  The extended Bose-Hubbard model for a double-well potential with atom-pair
tunneling is studied. Starting with a classical analysis we determine the
existence of three different quantum phases: self-trapping, phase-locking and
Josephson states. From this analysis we built the parameter space of quantum
phase transitions between degenerate and non-degenerate ground states driven by
the atom-pair tunneling. Considering only the repulsive case, we confirm the
phase transition by the measure of the energy gap between the ground state and
the first excited state. We study the structure of the solutions of the Bethe
ansatz equations for a small number of particles. An inspection of the roots
for the ground state suggests a relationship to the physical properties of the
system. By studying the energy gap we find that the profile of the roots of the
Bethe ansatz equations is related to a quantum phase transition.
