(a) Recorded optically measured spectra of the two fundamental mechanical resonances with frequencies f1 and f2 (PSD, power spectral density; optical power incident on sample: 11.3 nW). Grey noise spectra were recorded with the signal arm of the interferometer blocked. Black lines show the Lorentzian fit, used to determine the linewidths (full width at half maximum) shown in the figure. The displacement spectral density scale on the right-hand side assumes linear transduction of the known thermal motion of the structure at the cryostat temperature. (b) Detuning dependence of the measured transduced thermal motion, measured as the band power at f1, the lowest mechanical frequency, at room temperature and at 3 K. The black solid and dashed lines show fits with a Voigt lineshape squared (Methods section), with the widths (full width at half maximum) shown. (c) Schematic representation of thermal-motion-induced linewidth broadening: the thin orange lines represent the intrinsic cavity response at a few example detunings, while the thick brown line shows the overall response resulting from averaging over the fluctuating detuning. (d) Optical linewidth versus temperature. The solid line is a fit with a model that assumes a constant Lorentzian intrinsic linewidth κ convolved with a Gaussian with a width LG that depends on . The asymptotes (blue) of the fit function allow us to extract κ and the optomechanical coupling rates g0,i for the two mechanical modes (i=1, 2). Extracted values are shown in the figure.