Wannier states and Bose Hubbard parameters for 2D optical lattices

We consider the physical implementation of a 2D optical lattice with schemes involving three and four 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 neighbours, and quantify the extent that these different lattices affect the superfluid to Mott-insulator transition.