Bose-Hubbard model with attractive interactions

TL;DR

This paper investigates the behavior of the Bose-Hubbard model with attractive interactions, revealing a phase transition to macroscopic superpositions of atoms across degenerate lattice sites.

Contribution

It introduces the analysis of attractive interactions in the Bose-Hubbard model, showing the emergence of macroscopic superpositions at a critical interaction strength.

Findings

Phase transition to superposition states at critical interaction strength

Macroscopic superposition form involving all degenerate sites

Instability of phase-coherent condensate under attractive interactions

Abstract

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.