Paired phases and Bose-Einstein condensation of spin-one bosons with attractive interaction

We analyze paired phases of cold bosonic atoms with spin $S=1$ and with an attractive interaction. We derive mean-field self-consistent equations for the matrix order parameter describing such paired bosons on an optical lattice. The possible solutions are classified according to their symmetries. In particular, we find that the self-consistent equations for the SO(3) symmetric phase are of the same form as those for scalar bosons with attractive interactions. This singlet phase may exhibit either the BCS-type pairing instability (BCS phase) or the Bose-Einstein condensation (BEC) quasiparticle condensation together with the BCS-type pairing (BEC phase) for an arbitrary attraction ${U}_{0}$ in the singlet channel of the two-body interaction. We show that both condensate phases become stable if a repulsion ${U}_{2}$ in the quintet channel is above a critical value, which depends on ${U}_{0}$ and other thermodynamic parameters.