Ab-initio determination of Bose-Hubbard parameters for two ultracold atoms in an optical lattice

We calculate numerically the exact energy spectrum of the six-dimensional problem of two interacting Bosons in a three-well optical lattice. The particles interact via a full Born-Oppenheimer potential which can be adapted to model the behavior of the s-wave scattering length at Feshbach resonances. By adjusting the parameters of the corresponding Bose-Hubbard (BH) Hamiltonian the deviation between the numerical energy spectrum and the BH spectrum is minimized. This defines the optimal BH parameter set which we compare to the standard parameters of the BH model. The range of validity of the BH model with these parameter sets is examined, and an improved analytical prediction of the interaction parameter is introduced. Furthermore, the extended BH model and implications due to the energy dependence of the scattering length and couplings to higher Bloch bands at a Feshbach resonance are discussed.