Single-atom density of states of an optical lattice

TL;DR

This paper investigates how a harmonic trap affects the density of states of a single atom in an optical lattice, revealing a singular limit as the trap strength approaches zero, which differs from the translationally invariant case.

Contribution

It demonstrates that the limit of vanishing trap strength is singular, leading to qualitatively different density of states compared to an untrapped lattice.

Findings

Density of states differs qualitatively in the shallow trap limit

Wave functions are affected by the trap strength

The 5 0 limit is singular and non-trivial

Abstract

We consider a single atom in an optical lattice, subject to a harmonic trapping potential. The problem is treated in the tight-binding approximation, with an extra parameter \kappa denoting the strength of the harmonic trap. It is shown that the \kappa \to 0 limit of this problem is singular, in the sense that the density of states for a very shallow trap (\kappa \to 0) is {\it qualitatively different} from that of a translationally invariant lattice (\kappa = 0). The physics of this difference is discussed, and densities of states and wave functions are exhibited and explained.