Bound states of an ultracold atom interacting with a set of stationary impurities

TL;DR

This paper investigates the bound states of an ultracold atom interacting with static impurities, providing analytical and numerical insights into their energy spectra and band structures, relevant for quantum simulation and condensed matter analogs.

Contribution

It introduces analytical solutions for bound states with delta interactions and numerical methods for polarization potentials, extending understanding of atom-impurity systems in 1D.

Findings

Analytical bound state energies for delta potential interactions.

Numerical band structure calculations for polarization potentials.

Good agreement between analytical and numerical results at large impurity separations.

Abstract

In this manuscript we analyse properties of bound states of an atom interacting with a set of static impurities. We begin with the simplest system of a single atom interacting with two static impurities. We consider two types of atom-impurity interaction: (i) zero-range potential represented by regularized delta, (ii) more realistic polarization potential, representing long-range part of the atom-ion interaction. For the former we obtain analytical results for energies of bound states. For the latter we perform numerical calculations based on the application of finite element method. Then, we move to the case of a single atom interacting with one-dimensional (1D) infinite chain of static ions. Such a setup resembles Kronig-Penney model of a 1D crystalline solid, where energy spectrum exhibits band structure behaviour. For this system, we derive analytical results for the band structure of bound states assuming regularized delta interaction, and perform numerical calculations, considering polarization potential to model atom-impurity interaction. Both approaches agree quite well when separation between impurities is much larger than characteristic range of the interaction potential.