  We investigate how entanglement spreads along small Bose-Hubbard chains, with
only the first well initially occupied by a mesoscopic number of atoms, as the
number of sites increases. For two- and three-well chains in the
non-interacting case, we are able to obtain analytical solutions and show that
the presence of entanglement depends on having a sub-Poissonian state of the
atoms in the first well. In these cases, the correlations we calculate are
completely periodic. Restoring the collisional interactions or moving to a
four-well chain necessitates a numerical treatment, for which we use the fully
quantum positive-P representation. We examine two different correlations and
find that adding collisional interactions destroys the periodicity of the
correlations and causes them to degrade with time. This happens well before
there is a noticeable effect on the periodicity of the solutions for the number
of atoms in each well.
