Exact quantum phase model for mesoscopic Josephson junctions

Starting from the two-mode Bose-Hubbard model, we derive an exact version of the standard Mathieu equation governing the wave function of a Josephson junction. For a finite number of particles N, we find an additional $\cos 2\phi$ term in the potential. We also find that the inner product in this representation is nonlocal in $\phi .$ Our model exhibits phenomena, such as $\pi$ oscillations, which are not found in the standard phase model, but have been predicted from Gross-Pitaevskii mean-field theory.