Figure 1

(a) Schematic of the open Ising model in a transverse field. Internal states of trapped atoms or ions are described by spin-$1/2$ degrees of freedom on a lattice. The interaction between nearest neighbors has strength $V$ and there is an applied transverse field $\Omega$ (e.g., the Rabi frequency of a laser). Interaction with the radiation field leads to the incoherent emission of bath quanta (e.g., photons) with rate $\kappa$, which are detected and counted. (b) Quantum jump trajectory in a one-dimensional version of (a). Emitted photons are temporally (and spatially) resolved, so each point indicates where and when a quantum jump event took place. This particular trajectory shows intermittency, which manifests the coexistence of an active and an inactive dynamical phase (see the text).Reuse & Permissions

Figure 2

Mean-field phase diagram. Colored regions have a unique steady-state solution. In the gray region, delimited by the two spinodal lines $v_1$ and $v_2$ (see the text), two steady-state solutions with different magnetization exist. These solutions merge at $({\Omega }^{*};V^{*})=(2\kappa ;(3\sqrt{3}/z)\kappa )$. The small panels show a sketch of the behavior of the free energy $\theta \left(s\right)$ (solid line) and its first derivative $\langle k\rangle \left(s\right)$ (dotted line) in the vicinity of $s=0$. While in the region with a unique steady state both functions are smooth, the dynamical order parameter $\langle k\rangle \left(s\right)$ has a jump at $s=0$ for parameter values taken in the gray region. This indicates the existence of a first-order dynamical phase transition.Reuse & Permissions

Figure 3

Dynamical free energy $\theta \left(s\right)$ determined by exact diagonalization of the generalized master operator for the one-dimensional dissipative Ising model in a transverse field for $N=6$ spins with periodic boundary conditions. The plot shows $\theta \left(s\right)$ (in units of $\kappa$) for $V=100\kappa$ at three different values of the coupling strength $\Omega$ corresponding to an active (dashed red line) and inactive (solid blue line) dynamical phase and the behavior of the dynamical free energy under conditions of phase coexistence (dotted black line).Reuse & Permissions

Figure 4

Numerically determined (a) activity $\langle k\rangle /\kappa$ and (b) Mandel $Q$ parameter as a function of $\Omega /\kappa$ for the one-dimensional dissipative Ising model in a transverse magnetic field for $N=5,\cdots ,7$ sites and periodic boundary conditions. The lines show the results of an exact numerical diagonalization of the generalized master operator and the symbols show the results of quantum jump Monte Carlo simulations (ensembles of 1000 trajectories of length $t=200{\kappa }^{-1}$ obtained by slicing a much longer one of length $t_{\max }=2\times {10}^5{\kappa }^{-1}$).Reuse & Permissions

Figure 5

Analysis of data from quantum jump Monte Carlo simulations of a one-dimensional open Ising model in a transverse field with $N=12$ spins ($\kappa =0.1$). (a) Activity distributions (units of $\kappa$) of trajectories of length $t=10{\kappa }^{-1}$ (green) and $50{\kappa }^{-1}$ (transparent with solid lines) for three different values of $\Omega /\kappa$ for fixed $V=100\kappa$. These ensembles of trajectories are obtained by slicing longer trajectories of length $t_{\max }={10}^4{\kappa }^{-1}$. We also show sample trajectories at the corresponding conditions in which the emission events are site resolved. (b) Waiting-time distribution $p\left(\tau \right)$ (units of $\kappa$) for $\Omega =25\kappa$ and $V=100\kappa$ (crosses in green). The lines represent exponential waiting-time distributions with time constants $1/k_A$ (solid red line) and $1/k_I$ (dashed blue line). (c) $Q_{\tau }$ parameter as a function of the transverse magnetic field for $V=100\kappa$ extracted from the waiting-time distribution. The insets show samples of typical trajectories in each parameter regime. The maximum of $Q_{\tau }$ coincides with highly intermittent dynamics.Reuse & Permissions