Continuous-variable approach to the spectral properties and quantum states of the two-component Bose-Hubbard dimer

A bosonic gas formed by two interacting species trapped in a double-well potential features macroscopic localization effects when the interspecies interaction becomes sufficiently strong. A repulsive interaction spatially separates the species into different wells while an attractive interaction confines both species in the same well. We perform a fully analytic study of the transitions from the weak- to the strong-interaction regime by exploiting the semiclassical method in which boson populations are represented in terms of continuous variables. We find an explicit description of low-energy eigenstates and spectrum in terms of the model parameters which includes the neighborhood of the transition point. To test the effectiveness of the continuous-variable method we compare its predictions with the exact results found numerically. Numerical calculations confirm the spectral collapse evidenced by this method when the space localization takes place.