  We try to classify the spectrum of the two-qubit Dicke model by calculating
two quantum information measures of its eigenstates: the Wooters concurrence
and the mutual quantum information. We are able to detect four spectral sets in
each parity subspace of the model: one set is regular and given by the product
of a Fock state of the field times the singlet Bell state of the qubits; the
rest are fairly regular and related to the triplet states of the Bell basis.
The singlet states become trapping states when we couple the Dicke model to an
environment of harmonic oscillators, making them candidates for generating
maximally entangled states in experimental realizations of ion trap quantum
electrodynamics (QED) and circuit QED. Furthermore, they are robust and survive
the inclusion of driving and dipole-dipole interactions, pointing to their use
for storing quantum correlations, and it is straightforward to provide a
generalization of these trapping states to the Dicke model with even number of
qubits.
