The two-site Bose--Hubbard model

The two-site Bose--Hubbard model is a simple model used to study Josephson tunneling between two Bose--Einstein condensates. In this work we give an overview of some mathematical aspects of this model. Using a classical analysis, we study the equations of motion and the level curves of the Hamiltonian. Then, the quantum dynamics of the model is investigated using direct diagonalisation of the Hamiltonian. In both of these analyses, the existence of a threshold coupling between a delocalised and a self-trapped phase is evident, in qualitative agreement with experiments. We end with a discussion of the exact solvability of the model via the algebraic Bethe ansatz.