Ultracold atoms confined in an optical lattice plus parabolic potential: a closed-form approach

TL;DR

This paper provides an exact analytical solution for non-interacting atoms in an optical lattice with a parabolic trap, and explores collective oscillations, fermionization, and phase transitions in interacting systems.

Contribution

It introduces a closed-form Mathieu function approach for the non-interacting problem and combines it with numerical methods for interacting systems, advancing understanding of ultracold atoms in combined potentials.

Findings

Exact solutions for non-interacting systems using Mathieu functions.

Analytic and numerical analysis of collective oscillations.

Insights into fermionization and superfluid-Mott insulator transition.

Abstract

We discuss interacting and non-interacting one dimensional atomic systems trapped in an optical lattice plus a parabolic potential. We show that, in the tight-binding approximation, the non-interacting problem is exactly solvable in terms of Mathieu functions. We use the analytic solutions to study the collective oscillations of ideal bosonic and fermionic ensembles induced by small displacements of the parabolic potential. We treat the interacting boson problem by numerical diagonalization of the Bose-Hubbard Hamiltonian. From analysis of the dependence upon lattice depth of the low-energy excitation spectrum of the interacting system, we consider the problems of "fermionization" of a Bose gas, and the superfluid-Mott insulator transition. The spectrum of the noninteracting system turns out to provide a useful guide to understanding the collective oscillations of the interacting system, throughout a large and experimentally relevant parameter regime.