Derivation of some translation-invariant Lindblad equations for a quantum Brownian particle

TL;DR

This paper derives a translation-invariant Lindblad master equation describing the dynamics of a quantum Brownian particle with spin on an infinite lattice, coupled to bosonic baths, in the van Hove limit.

Contribution

It provides an explicit derivation of a translation-invariant Lindblad equation for a quantum particle with spin on a lattice, considering simultaneous small system-bath and hopping interactions.

Findings

Derived explicit Lindblad master equation for the system.

Established the conditions under which the equation applies.

Connected the microscopic model to the effective open system dynamics.

Abstract

We study the dynamics of a Brownian quantum particle hopping on an infinite lattice with a spin degree of freedom. This particle is coupled to free boson gases via a translation-invariant Hamiltonian which is linear in the creation and annihilation operators of the bosons. We derive the time evolution of the reduced density matrix of the particle in the van Hove limit in which we also rescale the hopping rate. This corresponds to a situation in which both the system-bath interactions and the hopping between neighboring sites are small and they are effective on the same time scale. The reduced evolution is given by a translation-invariant Lindblad master equation which is derived explicitly.