Mediated interaction between polarons in a one-dimensional Bose gas

TL;DR

This paper analytically investigates the induced interactions between impurities in a one-dimensional Bose gas, revealing how these interactions change with coupling strength and distance, and exploring the formation of bipolarons.

Contribution

It provides a comprehensive analytical study of impurity interactions in 1D Bose gases, including quantum fluctuation effects and their impact on bipolaron formation.

Findings

Interaction is mean-field-like at short distances and exponential decay at long distances.

Quantum fluctuations induce a long-range $1/r^3$ interaction at large distances.

At strong coupling, the interaction decays as $1/r$, indicating a crossover in behavior.

Abstract

We study a weakly-interacting one-dimensional Bose gas with two impurities coupled locally to the boson density. We derive analytical results for the induced interaction between the impurities at arbitrary coupling and separation $r$. At $r\lesssim \xi$, where $\xi$ denotes the healing length of the Bose gas, the interaction is well described by the mean-field contribution. Its form changes as the coupling is increased, approaching a linear function of $r$ at short distances in the regime of strong coupling. The mean-field contribution decays exponentially at arbitrary coupling for $r\gg\xi$. At such long distances, however, the effect of quantum fluctuations becomes important, giving rise to a long-ranged quantum contribution to the induced interaction. At longest distances it behaves as $1/r^3$, while at strong coupling we find an intermediate distance regime with a slower decay, $1/r$. The quantum contribution in the crossover regime is also calculated. The induced interaction between impurities (i.e., polarons) is attractive and leads to the formation of their bound state, known as bipolaron. We discuss its binding energy.