Wannier states and Bose-Hubbard parameters for 2D optical lattices

We consider the physical implementation of a 2D optical lattice with schemes involving 3 and 4 light fields. We illustrate the wide range of geometries available to the 3 beam lattice, and compare the general potential properties of the two lattice schemes. Numerically calculating the band structure we obtain the Wannier states and evaluate the parameters of the Bose-Hubbard models relevant to these lattices. Using these results we demonstrate lattices that realize Bose-Hubbard models with 2, 4, or 6 nearest neighbors, and quantify the extent that these different lattices effect the superfluid to Mott-insulator transition.