The Quantum Josephson Hamiltonian In The Phase Representation

The quantum Josephson Hamiltonian of two weakly linked Bose-Einstein condensates is written in an overcomplete phase representation, thus avoiding the problem of defining a Hermitian phase operator. We discuss the limit of validity of the standard, non-rigorous Mathieu equation, due to the onset of a higher order $\cos 2\varphi$ term in the Josephson potential, and also to the overcompleteness of the representation (the phase $\varphi$ being the relative phase between the two condensates). We thereby unify the Boson Hubbard and Quantum Phase models.