Generalised Kronig-Penney model for ultracold atomic quantum systems

TL;DR

This paper generalizes the Kronig-Penney model to include both even and odd scattering waves, providing analytical tools to study ultracold atomic systems interacting with structured potentials, with applications to atom-ion and neutral atom systems.

Contribution

It introduces a comprehensive analytical model extending the Kronig-Penney framework to complex scattering scenarios in ultracold atomic physics.

Findings

Analytical dispersion relation derived for generalized model

Excellent agreement with quantum defect theory for atom-ion systems

Derived Bose-Hubbard Hamiltonian for Bose gases in ion chains

Abstract

We study the properties of a quantum particle interacting with a one dimensional structure of equidistant scattering centres. We derive an analytical expression for the dispersion relation and for the Bloch functions in the presence of both even and odd scattering waves within the pseudopotential approximation. This generalises the well-known solid-state physics text-book result known as the Kronig-Penney model. Our generalised model can be used to describe systems such as degenerate Fermi gases interacting with ions or with another neutral atomic species confined in an optical lattice, thus enabling the investigation of polaron or Kondo physics within a simple formalism. We focus our attention on the specific atom-ion system and compare our findings with quantum defect theory. Excellent agreement is obtained within the regime of validity of the pseudopotential approximation. This enables us to derive a Bose-Hubbard Hamiltonian for a degenerate quantum Bose gas in a linear chain of ions.