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This article is concerned with a system of particles interacting with thequantized electromagnetic field (photons) in the non relativistic QuantumElectrodynamics (QED) framework and governed by the Pauli-Fierz Hamiltonian. Weare interested not only in deriving approximations of several quantities whenthe coupling constant is small but also in obtaining different controls of theerror terms. First, we investigate the time dynamics approximation in twosituations, the Markovian (Theorem 1.4 completed by Theorem 1.16) and nonMarkovian (Theorem 1.6) cases. These two contexts differ in particularregarding the approximation leading terms, the error control and the initialstates. Second, we examine two applications. The first application is the studyof marginal transition probabilities related to those analyzed by Bethe andSalpeter in \cite{B-S}, such as proving the exponential decay in the Markoviancase assuming the Fermi Golden Rule (FGR) hypothesis (Theorem 1.17 or Theorem1.15) and obtaining a FGR type approximation in the non Markovian case (Theorem1.5). The second application, in the non Markovian case, includes thederivation of Rabi cycles from QED (Theorem 1.7). All the results areestablished under the following assumptions at some steps of the proofs: anultraviolet and an infrared regularization are imposed, the quadratic terms ofthe Pauli-Fierz Hamiltonian are dropped, and the dipole approximation isassumed but only to obtain optimal error controls.

Time evolution for the Pauli-Fierz operator (Markov approximation and Rabi cycle)