  The Tavis-Cummings model is a paradigmatic central-mode model where a set of
two-level quantum emitters (spins) are coupled to a collective cavity mode.
Here we study the eigenstate spectrum, its localization properties and the
effect on dynamics, focusing on the two-excitation sector relevant for
nonlinear photonics. These models admit two sources of disorder: in the
coupling between the spins and the cavity and in the energy shifts of the
individual spins. While this model was known to be exactly solvable in the
limit of a homogeneous coupling and inhomogeneous energy shifts, we here
establish the solvability in the opposite limit of a homogeneous energy shift
and inhomogeneous coupling, presenting the exact solution and corresponding
conserved quantities. We identify three different classes of eigenstates,
exhibiting different degrees of multifractality and semilocalization closely
tied to the integrable points, and study their stability to perturbations away
from these solvable points. The dynamics of the cavity occupation number away
from equilibrium, exhibiting boson bunching and a two-photon blockade, is
explicitly related to the localization properties of the eigenstates and
illustrates how these models support a collective spin description despite the
presence of disorder.
