Tunable discrete quasi-time crystal from a single drive

Abstract

The search for exotic temporal orders in quantum matter, such as discrete quasi-time crystals (DQTCs), has become an important theme in nonequilibrium physics. However, realizing these phases has so far required complex protocols, such as drives with multiple incommensurate frequencies. Here, we present a significantly simpler mechanism: the emergence of DQTCs in a dissipative collective spin system subjected to only a single periodic drive. Remarkably, the characteristic frequencies of this novel phase are not fixed but can be continuously tuned by varying the strength of the drive. Even more strikingly, this tunability is punctuated by Arnold tongues, within which the response main frequency locks to rational fractions of the drive. Our model further provides a unified framework that also encompasses stationary, discrete time crystals and chaotic phases. This discovery simplifies the requirements for generating complex temporal orders and opens a viable route towards the experimental control and manipulation of quasi-time crystalline matter.

Figures (4)

• Figure 1: (a) Largest Lyapunov exponent (LLE) $\eta$ in the semiclassical dynamics. $\omega_{0}=1.5$ and $\kappa=1.0$. To compute the LLE, we evolve two initially nearby trajectories, $\vec{m}_1(0)=(0,0,1)$ and $\vec{m}_2(0)=\vec{m}_1(0)+\delta\vec{m}$, where the perturbation $\delta\vec{m}$ is random and very small ($|\delta\vec{m}|=10^{-8}$). (b) LLE as a function of $\omega_z$ for fixed $\omega_1=1.0$, corresponding to the black dashed line in (a). (c) The associated bifurcation diagram of $m^y$ along the same cut. The red dashed lines at $\omega_z \approx 0.9,1.42,2.15,2.4$ indicate dynamical transitions. As $\omega_z$ increases from $0$ to $3.2$, the system exhibits a sequence of dynamical regimes: limit-cycle (DQTC), chaos, a single fixed point, period-doubling with two stable points (DTC), and chaos again.
• Figure 2: The parameters are $\omega_{0}=1.5$, $\omega_{1}=1.0$, $\kappa=1.0$. Panels (a1)--(c1), (a2)--(c2), (a3)--(c3), and (a4)--(c4) correspond to $\omega_z=0.5$, $2.0$, $2.3$, and $3.0$, respectively. (a1)--(a4) The Poincaré sections resulting from the mean-field equations. Red dots show the stroboscopic evolution of the initial state $\vec{m}(0)=(0,0,1)$, while blue dots show the stroboscopic evolution of 300 randomly sampled initial states. (b1)--(b4) Liouvillian gap and Liouvillian spectrum. Empty blue symbols correspond to exact diagonalization results, while filled red symbols indicate results obtained using the matrix-free Arnoldi method SUppMaterial. (b1) The upper triangles represent the gap of the Liouvillian eigenvalues with the smallest real part. (b3) Circles denote the smallest gap among Liouvillian eigenvalues with $\arg(\lambda)=0$, and rectangles denote the smallest gap with $\arg(\lambda)=\pi$. Insets plot the Floquet-Liouvillian spectrum. (c1)--(c4) Connected autocorrelation function $G(t)$ for total spin ranging from $S=10$ (darkest) to $S=90$ (lightest). Insets display the scaling of $G(0)$ with system size $S$. In (c3), dashed vertical lines mark integer times $t=1$--$9$.
• Figure 3: Frequency response in the DQTC regime under varying kick strength. Curves correspond to system sizes from $S=10$ (darkest) to $S=80$ (lightest). The gray line represents the Fourier spectrum of the corresponding semi-classical dynamics. $\omega_{0}=1.5, \kappa=1.0$. The upper row corresponds to $\omega_z=0$: (a) $\omega_1=1.0$, (b) $\omega_1=2.0$, and (c) $\omega_1=3.0$. The lower row corresponds to $\omega_z=0.5$: (d) $\omega_1=1.0$, (e) $\omega_1=2.0$, and (f) $\omega_1=3.0$.
• Figure 4: Single quantum trajectory generated using the quantum jump method in the DQTCs regime. Parameters are fixed to $\omega_{0}=1.5$, $\kappa=1.0$, $\omega_{z}=0$ and $S=200$. The three columns correspond to increasing kick strength: (left) $\omega_{1}=1.0$, (middle) $\omega_{1}=2.0$, and (right) $\omega_{1}=3.0$. Top row: photon-count signal along the trajectory. Bottom row: corresponding Fourier spectra of the photon-count time series. Dashed blue lines indicate frequencies predicted from the semiclassical analysis.