Impurities in a one-dimensional Bose gas: the flow equation approach

TL;DR

This paper extends the flow equation method to compute various observables in one-dimensional Bose gases with impurities, validating mean-field approximations and analyzing impurity interactions.

Contribution

It introduces the use of flow equations for calculating densities and phase fluctuations, and compares mean-field results with flow equation outcomes for impurity interactions.

Findings

Mean-field approximation is accurate when phase coherence length exceeds healing length.

Flow equations validate mean-field results for single mobile impurity.

Impurity-impurity interactions are accurately captured by flow equations, revealing limitations of perturbation theory.

Abstract

A few years ago, flow equations were introduced as a technique for calculating the ground-state energies of cold Bose gases with and without impurities. In this paper, we extend this approach to compute observables other than the energy. As an example, we calculate the densities, and phase fluctuations of one-dimensional Bose gases with one and two impurities. For a single mobile impurity, we use flow equations to validate the mean-field results obtained upon the Lee-Low-Pines transformation. We show that the mean-field approximation is accurate for all values of the boson-impurity interaction strength as long as the phase coherence length is much larger than the healing length of the condensate. For two static impurities, we calculate impurity-impurity interactions induced by the Bose gas. We find that leading order perturbation theory fails when boson-impurity interactions are stronger than boson-boson interactions. The mean-field approximation reproduces the flow equation results for all values of the boson-impurity interaction strength as long as boson-boson interactions are weak.