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For systems of interacting, ultracold spin-zero neutral bosonic atoms,harmonically trapped and subject to an optical lattice potential, we derive anExtended Bose Hubbard (EBH) model by developing a systematic expansion for theHamiltonian of the system in powers of the lattice parameters and of a scaleparameter, the {\it lattice attenuation factor}. We identify the dominant termsthat need to be retained in realistic experimental conditions, up tonearest-neighbor interactions and nearest-neighbor hoppings conditioned by theon site occupation numbers. In mean field approximation, we determine the freeenergy of the system and study the phase diagram both at zero and at finitetemperature. At variance with the standard on site Bose Hubbard model, the zerotemperature phase diagram of the EBH model possesses a dual structure in theMott insulating regime. Namely, for specific ranges of the lattice parameters,a density wave phase characterizes the system at integer fillings, with domainsof alternating mean occupation numbers that are the atomic counterparts of thedomains of staggered magnetizations in an antiferromagnetic phase. We show aswell that in the EBH model, a zero-temperature quantum phase transition to pairsuperfluidity is in principle possible, but completely suppressed at lowestorder in the lattice attenuation factor. Finally, we determine the possibleoccurrence of the different phases as a function of the experimentallycontrollable lattice parameters.Comment: 18 pages, 7 figures, accepted for publication in Phys. Rev.

Extended Bose Hubbard model of interacting bosonic atoms in optical lattices: From superfluidity to density waves