Out-of-equilibrium evolution of kinetically constrained many-body quantum systems under purely dissipative dynamics

We explore the relaxation dynamics of quantum many-body systems that undergo purely dissipative dynamics through non-classical jump operators that can establish quantum coherence. Our goal is to shed light on the differences in the relaxation dynamics that arise in comparison to systems evolving via classical rate equations. In particular, we focus on a scenario where both quantum and classical dissipative evolution lead to a stationary state with the same values of diagonal or “classical” observables. As a basis for illustrating our ideas we use spin systems whose dynamics becomes correlated and complex due to dynamical constraints, inspired by kinetically constrained models (KCMs) of classical glasses. We show that in the quantum case the relaxation can be orders of magnitude slower than the classical one due to the presence of quantum coherences. Aspects of these idealized quantum KCMs become manifest in a strongly interacting Rydberg gas under electromagnetically induced transparency (EIT) conditions in an appropriate limit. Beyond revealing a link between this Rydberg gas and the rather abstract dissipative KCMs of quantum glassy systems, our study sheds light on the limitations of the use of classical rate equations for capturing the non-equilibrium behavior of this many-body system.

Figure 1

Relaxation of both classical and quantum dissipative East models for $\kappa /(1-\kappa )={10}^{-2}$ and a system size of $N=10$ spins (time in units of ${\lambda }^{-1}$). The upper panel shows the average excitation density, $\langle n\left(t\right)\rangle$. The lower panel shows the evolution of the coherences, represented by $\langle {\sigma }^x\left(t\right)\rangle$, in the quantum case. Inset: time-evolution of $\langle n_k\rangle$ for $k=1\cdots N$ in a single trajectory for $\theta =\pi /2$.

Figure 2

Waiting time distributions $t_{\mathrm{wait}}$ as a function of time in the dissipative quantum East model for $N=10$ and $\kappa /(1-\kappa )={10}^{-2}$ (all times in units of ${\lambda }^{-1}$). (a) When $\theta =\pi /2$, the plateaus in the excitation density give rise to the appearance of several peaks in the waiting time distributions. (b) When $\theta =\pi /20$, every jump brings the system away from equilibrium making the occurrence of a subsequent jump more likely.

Figure 3

Relaxation of both classical and quantum dissipative FA models for $\kappa /(1-\kappa )={10}^{-2}$ and a system size of $N=10$ spins (time in units of ${\lambda }^{-1}$). The upper panel shows the average excitation density, $\langle n\left(t\right)\rangle$. The lower panel shows the evolution of the coherences, represented by $\langle {\sigma }^x\left(t\right)\rangle$, in the quantum case. Inset: time-evolution of $\langle n_k\rangle$ for $k=1\cdots N$ in a single trajectory for $\theta =\pi /2$.

Figure 4

(a) Interacting atoms in a EIT configuration. Neighboring atoms in the Rydberg state, $|r\rangle$, interact with strength $V$. Strong decay, $\gamma$, of the intermediate level, $|p\rangle$, effectively gives rise to purely dissipative quantum dynamics. (b) Relaxation of the expectation value of the excitation density of the classical excluded volume model (red, dashed lines) and the quantum Rydberg lattice gas (black, solid lines) for $x={\Omega }_{\mathrm{p}}/{\Omega }_{\mathrm{c}}=0.1,1$, and 10 (system size $N=8$). (c) Relaxation of the expectation value of the coherences, represented by $\langle {\sigma }^x\left(t\right)\rangle$, for the same three values of the parameter $x$.