Extended Data Fig. 6: Optical spring shift and opto-thermal backaction.

a, Thermomechanical noise spectra of the first few mechanical modes imprinted on an unmodulated single drive/detection laser, as the laser’s frequency (ω_L) is swept across the cavity resonance. The four most intense peaks around frequencies ω_i/(2π) ≈ {3.7, 5.3, 12.8, 17.6} MHz correspond to flexural modes (labelled i) of the individual beam halves and show frequency-tuning characteristic to the optical spring effect, and the other peaks represent nonlinearly transduced harmonics of those modes.⁶⁵. PSD, power spectral density.b, Magnification of the PSD of the first four resonators. c, From the spectra in b, resonance frequencies ω_i (blue circles) and linewidths γ_i (red circles) are extracted. The resonance frequencies are fitted using the standard optical spring model (solid blue). Across all resonators, we find agreement in the fitted cavity resonance ω_c/(2π) = 195.62 THz and linewidth κ/(2π) = 320 GHz (Q factor, Q ≈ 600). The small sideband resolution ω_i/κ ≈ 10⁻⁵ suggests very little change in linewidth due to dynamical cavity backaction (dashed red). The linewidth modulations we observe suggest the presence of an opto-thermal retardation effect⁶⁶. Displayed errors correspond to fit uncertainty, smaller than plot markers on the fitted frequencies (Methods section ‘Error estimation’). On each panel, blue/red colour-coded arrows indicate the y scale for each plotted quantity. d, Drive laser frequency sweep, now using a separate, fixed-frequency, far-detuned detection laser. The fixed transduction of mechanical motion onto this detection laser allows a comparison of resonance peak area A_i(ω_L) versus linewidth γ_i(ω_L) as the drive laser frequency ω_L is varied (a.u., arbitrary units). The resonance peak area of mode i is proportional to the variance \(\langle {X}_{i}^{2}\rangle \) of its displacement x_i, which is proportional to its temperature T_i. Dynamical backaction modifies the effective mode temperature through \({T}_{i}={T}_{0}({\tilde{\gamma }}_{i}/{\gamma }_{i})\) (ref. ²⁹), where T₀ is the initial temperature and \({\tilde{\gamma }}_{i}\) is the mode’s intrinsic linewidth, determined by switching off the drive laser. Our data are well explained by linear fits of A_i(ω_L) versus \({\tilde{\gamma }}_{i}/{\gamma }_{i}({\omega }_{{\rm{L}}})\) (dashed), confirming the effective temperature model.