Wannier functions using a discrete variable representation for optical lattices
Saurabh Paul, Eite Tiesinga
We propose a numerical method using the discrete variable representation (DVR) for constructing real-valued approximate 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 for a symmetric lattice with in version symmetry, we construct Wannier functions as eigen states of the position operators 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.