  The Mott insulator-superfluid transition for ultracold bosonic atoms in an
optical lattice has been extensively studied in the framework of the
Bose-Hubbard model with two-body on-site interactions. In this paper, we
analyze the additional effect of the three-body on-site interactions on this
phase transition in optical lattice and the transitions between the various
phases that arise in an optical superlattice. Using the mean-field theory and
the density matrix renormalization group method, we find the phase diagrams
depicting the relationships between various physical quantities in an optical
lattice and superlattice. We also suggest possible experimental signatures to
observe the three-body interactions.
