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 of incoherent pumping and including detuning and dephasing. This system displays a striking quantum to classical transition. On the grounds of several approximations that reproduce to various degrees exact results obtained numerically, 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 manifests itself as a Mollow triplet structure in the direct emitter photoluminescence spectrum, which embeds fundamental features of the full-field quantization description of light-matter interactions.
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