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We construct a non-perturbative approach based on quantum averaging combinedwith resonant transformations to detect the resonances of a given Hamiltonianand to treat them. This approach, that generalizes the rotating-waveapproximation, takes into account the resonances at low field and also at highfield (non-linear resonances). This allows to derive effective Hamiltoniansthat contain the qualitative features of the spectrum, i.e. crossings andavoided crossings, as a function of the coupling constant. At a second stagethe precision of the spectrum can be improved quantitatively by standardperturbative methods like contact transformations. We illustrate this method todetermine the spectrum of a two-level atom interacting with a single mode of aquantized field.

Quantum averaging and resonances: Two-level atom in a one-mode quantized field