Fermionic quantum walkers coupled to a bosonic reservoir

TL;DR

This paper models the dynamics of non-interacting fermions coupled to a bosonic reservoir via a quantum walk, deriving explicit fermionic observable dynamics and showing convergence to an infinite-temperature state over time.

Contribution

It introduces a detailed model of fermion-boson coupling via quantum walks and provides explicit dynamics and long-term state convergence analysis.

Findings

Explicit Heisenberg dynamics for fermionic observables.

Large-coupling regime expansion controlled by spectral methods.

Fermionic reduced state converges to an infinite-temperature Gibbs mixture.

Abstract

We analyse the discrete-time dynamics of a model of non-interacting fermions coupled to an infinite reservoir formed by a bosonic quantum walk on ${\mathbb Z}$. This dynamics consists of consecutive applications of free evolutions of the fermions and bosons followed by a local coupling between them. The unitary operator implementing this coupling accounts for energy exchanges between the system and reservoir while it preserves the number of fermions. The free fermion evolution is given by a second-quantized single-particle unitary operator satisfying some genericity assumptions. The free boson evolution is given by the second-quantized shift operator on ${\mathbb Z}$. We derive explicitly the Heisenberg dynamics of fermionic observables and obtain a systematic expansion in the large-coupling regime, which we control by using spectral methods. We also prove that the reduced state of the fermions converges in the large-time limit to a mixture of infinite-temperature Gibbs states in each particle sector.