Entanglement via a three-well Bose-Hubbard system and via an optical beamsplitter

We compare and contrast the entangling properties of a three-well Bose-Hubbard model and an optical beamsplitter. The coupling between the different modes is linear in both cases, and we may identify two output modes. Obvious differences are that our Bose-Hubbard model, with only the middle well initially occupied, does not have a vacuum input port, there is no equivalent of a collisional, $\chi^{(3)}$ nonlinearity with the beamsplitter, and the results of the Bose-Hubbard model show a time-dependence. In the non-interacting case, we obtain analytic solutions and show that, like a beamsplitter, the Bose-Hubbard system will not produce entanglement for classical initial states. We also show that whether inseparability or entanglement are detected depends sensitively on the criteria measured, with different criteria giving contradictory predictions.