Density profile of a semi-infinite one-dimensional Bose gas and bound states of the impurity

TL;DR

This paper analyzes how a boundary affects a one-dimensional Bose gas's density profile and explores impurity localization, revealing quantum fluctuations' significant role and a Casimir-like interaction.

Contribution

It provides an analytic density profile near the boundary and exact results for impurity bound states, including quantum fluctuation effects and long-range interactions.

Findings

Density is suppressed at the boundary and recovers with inverse square law.

Exact bound state energies and wave functions are derived at the mean-field level.

Quantum fluctuations induce small corrections and enable Casimir-like interactions.

Abstract

We study the effect of the boundary on a system of weakly interacting bosons in one dimension. It strongly influences the boson density which is completely suppressed at the boundary position. Away from it, the density is depleted over the distances on the order of the healing length at the mean-field level. Quantum fluctuations modify the density profile considerably. The local density approaches the average one as an inverse square of the distance from the boundary. We calculate an analytic expression for the density profile at arbitrary separations from the boundary. We then consider the problem of localization of a foreign quantum particle (impurity) in the potential created by the inhomogeneous boson density. At the mean-field level, we find exact results for the energy spectrum of the bound states, the corresponding wave functions, and the condition for interaction-induced localization. The quantum contribution to the boson density gives rise to small corrections of the bound state energy levels. However, it is fundamentally important for the existence of a long-range Casimir-like interaction between the impurity and the boundary.