  For systems of interacting, ultracold spin-zero neutral bosonic atoms,
harmonically trapped and subject to an optical lattice potential, we derive an
Extended Bose Hubbard (EBH) model by developing a systematic expansion for the
Hamiltonian of the system in powers of the lattice parameters and of a scale
parameter, the {\it lattice attenuation factor}. We identify the dominant terms
that need to be retained in realistic experimental conditions, up to
nearest-neighbor interactions and nearest-neighbor hoppings conditioned by the
on site occupation numbers. In mean field approximation, we determine the free
energy of the system and study the phase diagram both at zero and at finite
temperature. At variance with the standard on site Bose Hubbard model, the zero
temperature phase diagram of the EBH model possesses a dual structure in the
Mott insulating regime. Namely, for specific ranges of the lattice parameters,
a density wave phase characterizes the system at integer fillings, with domains
of alternating mean occupation numbers that are the atomic counterparts of the
domains of staggered magnetizations in an antiferromagnetic phase. We show as
well that in the EBH model, a zero-temperature quantum phase transition to pair
superfluidity is in principle possible, but completely suppressed at lowest
order in the lattice attenuation factor. Finally, we determine the possible
occurrence of the different phases as a function of the experimentally
controllable lattice parameters.
