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The objective of this paper is the theoretical description of theMott-insulator to superfluid quantum phase transition of a Bose gas in anoptical lattice. In former works the Rayleigh-Schr\"odinger perturbation theorywas used within a mean-field approach, which yields partially non-physicalresults since the degeneracy between two adjacent Mott lobes is not taken intoaccount. In order to correct such non-physical results we apply theBrillouin-Wigner perturbation theory to the mean-field approximation of theBose-Hubbard model. Detailed explanations of how to use the Brillouin-Wignertheory are presented, including a graphical approach that allows to efficientlykeep track of the respective analytic terms. To prove the validity of thiscomputation, the results are compared with other works. Besides the analyticcalculation of the phase boundary from Mott-insulator to superfluid phase, thecondensate density is also determined by simultaneously solving two algebraicequations. The analytical and numerical results turn out to be physicallymeaningful and can cover a region of system parameters inaccessible until now.Our results are of particular interest provided an harmonic trap is added tothe former calculations in an homogeneous system, in view of describing anexperiment within the local density approximation. Thus, the paper representsan essential preparatory work for determining the experimentally observedwedding-cake structure of particle-density profile at both finite temperatureand hopping.

Improving Mean-Field Theory for Bosons in Optical Lattices via Degenerate Perturbation Theory