Analytical study of a finite-range impurity in a one-dimensional Bose gas

TL;DR

This paper analytically investigates a finite-range impurity in a one-dimensional Bose gas, revealing finite polaron energy and mass in strong coupling, contrasting with contact potential models, and characterizing properties via the impurity interaction range.

Contribution

It introduces an analytical solution for a finite-range impurity in a 1D Bose gas, improving understanding of polaron properties beyond contact potential approximations.

Findings

Polaron energy and effective mass remain finite at strong coupling.

Polaron properties scale with the ratio of interaction range to coherence length.

Attractive and repulsive polarons exhibit distinct scaling behaviors.

Abstract

One-dimensional Bose gases present an interesting setting to study the physics of Bose polarons, as density fluctuations play an enhanced role due to reduced dimensionality. Theoretical descriptions of this system have predominantly relied on contact pseudopotentials to model the impurity-bath interaction, leading to unphysical results in the strongly coupled limit. In this work, we analytically solve the Gross-Pitaevskii equation, using a square well potential instead of a zero-range potential, for the ground-state wave function of a static impurity. We compute perturbative corrections arising from infinitesimally slow impurity motion. The polaron energy and effective mass remain finite in the strongly coupled regime, in contrast to the divergent behavior obtained using a contact potential. In this limit, we characterize the polaron properties in terms of the dimensionless ratio $\bar{w}\equiv w/\xi$ between the interaction range $w$ of the impurity-bath potential and the coherence length $\xi$ of the Bose gas. The effective mass exhibits a $1/\bar{w}$ scaling. The energy of the attractive polaron scales as $-1/\bar{w}^3$, whereas the repulsive polaron features subleading corrections to the dark soliton energy at the order $\bar{w}^3$.