Spinor Bose condensate and $\mathrm{su}(1,1)$ Richardson model

A dilute gas of trapped bosonic atoms with hyperfine spin $F=1$ interacting with the classical laser field is considered. In the single mode approximation the Hamiltonian of the model may be expressed in terms of the boson realization of the $\mathrm{su}(1,1)$ algebra. The model is solved by the Algebraic Bethe Ansatz approach.