Filling an empty lattice by local injection of quantum particles

Abstract

We study the quantum dynamics of filling an empty lattice of size L by connecting it locally with an equilibrium thermal bath that injects noninteracting bosons or fermions. We adopt four different approaches, namely, (i) direct exact numerics, (ii) Redfield equation, (iii) Lindblad equation, and (iv) quantum Langevin equation, which are unique in their ways for solving the time dynamics and the steady state. In this simple setup we investigate open quantum dynamics and subsequent approach to thermalization. The quantities of interest that we consider are the spatial density profile and the total number of bosons and fermions in the lattice. The spatial spread is ballistic in nature and the local occupation eventually settles down owing to equilibration. The ballistic spread of local density admits a universal scaling form. We show that this universality is only seen when the condition of detailed balance is satisfied by the baths. The difference between bosons and fermions shows up in the early time growth rate and the saturation values of the profile. The techniques developed here are applicable to systems in arbitrary dimensions and for arbitrary geometries.