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Starting from integrable $su(2)$ (quasi-)spin Richardson-Gaudin XXZ models wederive several properties of integrable spin models coupled to a bosonic mode.We focus on the Dicke-Jaynes-Cummings-Gaudin models and the two-channel$(p+ip)$-wave pairing Hamiltonian. The pseudo-deformation of the underlying$su(2)$ algebra is here introduced as a way to obtain these models in thecontraction limit of different Richardson-Gaudin models. This allows for theconstruction of the full set of conserved charges, the Bethe Ansatz state, andthe resulting Richardson-Gaudin equations. For these models an alternative andsimpler set of quadratic equations can be found in terms of the eigenvalues ofthe conserved charges. Furthermore, the recently proposed eigenvalue-baseddeterminant expressions for the overlaps and form factors of local operatorsare extended to these models, linking the results previously presented for theDicke-Jaynes-Cummings-Gaudin models with the general results forRichardson-Gaudin XXZ models.

Eigenvalue-based determinants for scalar products and form factors in Richardson-Gaudin integrable models coupled to a bosonic mode