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Superradiant phase transitions with three-level systems

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.