Quantum averaging for driven systems with resonances

We discuss the effects of resonances in driven quantum systems within the context of quantum averaging techniques in the Floquet representation. We consider in particular iterative methods of KAM type and the extensions needed to take into account resonances. The approach consists in separating the coupling terms into resonant and nonresonant components at a given scale of time and intensity. The nonresonant part can be treated with perturbative techniques, which we formulate in terms of KAM-type unitary transformations that are close to the identity. These can be interpreted as averaging procedures with respect to the dynamics defined by effective uncoupled Hamiltonians. The resonant parts are treated with a different kind of unitary transfomations that are not close to the identity, and are adapted to the structure of the resonances. They can be interpreted as a renormalization of the uncoupled Hamiltonian, that yields an effective dressed Hamiltonian, around which a perturbation expansion can be developed. The combination of these two ingredients provides a strongly improved approximation technique.