Boundary Time Crystals Induced by Local Dissipation and Long-Range Interactions

Abstract

Driven-dissipative many-body system supports nontrivial quantum phases absent in equilibrium. As a prominent example, the interplay between coherent driving and collective dissipation can lead to a dynamical quantum phase that spontaneously breaks time-translation symmetry. This so-called boundary time crystal (BTC) is fragile in the presence of local dissipation, which can easily relax the system to a stationary state. In this work, we demonstrate a robust BTC that is intrinsically induced by local dissipation. We provide extensive numerical evidences to support existence of the BTC and study its behaviors in different regimes. In particular, with decreasing interaction range, we identify a transition from classical limit cycles to quantum BTCs featuring sizable spatial correlations. Our studies significantly broaden the scope of nonequilibrium phases and shed new light on experimental search for dynamical quantum matter.

Figures (5)

• Figure 1: (a) and (b) show the schematic of the superradiance-based BTC and its mean-field evolution of the magnetization $m_z$, respectively. Here, a small local decay $\gamma=0.01\kappa$ (with $\kappa$ the collective decay rate and $\Omega=1.5\kappa$) can destroy the oscillation. (c) and (d) show the proposed spin-1 BTC model and its mean-field evolution of the order parameter $s_z$, respectively. Here, the oscillation is induced solely by local dissipation and persists within a finite range of the decay rate $\gamma$.
• Figure 2: (a) Mean-field phase diagram in the $\chi$-$\gamma$ plane. (b) Classical trajectories of the system in the $s_z$-$n$ plane. The three panels correspond to the three data points indicated in (a). The system is initially in a product state $\rho_0=\prod_i|{0}\rangle_i\langle{0}|$.
• Figure 3: Characterization of the BTC with all-to-all interactions. (a) Evolution of several low-lying Liouvillian eigenvalues $\lambda_k$ with $N$ along the real axis. The data are fitted by an algebraic function $c_k+N^{-\beta_k}$. The inset shows the spectrum in the complex plane for $N=20$. (b) The two-time correlation function $G(t)$ for different system sizes.
• Figure 4: Scaling of the steady-state fluctuations $F^2$ with $N$. (a) and (b) show results for the BTC phase ($\Delta=-7$) and the SP ($\Delta=-1$), respectively. The insets show the time evolution of the fluctuations ($N=80$ for the BTC and $N=40$ for the SP), obtained from the MPS (red curves) and the CME (blue curves). The errorbars indicate the temporal fluctuations caused by finite trajectory realizations.
• Figure 5: (a) A phase transition driven by the factor $\alpha$. (b)-(d) show the lifetime $\tau$, the steady-state off-diagonal correlation $\mathcal{C}_\mathrm{off}$, and the mean correlation $\mathcal{C}_\mathrm{off}^{r}$ ($r=30$ and averaged over one period at $t=30$) as a function of $\alpha$ for the indicated system sizes ($N=500,1000,1500$). The results are obtained from the CME. (e) A single-trajectory MPS simulation of the dynamics for $\alpha=1.1$ (green) and $1.6$ (grey), whose Fourier spectra are shown in the inset.