# Momentum coupling in non-Markovian Quantum Brownian motion

## TL;DR

This paper analyzes a non-Markovian quantum Brownian motion model with position and momentum coupling, deriving the master equation, solving for Gaussian states, and showing how momentum coupling affects dissipation and non-Markovianity.

## Contribution

It introduces a detailed model of non-Markovian quantum Brownian motion with momentum coupling and derives the corresponding master equation and dynamical properties.

## Key findings

- Momentum coupling enhances dissipation and accelerates relaxation.
- The asymptotic state is modified by momentum coupling.
- Non-Markovianity is characterized via positivity of master equation coefficients.

## Abstract

We consider a model of non-Markovian Quantum Brownian motion that consists of an harmonic oscillator bilinearly coupled to a thermal bath, both via its position and momentum operators. We derive the master equation for such a model and we solve the equations of motion for a generic Gaussian system state. We then investigate the resulting evolution of the first and second moments for both an Ohmic and a super-Ohmic spectral density. In particular, we show that, irrespective of the specific form of the spectral density, the coupling with the momentum enhances the dissipation experienced by the system, accelerating its relaxation to the equilibrium, as well as modifying the asymptotic state of the dynamics. Eventually, we characterize explicitly the non-Markovianity of the evolution, using a general criterion which relies on the positivity of the master equation coefficients.