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Self-consistent approach for Bose-condensed atoms in optical lattices

Bose atoms in optical lattices are considered at low temperatures and weak interactions, when Bose-Einstein condensate is formed. A self-consistent approach, based on the use of a representative statistical ensemble, is employed, ensuring a gapless spectrum of collective excitations and the validity of conservation laws. In order to show that the approach is applicable to both weak and tight binding, the problem is treated in the Bloch as well as in the Wannier representations. Both these ways result in similar expressions that are compared for the self-consistent Hartree-Fock-Bogolubov approximation. A convenient general formula for the superfluid fraction of atoms in an optical lattice is derived.