Filling an empty lattice by local injection of quantum particles

TL;DR

This paper investigates how quantum particles fill an empty lattice through local thermal baths, revealing universal ballistic spreading and differences between bosons and fermions, using multiple theoretical approaches.

Contribution

It introduces a comprehensive analysis of quantum lattice filling dynamics with a unified framework applicable to various dimensions and geometries.

Findings

Density profile spreads ballistically with universal scaling.

Universal behavior occurs only when baths satisfy detailed balance.

Bosons and fermions differ in early growth and saturation profiles.

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 non-interacting 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. Our setup offers a simplistic platform to understand fundamental aspects of dynamics and approach to thermalization. The quantities of interest that we consider are the spatial density profile and the total number of bosons/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.