  In this work, we construct an alternative formulation to the traditional
Algebraic Bethe ansatz for quantum integrable models derived from a generalised
rational Gaudin algebra realised in terms of a collection of spins 1/2 coupled
to a single bosonic mode. The ensemble of resulting models which we call
Dicke-Jaynes-Cummings- Gaudin models are particularly relevant for the
description of light-matter interaction in the context of quantum optics.
Having two distinct ways to write any eigenstate of these models we then
combine them in order to write overlaps and form factors of local operators in
terms of partition functions with domain wall boundary conditions. We also
demonstrate that they can all be written in terms of determinants of matrices
whose entries only depend on the eigenvalues of the conserved charges. Since
these eigenvalues obey a much simpler set of quadratic Bethe equations, the
resulting expressions could then offer important simplifications for the
numerical treatment of these models.
