Dynamics of a Many-Body-Localized System Coupled to a Bath

Coupling a many-body-localized system to a dissipative bath necessarily leads to delocalization. Here, we investigate the nature of the ensuing relaxation dynamics and the information it holds on the many-body-localized state. We formulate the relevant Lindblad equation in terms of the local integrals of motion of the underlying localized Hamiltonian. This allows us to map the quantum evolution deep in the localized state to tractable classical rate equations. We consider two different types of dissipation relevant to systems of ultracold atoms: dephasing due to inelastic scattering on the lattice lasers and particle loss. Our approach allows us to characterize their different effects in the limiting cases of weak and strong interactions.

Figure 1

Decay of an initially prepared imbalance ${\mathcal{I}}_0\sim 0.9$ for a disorder strength $h=10J$ under dephasing, i.e., $L_i=n_i$, for a system of size $N=20$ for 150 disorder realizations. The shaded regions denote standard errors.

Figure 2

Collapse of the imbalance decay for disorder strengths $h/J=10$, 15, 20 (solid, dashed, and dash-dotted lines and 150 disorder realizations each) under dephasing with $\tilde{t}=t\gamma (J/h{)}^2$. The dashed black line denotes $\exp [-(\tilde{t}/t_0{)}^{\beta }]$ with $\beta \approx 0.38$ and $t_0\approx 0.02$. The inset shows the size dependence using the diagonal rate equations.

Figure 3

Comparison of the dynamics from the classical rate equation (solid line, 250 disorder realizations) and the numerical simulation (dashed lines, $\gamma /J=0.1$, 0.5, 1.0, and 100 disorder realizations) for a system of size $N=20$, $U=0.5J$, and $h=10J$. The shaded regions denote standard errors.

Figure 4

Imbalance decay in the presence of interactions for particle loss calculated using the classical rate equations for a system of size $N=20$, $h=10J$, $\gamma =0.1J$, and 250 disorder realizations. The inset shows the induced hopping processes that lead to diffusion as described by Eq. (13).