Coalescence of Two Impurities in a Trapped One-dimensional Bose Gas

TL;DR

This paper investigates the ground state properties of a 1D trapped Bose gas with two impurities, revealing impurity clustering and comparing effective potentials, advancing understanding of multi-impurity effects in cold-atom systems.

Contribution

The authors develop a variational method validated against exact results, providing new insights into impurity clustering and impurity-impurity interactions in 1D Bose gases beyond the polaron model.

Findings

Bosonic non-interacting impurities tend to cluster.

Induced impurity-impurity potentials are calculated in a harmonic trap.

Comparison of mean-field and phonon-exchange potentials reveals differences.

Abstract

We study the ground state of a one-dimensional (1D) trapped Bose gas with two mobile impurity particles. To investigate this set-up, we develop a variational procedure in which the coordinates of the impurity particles are slow-like variables. We validate our method using the exact results obtained for small systems. Then, we discuss energies and pair densities for systems that contain of the order of one hundred atoms. We show that bosonic non-interacting impurities cluster. To explain this clustering, we calculate and discuss induced impurity-impurity potentials in a harmonic trap. Further, we compute the force between static impurities in a ring ({\it {\`a} la} the Casimir force), and contrast the two effective potentials: the one obtained from the mean-field approximation, and the one due to the one-phonon exchange. Our formalism and findings are important for understanding (beyond the polaron model) the physics of modern 1D cold-atom systems with more than one impurity.