Fluctuation-induced potential for an impurity in a semi-infinite one-dimensional Bose gas

TL;DR

This paper investigates the interaction potential for an impurity near the boundary of a semi-infinite one-dimensional Bose gas, revealing a universal $1/r^2$ decay at large distances and implications for impurity localization.

Contribution

It provides a detailed calculation of the impurity-wall interaction potential, including the crossover from exponential to universal power-law decay, and discusses impurity localization and Casimir-like effects.

Findings

Interaction potential is attractive for repulsive impurity-boson interactions.

Potential decays exponentially at short distances and as $1/r^2$ at large distances.

Universal $1/r^2$ behavior applies to slowly moving impurities.

Abstract

We consider an impurity in a semi-infinite one-dimensional system of weakly-interacting bosons. We calculate the interaction potential for the impurity due to the end of the system, i.e., the wall. For local repulsive (attractive) interaction between the impurity and the Bose gas, the interaction potential is attractive (repulsive). At short distances from the wall it decays exponentially crossing over into a universal $1/r^2$ behavior at separations $r$ above the healing length. Our results can also be interpreted as a Casimir-like interaction between two impurities, where one of them is infinitely strongly coupled to the Bose gas. We discuss various scenarios for the induced interaction between the impurities using the scattering approach. We finally address the phenomenon of localization of the impurity near the wall. In the paper we mainly study the case of a static impurity, however the universal $1/r^2$ interaction also holds for a slowly moving impurity.