  We study inter-well coherent quantum tunneling of Bose-Einstein condensates
(BEC's) in optical lattices. Irrespective of whether the BEC is a superfluid in
the whole lattice, it may or may not be a superfluid within each well of the
lattice depending on whether the ratio of the well kinetic energy to the
interaction energy is small or large. When the bosons in each well are
superfluid, we have Josephson tunneling among the wells driven by the phase
difference between the superfluids in the different wells. For typical lattice
parameters the Josephson tunneling rate will be somewhat enhanced from the
naive estimate of (tunneling energy scale)$/\hbar$. Whether it is enhanced or
not will depend on lattice parameters, but it can never be zero. When the
bosons in each well are not superfluid, there is coherent quantum tunneling of
essentially free bosons between the wells. This tunneling is proportional to
the boson number difference between the wells.
