# Bose polaron as an instance of quantum Brownian motion

## TL;DR

This paper models a Bose polaron as an open quantum system using quantum Brownian motion, deriving a generalized Langevin equation to analyze impurity dynamics, including super-diffusion and position squeezing, with implications for experiments.

## Contribution

It introduces a generalized Langevin equation framework for Bose polarons, capturing memory effects and noise, and predicts phenomena like super-diffusion and position squeezing.

## Key findings

- Impurity exhibits super-diffusive behavior at long times.
- Back-flow of energy occurs between impurity and environment.
- Position squeezing is observed for trapped impurities.

## Abstract

We study the dynamics of a quantum impurity immersed in a Bose-Einstein condensate as an open quantum system in the framework of the quantum Brownian motion model. We derive a generalized Langevin equation for the position of the impurity. The Langevin equation is an integrodifferential equation that contains a memory kernel and is driven by a colored noise. These result from considering the environment as given by the degrees of freedom of the quantum gas, and thus depend on its parameters, e.g. interaction strength between the bosons, temperature, etc. We study the role of the memory on the dynamics of the impurity. When the impurity is untrapped, we find that it exhibits a super-diffusive behavior at long times. We find that back-flow in energy between the environment and the impurity occurs during evolution. When the particle is trapped, we calculate the variance of the position and momentum to determine how they compare with the Heisenberg limit. One important result of this paper is that we find position squeezing for the trapped impurity at long times. We determine the regime of validity of our model and the parameters in which these effects can be observed in realistic experiments.