@article {1823,
title = {Wannier functions using a discrete variable representation for optical lattices},
journal = {Physical Review A},
volume = {94},
year = {2016},
month = {2016/09/07},
pages = {033606},
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 S. Kivelson [Phys. Rev. B\ 26, 4269 (1982)], for a symmetric lattice with inversion symmetry, we construct Wannier functions as eigenstates of the position operators\ x\ˆ,\ y\ˆ, and\ z\ˆ\ 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 eigenstates 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 10 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 model and show the dependence of these energies on lattice asymmetry.

},
doi = {http://dx.doi.org/10.1103/PhysRevA.94.033606},
url = {http://journals.aps.org/pra/abstract/10.1103/PhysRevA.94.033606},
author = {Saurabh Paul and Eite Tiesinga}
}