Stochastic Representation of a Class of Non-Markovian Completely Positive Evolutions

TL;DR

This paper introduces a stochastic framework for modeling non-Markovian quantum evolutions, ensuring complete positivity, and explores how different renewal process statistics influence decay behaviors in open quantum systems.

Contribution

It generalizes classical continuous time random walks to quantum systems, deriving non-Markovian master equations with guaranteed complete positivity and analyzing their dynamics.

Findings

Non-exponential decay behaviors can be modeled.

Explicit solutions for simple systems are provided.

Positivity issues in memory-affected quantum master equations are clarified.

Abstract

By modeling the interaction of an open quantum system with its environment through a natural generalization of the classical concept of continuous time random walk, we derive and characterize a class of non-Markovian master equations whose solution is a completely positive map. The structure of these master equations is associated with a random renewal process where each event consist in the application of a superoperator over a density matrix. Strong non-exponential decay arise by choosing different statistics of the renewal process. As examples we analyze the stochastic and averaged dynamics of simple systems that admit an analytical solution. The problem of positivity in quantum master equations induced by memory effects [S.M. Barnett and S. Stenholm, Phys. Rev. A 64, 033808 (2001)] is clarified in this context.