Synchronization in rotating supersolids

Abstract

Synchronization is ubiquitous in nature at various scales and fields. This phenomenon not only offers a window into the intrinsic harmony of complex systems, but also serves as a robust probe for many-body quantum systems. One such system is a supersolid: an exotic state that is simultaneously superfluid and solid. Here, we show that putting a supersolid under rotation leads to a synchronization of the crystal's motion to an external driving frequency triggered by quantum vortex nucleation, revealing the system's dual solid-superfluid response. Benchmarking the theoretical framework against experimental observations, we exploit this model as a novel method to investigate the critical frequency required for vortex nucleation. Our results underscore the utility of synchronization as a powerful probe for quantum systems.

Figures (10)

• Figure 1: Synchronization and concurrent vortex nucleation. (a) Rotating supersolid from eGPE simulation, the isosurfaces are at $20\%$ (red) and $0.8\%$ (beige) of the maximum density, representing three droplets (one droplet highlighted in plain red color) and the halo, respectively. (b) In-plane trajectory of the droplet's tip (black) together with its decomposition in guiding center (green) and cyclotron (orange) motions. The light-blue shaded area highlights the time window in which a vortex is detected inside the system. The 3D isosurfaces (i)-(iv) correspond to different density frames during the synchronization process, with the black tubes corresponding to vortices. The insets show a schematic illustration of the droplet epitrochoidal (b$_1$) and circular (b$_2$) trajectory followed by the droplet. (c) Schematic representation of the decomposition in guiding center and cyclotron coordinates. The results are obtained for parameters: $a_\text{dd}\,{=}\,130.8\,a_0$, trap frequencies $[\omega_\perp,\omega_z] = 2\pi \times[50,95]$ Hz, atom number $N=50000$, $a_s=95\,a_0$, magnetic field tilt angle $\theta = 30^\circ$, $\Omega = 2\pi\times 15$ Hz, and dissipation constant $\gamma=0$.
• Figure 2: Quantification of synchronization. Time evolution of the cyclotron (a) and guiding center (b) coordinates. (c) Two exemplar frames showing the column density and the central phase slice of the rotating supersolid at $t=96.2$ ms and $t=670.7$ ms. The solid and dash-dotted circles mark the cutoff radii $r^*=6\,\mu$m and $r^*=4.5\,\mu$m used to count the vortex number (d) time averaged over 35 ms, with the same linestyle as the circles. (e) Frequency alignment $\kappa$, as defined in the main text. (f) Total angular momentum $\langle L_z\rangle$ and angular momentum of the droplets $L_\text{droplets}$, where (i)-(iv) refer to the ones of Fig. \ref{['fig:fig1']}(b). Across all subplots, the blue shaded region highlights when vortices enter within $r^*=4.5\,\mu$m. Parameters as in Fig. \ref{['fig:fig1']}.
• Figure 3: Independent droplet regime. (a) Time evolution of the droplet's edge trajectory and corresponding cyclotron (b) and guiding center coordinate (c). (d) Frequency alignment $\kappa$. All the results are obtained for the same parameters of Fig. \ref{['fig:fig1']}, except for $a_s=90\,a_0$.
• Figure 4: Experimental observation of the synchronization process. Angular position of the droplets (a$_1$, b$_1$) in the rotating frame as a function of time in experiment (a) and simulation (b). The orange and green lines are guides to the eye for the unsynchronized and synchronized cases, respectively. (a$_2$, a$_3$) 2D Fourier transform of the experimental droplet angular position for early [$0,50$] ms and late [$60,110$] ms time intervals, respectively. (b$_2$, b$_3$) 2D Fourier transform of the theoretical droplet angular position for early [$0,200$] ms and late [$200,400$] ms time intervals, respectively. The colorbars in (a$_2$, b$_2$) and (a$_3$, b$_3$) go from white to orange and white to green, respectively, with the three largest peaks highlighted in black. The experimental data is taken for $\Omega=2\pi\times9$ Hz, trap frequencies $[\omega_\perp,\omega_z]=2\pi\times[50.5(6), 137(3)]\,Hz$, B=18.24(2) G, $N \approx 69000$. Theoretical simulations are done for $\Omega=2\pi\times9$ Hz, $N=60000$, $a_s=90\,a_0$, trap frequencies $[\omega_\perp,\omega_z] = 2\pi\times[50,149]$ Hz, dissipation parameter $\gamma=0.08$.
• Figure 5: Synchronization during a slow ramp of the driving rotation frequency. (a) $\phi_{\mathrm{rot}}$ measured at different points of the slow ramp of $\Omega$ from 0 to $2\pi\times$8Hz in 200 ms. The orange and green lines are a guide to the eye for the unsynchronized and synchronized cases, respectively. Experimental parameters: trap frequency $[\omega_\perp,\omega_z]=2\pi\times[50.3(4), 140.1(5)]\,Hz$, $N\approx69000$, B=18.30(2) G. (b) $\langle \hat{L}_z\rangle$ for ground states in the rotating frame varying $\Omega$ together with exemplary density isosurfaces. Simulation parameters: $N=70000$, $a_s=92\,a_0$, trap frequencies $2\pi\times[50,140]$ Hz.