A Matrix Model of Relaxation

TL;DR

This paper models a two-level quantum system coupled to a large, random matrix-based environment, deriving an integral representation for its reduced density matrix and establishing conditions for thermalization and Markovian dynamics.

Contribution

It introduces a new matrix model for the environment that exhibits thermalization properties and derives a master equation in the van Hove limit.

Findings

Derives an integral representation for the mean reduced density matrix.

Identifies a model environment with properties of macroscopic thermal reservoirs.

Obtains a Markovian master equation in the appropriate limit.

Abstract

We consider a two level system, $\mathcal{S}_{2}$, coupled to a general $n$ level system, $\mathcal{S}_{n}$, via a random matrix. We derive an integral representation for the mean reduced density matrix ${\rho} (t)$ of $\mathcal{S}_{2}$ in the limit $n\to \infty $, and we identify a model of $\mathcal{S}_{n}$ which possesses some of the properties expected for macroscopic thermal reservoirs. In particular, it yields the Gibbs form for ${\rho} (\infty)$. We consider also an analog of the van Hove limit and obtain a master equation (Markov dynamics) for the evolution of $\rho (t)$ on an appropriate time scale.