Time crystal optomechanics

Abstract

Time crystals are an enigmatic phase of matter in which a quantum mechanical system displays repetitive, observable motion - they spontaneously break the time translation symmetry. On the other hand optomechanical systems, where mechanical and optical degrees of freedom are coupled, are well established and enable a range of applications and measurements with unparalleled precision. Here, we connect a time crystal formed of magnetic quasiparticles, magnons, to a mechanical resonator, a gravity wave mode on a nearby liquid surface, and show that their joint dynamics evolves as a cavity optomechanical system. Our results pave way for exploiting the spontaneous coherence of time crystals in an optomechanical setting and remove the experimental barrier between time crystals and other phases of condensed matter.

Figures (5)

• Figure 1: Time crystal optomechanics.a The time crystals are formed of magnons (represented by operator $\hat{a}$) that are spatially trapped by the combined effect of the spin-orbit energy related to order parameter distribution (radial direction) and Zeeman energy controlled by the magnetic field profile (axial direction). The magnetic field profile ${\bf H}$ is used to move the time crystal against the free surface (red blob inside the container) or within the bulk liquid (blue blob inside the container). The time crystals couple to externally driven surface wave mode (represented by the position operator $\hat{x}$). b The precession of magnetisation within the time crystal is observed as an induced, decaying sinusoidal voltage in the pick-up coils. c A sliding windowed fast Fourier transform (FFT) of the signal with on-resonance mechanical forcing reveals that the time crystal signal is accompanied by sidebands. The sidebands result from the frequency modulation of the time crystal signal, caused by the motion of the free surface. The red dash line indicates the mean frequency of the time crystal in the limit of vanishing magnon number $\omega_{\rm TC}^\infty = \langle \omega_{\rm TC} (t \rightarrow \infty) \rangle$. The measurements of the optomechanical coupling are carried out in this limit. The large inset shows a snapshot of the signal including the fitted spectrum shape and parameter values in the region where the time crystal's frequency has ceased changing. The fundamental surface wave mode driven in this Article is depicted in the small inset.
• Figure 1: Calibration of the excitation amplitude To reduce noise in determination of the mechanical excitation amplitude, we estimate the functional dependence of the real excitation amplitude, measured by the geophone voltage $V_{\rm gp}$, on the nominal excitation amplitude $A_{\rm nom}$. The calibration function is displayed in the legend.
• Figure 2: The mechanical mode.a The frequency modulation amplitude $G = g \theta_{\rm max}^2$ of the bulk time crystal measured as a function of the mechanical forcing frequency $\omega_{\rm exc}$. The solid lines are fits to a driven and damped harmonic oscillator response, from which the resonance frequency and width of the mechanical surface wave mode are determined. The thermometer fork width $\Delta f$ relative to the intrinsic width $\Delta f_0$ of the device for the two data sets is given in the legend. In the shown temperature interval the on-resonance amplitude $G$ remains approximately constant despite exponentially increasing resonance width, suggesting that coupling increases with the same exponent as a function of temperature. Here, unlike other measurements in this Article, the magnon number was kept constant by applying continuous pumping to an excited state in the confining trapPhysRevLett.120.215301, as extremely long lifetime of bulk time crystal makes pulsed spectral measurements impractical. b Data measured for the surface time crystal are extracted from the end of the decay after an RF excitation pulse, where the time crystal's frequency has stopped changing. c The extracted width of the surface wave resonance (bulk: blue triangles, surface: red diamonds) scales linearly with the the thermometer fork resonance width (solid line). This confirms that the damping of the mechanical mode that is coupled to the time crystal (mainly) originates from scattering of thermal excitations in the superfluid. The y axis intercept corresponds to the mechanical mode dissipation in the absence of superfluid thermal excitations, which may be caused by friction on the container walls, or quasiparticle states bound on the free surfaceforstner2024dynamicinteractionchiralcurrents.
• Figure 2: Tilt angle calibration. ( A) Applied surface wave excitation leads to additional heating of the sample, measured both via relaxation rate of the bulk time crystal (red points), as well as via the thermometer fork (blue points). The observed temperature between the two location separated by 14$\,$cm is a proxy for dissipated power. ( B) The measured temperature difference is converted to tilt angle using Eq. \ref{['eq:tiltcalib']} (magenta points) and fitted with a single parameter linear fit resulting in $\theta_{\rm max}^2 {\rm (deg)}^2 \approx 2.6182\,A_{\rm exc}$ (black dashed line).
• Figure 3: The optomechanical coupling mechanisms.a With on-resonance mechanical forcing, the fitted time crystal frequency modulation amplitude $G$ (bulk: red points, surface: blue points) is found to depend linearly on $\theta_\mathrm{max}^2$ (fitted lines). b The mean frequency shift $\Delta \omega_{\rm TC}^\infty = (\omega_{\rm TC}^\infty(\theta_{\rm max}) - \omega_{\rm TC}^\infty(\theta_{\rm max} = 0))$ is half of the frequency modulation amplitude, in agreement with Eq. (\ref{['eq:CavityFreq']}). c The product of the fitted asymmetry $\Theta$ and $\theta_{\rm max}$ (points) is found to be independent of the mechanical motion amplitude, as expected if $\theta_0$ 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 tilt angle, the surface time crystal frequency shift (red points) is consistent with the coupling constant $g_{\rm surf} = (2.2\div12.0)$ Hz deg$^{-2}$ determined from dynamic measurements (red shaded region). The static shift for the bulk time crystal (blue points) is smaller than expected from the dynamic measurements (blue shaded region) $g_{\rm bulk} = (9.8\div54.0)\,$ 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 $g_{\rm surf}=3.74$ Hz deg$^{-2}$. The horizontal error bars correspond to the value of the static tilt from panel c and the vertical error bars correspond to one standard deviation for measurements performed at different minimum coil currents ($N_{\rm surface} = 8$, $N_{\rm bulk}=3$). eBottom: 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.