  We determine the phase diagram of $N$ identical three-level systems
interacting with a single photonic mode in the thermodynamical limit ($N \to
\infty$) by accounting for the so-called diamagnetic term and the inequalities
imposed by the Thomas-Reich-Kuhn (TRK) oscillator strength sum rule. The key
role of transitions between excited levels and the occurrence of first-order
phase transitions is discussed. We show that, in contrast to two-level systems,
in the three-level case the TRK inequalities do not always prevent a
superradiant phase transition in presence of a diamagnetic term.

Quantum Physics;Optics