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We study a two-level system (atom, superconducting qubit or quantum dot)strongly coupled to the single photonic mode of a cavity, in the presence ofincoherent pumping and including detuning and dephasing. This system displays astriking quantum to classical transition. On the grounds of severalapproximations that reproduce to various degrees exact results obtainednumerically, we separate five regimes of operations, that we term "linear","quantum", "lasing", "quenching" and "thermal". In the fully quantized picture,the lasing regime arises as a condensation of dressed states and manifestsitself as a Mollow triplet structure in the direct emitter photoluminescencespectrum, which embeds fundamental features of the full-field quantizationdescription of light-matter interactions.

Regimes of strong light-matter coupling under incoherent excitation