Bose-Hubbard model with attractive interactions

We consider the Bose-Hubbard model of atoms in an optical lattice potential when the atom-atom interactions are attractive. If the lowest energy lattice sites are degenerate (such as in the homogeneous case), then, at a critical value of the interaction strength, a phase-coherent condensate becomes unstable to a quantum superposition such that the number distribution of each of the degenerate sites becomes double peaked. In the limit when the interaction dominates, the superposition becomes macroscopic and has the form $|\psi>\propto\sum_{j} e^{i\phi_{j}}\hat{b}^{\dagger N}_{j}|{\rm vac}>$, where $N$ is the total number of atoms and the sum ranges over the energy-degenerate sites.