# Master equation for non-Markovian quantum Brownian motion: The emergence of lateral coherences

## TL;DR

This paper develops a non-Markovian quantum master equation for Brownian motion, revealing transient lateral coherences and negative entropy production, advancing understanding of non-Markovian effects in open quantum systems.

## Contribution

It introduces a high-temperature non-Markovian extension of the Caldeira-Leggett model with an analytical solution for a free particle, highlighting the emergence of lateral coherences.

## Key findings

- Transient lateral coherences observed at intermediate times.
- Identification of negative entropy production rate as a non-Markovian signature.
- Analytical solution derived for the free particle case.

## Abstract

Understanding the behaviour of a quantum system coupled to its environment is of fundamental interest in the general field of quantum technologies. It also has important repercussions on foundational problems in physics, such as the process of decoherence and the so-called quantum measurement problem. There have been many approaches to explore Markovian and non-Markovian dynamics within the framework of open quantum systems, but the richness of the ensuing dynamics is still not fully understood. In this paper we develop a non-Markovian extension of the standard Caldeira-Leggett model, based on expanding the dynamics of the reduced system at high temperature in inverse powers of the high-frequency cut-off of the Ohmic spectral density of the environment, and derive a non-Markovian master equation for the reduced density matrix for the case of a general potential. We also obtain a fully analytical solution in the free particle case. While the short-time behaviour of this solution does not diverge substantially from the Markovian behaviour, at intermediate times we find a resurgence of coherence, which we name lateral coherence. We identify this with a corresponding transient negative entropy production rate, which is understood to be characteristic of non-Markovian dynamics. We also analyze the positivity of the reduced density matrix, and derive the corresponding Fokker-Planck equation in the classical limit.