious. As we now discuss, the Fano lineshape arises from interference between the membrane’s response to classical laser noise and the classical laser noise itself, an effect similar to what is seen in single-sideband measurements in other optomechanical systems [@Rocheleau2010Preparation; @Teufel2011Sideband; @Gavartin2012Hybrid].

General Model of Optomechanics with Classical 