13859nas a2200145 45000080041000002450084000412100069001252600015001943000011002094900007002205201338100227100001813608700001913626856006813645 2016 eng d00aWannier functions using a discrete variable representation for optical lattices0 aWannier functions using a discrete variable representation for o c2016/09/07 a0336060 v943 aWe 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 xˆ, yˆ, and zˆ 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.

1 aPaul, Saurabh1 aTiesinga, Eite uhttp://journals.aps.org/pra/abstract/10.1103/PhysRevA.94.033606