Fig. 3: The optomechanical coupling mechanisms.

a With on-resonance mechanical forcing, the fitted time crystal frequency modulation amplitude G (points) is found to depend linearly on \({\theta }_{\max }^{2}\) (fitted lines). b The mean frequency shift \(\Delta {\omega }_{{{\rm{TC}}}}^{\infty }=({\omega }_{{{\rm{TC}}}}^{\infty }({\theta }_{\max })-{\omega }_{{{\rm{TC}}}}^{\infty }({\theta }_{\max }=0))\) is half of the frequency modulation amplitude, in agreement with Eq. (1). c The product of the fitted asymmetry Θ and \({\theta }_{\max }\) (points) is found to be independent of the mechanical motion amplitude, as expected if θ₀ originates from misalignment of the surface normal and container axis. Solid horizontal lines are a guide to the eye. d As a function of the static tilt angle, the surface time crystal frequency shift (points) is consistent with the coupling constant g_surf  between 2.2 and 12.0 Hz deg⁻² determined from dynamic measurements (red and magenta shaded regions). The static shift for the bulk time crystal (points) is smaller than expected from the dynamic measurements (blue and magenta shaded regions) with g_bulk in the range from 9.8 to 54.0 Hz deg⁻², showing that the optomechanical coupling is enhanced significantly in the dynamic case. The red dash line is a fit to points with g_surf = 3.74 Hz deg⁻². The horizontal error bars correspond to the value of the static tilt from c and the vertical error bars correspond to one standard deviation for measurements performed at different minimum coil currents (N_surface = 8, N_bulk = 3). e Bottom: Larger mechanical forcing amplitude results in larger frequency modulation, reflected in the number and amplitude of the side bands seen in the Fourier spectrogram of the time crystal signal. Top: For the largest free surface motion amplitudes the time crystal frequency modulation becomes large enough to completely diminish the central frequency band corresponding to \({\omega }_{{{\rm{TC}}}}^{\infty }\). The black dashed line in the bottom panel shows where this individual frequency spectrum lies in the plot below.