Discrete Time Crystals in the Absence of Manifest Symmetries or Disorder in Open Quantum Systems

We establish a link between metastability and a discrete time-crystalline phase in a periodically driven open quantum system. The mechanism we highlight requires neither the system to display any microscopic symmetry nor the presence of disorder, but relies instead on the emergence of a metastable regime. We investigate this in detail in an open quantum spin system, which is a canonical model for the exploration of collective phenomena in strongly interacting dissipative Rydberg gases. Here, a semiclassical approach reveals the emergence of a robust discrete time-crystalline phase in the thermodynamic limit in which metastability, dissipation, and interparticle interactions play a crucial role. We perform numerical simulations in order to investigate the dependence on the range of interactions, from all to all to short ranged, and the scaling with system size of the lifetime of the time crystal.

Figure 1

DTC in a metastable open quantum system. (a) The phase space of the system is divided into two basins of attraction (bright and dark areas), associated with two stationary states, ${\rho }_1^{\mathrm{ss}}$ (blue dot) and ${\rho }_2^{\mathrm{ss}}$ (red dot), respectively. The combination of an appropriate transformation [${\mathcal{L}}^{(R)}$] and dissipative dynamics [${\mathcal{L}}^{(0)}$] during a time interval $T$ maps one of the stationary state into the other one. The repetition of this composed transformation makes the state oscillate with a doubled period $2T$. (b) Time-evolution of a typical observable in the DTC phase. Here, we show the expectation value of $S^x(t)$ as a function of time $t$ for a dissipative Rydberg gas (see text for details). The period of oscillations is twice the one of the driving (with shaded areas corresponding to different basins of attraction). Here, ${\mathrm{\Omega }}_x^{(0)}=0.7\mathrm{\Gamma }$, $V=12\mathrm{\Gamma }$, ${\mathrm{\Delta }}^{(0)}=-3.5\mathrm{\Gamma }$, $T=2{\mathrm{\Gamma }}^{-1}$, and $t_R={10}^{-2}{\mathrm{\Gamma }}^{-1}$.

Figure 2

(a) Mean-field phase diagram of the normalized Fourier component $f_2$, evaluated numerically over $K=30$ periods, as a function of ${\mathrm{\Omega }}_x^{(0)}$ and $V$. The region delimited by the two gray lines corresponds to the bistable regime, with the dark zone associated with the DTC phase and the bright one to the normal phase. The result suggests that $f_2$ vanishes continuously at the right boundary, whereas it undergoes a discontinuous jump at the left one. (b) Density plot of $|{\lambda }_2{|}^{-1}$ as a function of ${\mathrm{\Omega }}_x^{(0)}$ and $V_0$ for the Rydberg fully connected model. Lines denote the corresponding mean-field bistability region for comparison. (c) Stroboscopic time evolution of $S^x(t)$ for a system of $N=28$ (blue), $N=20$ (green), $N=12$ (orange) particles, and ${\mathrm{\Omega }}_x^{(0)}=0.7\mathrm{\Gamma }$, $V_0=6\mathrm{\Gamma }$. At $t=0$ the system is in the state with all spins pointing down in the $z$ direction and the transformation parameters are obtained from Eq. (4) with $V=V_0$. Red dashed lines represent a fit of the stroboscopic data for $N=28$ with the functions $g_{\pm }(t)=a\pm b{e}^{-t/{\tau }_{\mathrm{DTC}}}$. Inset: lifetime of the DTC oscillations, ${\tau }_{\mathrm{DTC}}$, extracted from the stroboscopic dynamics of $S^{x/y/z}(t)$ as a function of $N$. In this range of $N$ its functional behavior is well captured by the power law ${\tau }_{\mathrm{DTC}}(N)\sim N^{\alpha }$, with $\alpha \approx 0.5$. In all panels, ${\mathrm{\Delta }}^{(0)}=-3.5\mathrm{\Gamma }$, $T=2{\mathrm{\Gamma }}^{-1}$, and $t_R={10}^{-2}{\mathrm{\Gamma }}^{-1}$.

Figure 3

Lifetime ${\tau }_{\mathrm{DTC}}$ of the oscillations with period $2T$ as a function of $N$ in a 1D Rydberg gas with nearest-neighbor interactions extracted from the stroboscopic dynamics of $S^{x/y/z}(t)$. The state at $t=0$ is with all spins pointing down in the $z$ direction, while the transformation parameters are given by Eq. (4) with $d_x=0.1387$, $d_y=0.6824$, and $d_z=-0.7177$. The black dashed line represents the asymptotic value of ${\tau }_{\mathrm{DTC}}$ obtained by fitting with the function $g(N)=a-b/N^c$. Inset: gap $|{\lambda }_2|$ as a function of $N$ associated with the ${\mathcal{L}}^{(0)}$ dynamics of the main panel. Data points are obtained by fitting the long-time decay of $S^x(t)$ with an exponential decay $\propto e^{-{\lambda }_{2}t}$. The black dashed line is the asymptotic value obtained by a fit with $g(N)$ as before. Here, ${\mathrm{\Omega }}_x^{(0)}=\mathrm{\Gamma }$, $V_0=1.6\mathrm{\Gamma }$, ${\mathrm{\Delta }}^{(0)}=-3.5\mathrm{\Gamma }$, $T=2{\mathrm{\Gamma }}^{-1}$, and $t_R={10}^{-2}{\mathrm{\Gamma }}^{-1}$.