Spreading of entanglement and steering along small Bose-Hubbard chains

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.