  Spectral characterization is a fundamental step in the development of useful
quantum technology platforms. Here, we study an ensemble of interacting qubits
coupled to a single quantized field mode, an extended Dicke model that might be
at the heart of Bose-Einstein condensate in a cavity or circuit-QED experiments
for large and small ensemble sizes, respectively. We present a semi-classical
and quantum analysis of the model. In the semi-classical regime, we show
analytic results that reveal the existence of a third regime, in addition of
the two characteristic of the standard Dicke model, characterized by one
logarithmic and two jump discontinuities in the derivative of the density of
states. We show that the finite quantum system shows two different types of
clustering at the jump discontinuities, signaling a precursor of two excited
quantum phase transitions. These are confirmed using Peres lattices where
unexpected order arises around the new precursor. Interestingly, Peres
conjecture regarding the relation between spectral characteristics of the
quantum model and the onset of chaos in its semi-classical equivalent is valid
in this model as a revival of order in the semi-classical dynamics occurs
around the new phase transition.
