  We consider an important generalization of the Dicke model in which
multi-level atoms, instead of two-level atoms as in conventional Dicke model,
interact with a single photonic mode. We explore the phase diagram of a broad
class of atom-photon coupling schemes and show that, under this generalization,
the Dicke model can become multicritical. For a subclass of experimentally
realizable schemes, multicritical conditions of arbitrary order can be
expressed analytically in compact forms. We also calculate the atom-photon
entanglement entropy for both critical and non-critical cases. We find that the
order of the criticality strongly affects the critical entanglement entropy:
higher order yields stronger entanglement. Our work provides deep insight into
quantum phase transitions and multicriticality.
