determined from Fano fits (similar to **a**) for different values of $\Delta_\text{p}/2\pi$. Solid lines represent a simultaneous fit of these two data sets to optomechanical theory.[]{data-label="fig2"}](figure2.pdf){width="\textwidth"}

The cooling beam adds a significant optomechanical damping and spring to the membrane, so the linewidth and center frequency of the sidebands in Fig. \[fig1\]d depend on its detuning $\Delta_\text{p}$ and power $P_\text{p}$. Figure \[fig2\]a shows typical heterodyne spectra for the cooling beam red-detuned by $\Delta_\text{p}/2\pi = -250$ kHz. As the cooling power $P_\text{p}$ is increased (from $P_\text{p}=0$ in Fig. \[fig1\]a), the membrane’s vibrations are laser cooled; the linewidth increases and the integrated area under the curve decreases qualitatively as expected. At high $P_\text{p}$ the red and blue sidebands exhibit a large asymmetry. We find the spectra are always well-fit by a Fano lineshape.

Figure \[fig2\]b shows the membrane’s mechanical frequency and damping as a function of $\Delta_\text{p}$. We simultaneously fit the frequency and damping to the theory described in Ref. [@Marquardt2007Quantum] (and outlined in section \[section-model\] below), allowing four parameters to vary: the free spectral range FSR, the ratio between the cavity’s loss through the entrance mirror to the total cavity loss $\kappa_{\text{ext}}/\kappa$, the bare mechanical frequency $\omega_\text{m}$, and and the signal beam 