  Hybrid quantum systems consisting of an ensemble of two--level systems
interacting with a single--mode electromagnetic field are important for the
development of quantum information processors and other quantum devices. These
systems are characterized by the set of energy level hybridizations, split by
collective Lamb shifts, that occur when the ensemble and field mode interact
coherently with high cooperativity. Computing the full set of Lamb shifts is
generally intractable given the high dimensionality of many devices. In this
work, we present a set of techniques that allow a compact description of the
Lamb shift statistics across all collective angular momentum subspaces of the
ensemble without using restrictive approximations on the state space. We use
these techniques to both analyze the Lamb shift in all subspaces and excitation
manifolds and to describe the average observed Lamb shift weighted over the
degeneracies of all subspaces.
