Nonequilibrium Dark Space Phase Transition

We introduce the concept of dark space phase transition, which may occur in open many-body quantum systems where irreversible decay, interactions, and quantum interference compete. Our study is based on a quantum many-body model that is inspired by classical nonequilibrium processes which feature phase transitions into an absorbing state, such as epidemic spreading. The possibility for different dynamical paths to interfere quantum mechanically results in collective dynamical behavior without classical counterpart. We identify two competing dark states, a trivial one corresponding to a classical absorbing state and an emergent one which is quantum coherent. We establish a nonequilibrium phase transition within this dark space that features a phenomenology which cannot be encountered in classical systems. Such emergent two-dimensional dark space may find technological applications, e.g., for the collective encoding of a quantum information.

Figure 1

Dark state phase transition. (a) Three-level quantum system with basis states $|•⟩$,$|*⟩$ representing contagious and noncontagious infected units, and the healthy state $|\circ ⟩$. (b) The dynamics consists of coherent transitions between infected states $|•⟩\leftrightarrow |*⟩$ (rate ${\mathrm{\Omega }}_1$) and between states $|\circ ⟩\leftrightarrow |*⟩$. This (infection) process must be facilitated by the presence of a contagious neighbor. For each contagious neighbor, the rate of the process is enhanced by a factor ${\mathrm{\Omega }}_2/z$, where $z$ is the coordination number of the lattice. (c) Illustrative trajectories for a 1D quantum system with 50 sites. Top: approach to the dark state $|D⟩$ for the model depicted in (a). Middle: fluctuating phase which typically emerges in classical and quantum models with absorbing states. The example shown is for the quantum contact process [9]. Bottom: emergence of the dark state $|D_e⟩$ for the model in depicted in (a).

Figure 2

Infinite-dimensional lattice. (a) Stationary behavior of the density of contagious sites ${\varrho }_{•}$, which is an order parameter for the dark space phase transition. One phase is dominated by the classical jumps $|*⟩\to |\circ ⟩$, bringing the system toward the state $|D⟩$. The other is dominated by the “no jump” dynamics under $H_{\mathrm{eff}}$, which drives the system toward the dark state $|D_e⟩$. (b) Finite expectation values of ${\lambda }_4$ demonstrate that $|D_e⟩$ features coherence between states $|•⟩$ and $|\circ ⟩$. (c) Log-log plot of the gap ($r_{\mathrm{gap}}$) of $H_{\mathrm{eff}}$ for ${\mathrm{\Omega }}_1=\gamma$ as a function of ${\mathrm{\Omega }}_2/\gamma$. This quantity is half the escape rate from the dark state $|D_e⟩$ and rapidly vanishes for increasing $N$, when ${\mathrm{\Omega }}_2>2{\mathrm{\Omega }}_1$. In the inset we show the behavior of $r_{\mathrm{gap}}$ as a function of $N$ for the critical rate ${\mathrm{\Omega }}_2=2$. The curve $N^{-0.3}$ (dashed line) is shown for comparison. (d) Average density ${\varrho }_{•}$ for the eigenvector of $H_{\mathrm{eff}}$ associated with the gap. This shows a phase transition, from a region where the density is zero to a region where the density is finite. For increasing $N$, the numerical results approach the analytic prediction. Highlighted in red, the region where the average dynamics starting from $|U⟩$ approaches the dark state $|D_e⟩$. The finite $N$ curves are for $N=20,30,40,\dots ,120$.

Figure 3

One-dimensional lattice. (a),(b) Simulations of quantum jump trajectories, for a 1D system with $N=6$. Each data point is obtained by averaging over 100 trajectories. The panels show the behavior of the density of contagious sites ${\varrho }_{•}$ and of the coherence $|⟨{\lambda }_4⟩|$, for $t=20/\gamma$. Two different phases emerge: one is the dark state $|D⟩$, while in the other the system approaches the emergent dark state $|D_e⟩$. (c) Order parameter ${\varrho }_{•}$ as a function of ${\mathrm{\Omega }}_2$, for ${\mathrm{\Omega }}_1=\gamma$ in the emergent dark state. For finite systems, this is estimated by looking at properties of the state associated with the gap of the effective Hamiltonian (solid lines for $N=3,4,\dots ,9$). For infinite systems, we exploit matrix product state methods to target the fixed point of the dynamics in Eq. (4) (black circles). This shows a transition from $|D⟩$ to $|D_e⟩$. We have further obtained values of the density ${\varrho }_{•}$ in the steady state of the Lindblad dynamics Eq. (1) (red crosses), which are in agreement with the prediction made for $H_{\mathrm{eff}}$. In the shaded region, our MPS algorithms did not converge.