Wannier functions using a discrete variable representation for optical lattices

TL;DR

This paper introduces a numerical DVR-based method for constructing real-valued Wannier functions in optical lattices, enabling accurate modeling of ultracold atoms and their interactions in symmetric and asymmetric potentials.

Contribution

The paper presents a novel DVR-based approach for constructing real-valued Wannier functions in optical lattices, applicable to both symmetric and asymmetric potentials, with high numerical accuracy.

Findings

Wannier functions constructed with better than ten significant digits accuracy.

Localized functions effectively used to compute two-body interaction energies.

Method applicable to complex lattice geometries with tunable asymmetry.

Abstract

We propose a numerical method using the discrete variable representation (DVR) for constructing real-valued Wannier functions localized in a unit cell for both symmetric and asymmetric periodic potentials. We apply these results to finding Wannier functions for ultracold atoms trapped in laser-generated optical lattices. Following Kivelson \cite{kivelson_wannier_1982}, for a symmetric lattice with inversion symmetry, we construct Wannier functions as eigen states of the position operators $\hat x$, $\hat y$ and $\hat z$ restricted to single-particle Bloch functions belonging to one or more bands. To ensure that the Wannier functions are real-valued, we numerically obtain the band structure and real-valued eigen states using a uniform Fourier grid DVR. We then show by a comparison of tunneling energies, that the Wannier functions are accurate for both inversion symmetric and asymmetric potentials to better than ten significant digits when using double-precision arithmetic. The calculations are performed for an optical lattice with double-wells per unit cell with tunable asymmetry along the $x$ axis and a single sinusoidal potential along the perpendicular directions. Localized functions at the two potential minima within each unit cell are similarly constructed, but using a superposition of single-particle solutions from the two lowest bands. We finally use these localized basis functions to determine the two-body interaction energies in the Bose-Hubbard (BH) model, and show the dependence of these energies on lattice asymmetry.