Time Crystal in a Single-Mode Nonlinear Cavity

Time crystal is a class of nonequilibrium phases with broken time-translational symmetry. Here, we demonstrate the time crystal in a single-mode nonlinear cavity. The time crystal originates from the self-oscillation induced by a linear gain and is stabilized by a nonlinear damping. We show in the time crystal phase there are sharp dissipative gap closing and pure imaginary eigenvalues of the Liouvillian spectrum in the thermodynamic limit. Dynamically, we observe a metastable regime with the emergence of quantum oscillation, followed by a dissipative evolution with a timescale much longer than the oscillating period. Moreover, we show there is a dissipative phase transition at the Hopf bifurcation, which can be characterized by the photon number fluctuation in the steady state. These results pave a new promising way for further experiments and deepen our understanding of time crystals.

Figure 1

Schematic representation of our setup: a cavity mode $a$ under driving with strength $ϵ$. There are both linear (single-photon) and nonlinear (two-photon) damping with rates $\kappa$ and $\eta$, respectively. There is also a linear gain with rate $g$.

Figure 2

Dissipative gap closing and stable quantum oscillation. (a),(b) Two dissipative gaps as a function of rescaled driving strength $ϵ\sqrt{\eta }$ for different nonlinear damping rate $\eta$. The black dashed line indicates the Hopf bifurcation of the reduced classical model, and the pink (blue) area indicates the limit cycle (equilibrium) phase. The fifty lines in (b) from the bottom up correspond to equally spaced $\eta$ from 0.001 to 0.05. The dashed red line in (b) indicates the nonlinear fitting curve of the minimums. (c),(d) The time evolution of rescaled photon number $\eta N_{\mathrm{a}}$ (c) and state purity $\mathrm{Tr}{\rho }^2$ (d). The black dashed line in (c) indicates the time evolution of the classical model. The inset in (d) shows the average purity in a period around $t=10$ (red dashed line) and the steady-state purity for $t\to \infty$ (black solid line) as a function of the nonlinear damping rate. The initial state is the vacuum state, and the rescaled driving strength is chosen as $ϵ\sqrt{\eta }=2$. Other parameters are $\mathrm{\Delta }=10$, $\kappa =0.1$, and $g=1$.

Figure 3

Dephasing in the quantum and classical evolution. (a) Maximal values of the quantum Husimi distribution as a function of time. The pink area denotes the metastable regime. The inset is the detailed quantum Husimi distribution at $t=10$. (b) Classical Husimi distribution at $t=10$. The initial states are both the vacuum state. (c) Oscillating phases in the classical trajectory at $t=10$. The oscillating phase depends on the start point in the phase space. There is a singular point where the oscillating phase is not continuous. The parameters are $\mathrm{\Delta }=10$, $\kappa =0.1$, $g=1$, $\eta =0.005$, and $ϵ\sqrt{\eta }=2$.

Figure 4

Liouvillian spectra for $ϵ\sqrt{\eta }=0$ (a), $ϵ\sqrt{\eta }=4.8$ (b), and $ϵ\sqrt{\eta }=7.2$ (c), corresponding to time crystal phase, critical regime and equilibrium phase. Red points are the eigenvalues, and the horizontal blue lines denote a frequency scale equal to the detuning $\mathrm{\Delta }$. (d) The oscillating frequencies as a function of the rescaled driving strength. The quantum results (red points) are the imaginary part of the eigenvalues with the largest real part and nonzero imaginary part. The classical results (blue points) are obtained by solving the limit cycle through the Poincaré map. Other parameters are $\mathrm{\Delta }=10$, $\kappa =0.1$, $g=1$, and $\eta =0.01$.

Figure 5

Quantum limit cycle and photon statistics in the steady state. (a) Steady-state quantum Husimi distributions for $ϵ\sqrt{\eta }=3.2$, 4.8, 6.4 from left to right. The nonlinear damping rate is $\eta =0.01$. (b) The rescaled photon number fluctuation as a function of rescaled driving strength $ϵ\sqrt{\eta }$. From the bottom up, the nonlinear damping rate $\eta$ is increased from 0.001 to 0.05 linearly. The pink (blue) area indicates the limit cycle (equilibrium) phase. (c) The photon distributions for $\eta =0.001$ and five different driving strengths. Other parameters are $\mathrm{\Delta }=10$, $\kappa =0.1$, and $g=1$.