Observation of a Time Quasicrystal and Its Transition to a Superfluid Time Crystal

We report experimental realization of a quantum time quasicrystal and its transformation to a quantum time crystal. We study Bose-Einstein condensation of magnons, associated with coherent spin precession, created in a flexible trap in superfluid ${}^3\mathrm{He}\text{−}B$. Under a periodic drive with an oscillating magnetic field, the coherent spin precession is stabilized at a frequency smaller than that of the drive, demonstrating spontaneous breaking of discrete time translation symmetry. The induced precession frequency is incommensurate with the drive, and hence, the obtained state is a time quasicrystal. When the drive is turned off, the self-sustained coherent precession lives a macroscopically long time, now representing a time crystal with broken symmetry with respect to continuous time translations. Additionally, the magnon condensate manifests spin superfluidity, justifying calling the obtained state a time supersolid or a time supercrystal.

Figure 1

Time supercrystal in superfluid ${}^3\mathrm{He}\text{−}B$ emerging from spontaneous coherence in freely precessing magnetization. (Top) The coherent precession of magnetization, $M_x+i{M}_y=\gamma S_{\perp }{e}^{i\omega t}$, is established with time constant ${\tau }_E$ after the excitation pulse at $t=0$. The signal is picked up by the NMR coils (Fig. 2) and down-converted to lower frequency (with reference at 834 kHz). (Bottom) Since the magnetic relaxation in superfluid ${}^3\mathrm{He}$ is small, the number of magnons $N$ slowly decreases with timescale ${\tau }_N\gg {\tau }_E$. During relaxation the precession remains coherent, and the state represents Bose-Einstein condensate of magnons until the number of magnons drops below the critical value (which in these experiments corresponds to a signal below the noise level).

Figure 2

Experimental setup: Sample container is made from quartz glass and filled with superfluid ${}^3\mathrm{He}\text{−}B$. NMR coils create a pumping field ${\mathbf{H}}_{\mathrm{rf}}$ and pick up the induction signal from coherently precessing magnetization $\mathbf{M}$ of the magnon BEC. The trapping potential for the condensate is created by a spatial distribution of the orbital anisotropy axis of ${}^3\mathrm{He}\text{−}B$ (small green arrows) and an axial minimum in the static magnetic field produced by a pinch coil.

Figure 3

Time crystal and time quasicrystal. The measured signal is analyzed using time-windowed Fourier transformation, revealing two distinct states of coherent precession seen as sharp peaks. Driving field ${\mathbf{H}}_{\mathrm{rf}}$ with the frequency ${\omega }_{\text{drive}}$, applied at $t<0$, excites magnons on the second radial excited level in the harmonic trap. The majority of these magnons moves to the ground level, where they form the magnon BEC. Their precession is seen as the signal at the smaller frequency, ${\omega }_{\text{coherent}}<{\omega }_{\text{drive}}$. The total signal is quasiperiodic in time, see Fig. 4. At $t=0$, the rf pumping is switched off and the (2,0) level quickly depopulates. Only the periodic signal from the precession of the ground-level magnon BEC remains, like in Fig. 1. Its frequency is slowly increasing in time in the flexible trap, following the decay of the magnon number $N$. This emphasizes the robust nature of the breaking of continuous time translation symmetry and signifies the time crystal.

Figure 4

Quasiperiodic signal from a time quasicrystal at $t<0$ in Fig. 3 is shown in a time domain. The section plotted here begins at $t=-4.97\mathrm{s}$. The amplified signal from NMR coils is down-converted to lower frequency using a Larmor frequency as a reference and digitized at 44 kHz rate. From the total signal, the drive (lower plot) is subtracted, as calibrated in a separate measurement without magnon BEC. The quasiperiodic signal (middle blue line in the upper plot) is then revealed. It is further splitted using bandpass filters to the signal from the ground-level magnon BEC (upper green line) at $({\omega }_{\text{coherent}}-{\omega }_{\mathrm{L}})/2\pi \approx 290\mathrm{Hz}$ and the signal from the (2,0) level at $({\omega }_{\text{drive}}-{\omega }_{\mathrm{L}})/2\pi \approx 980\mathrm{Hz}$ (lower red line). These lines have been shifted vertically for clarity. Error bars on the right denote measurement noise amplitude in the respective frequency bands. Oscillations at the (2,0) level proceed at exactly the drive frequency as shown by the dashed vertical lines. The frequency locking occurs automatically by self-tuning of the steady-state number of magnons [37].