  This article is concerned with a system of particles interacting with the
quantized electromagnetic field (photons) in the non relativistic Quantum
Electrodynamics (QED) framework and governed by the Pauli-Fierz Hamiltonian. We
are interested not only in deriving approximations of several quantities when
the coupling constant is small but also in obtaining different controls of the
error terms. First, we investigate the time dynamics approximation in two
situations, the Markovian (Theorem 1.4 completed by Theorem 1.16) and non
Markovian (Theorem 1.6) cases. These two contexts differ in particular
regarding the approximation leading terms, the error control and the initial
states. Second, we examine two applications. The first application is the study
of marginal transition probabilities related to those analyzed by Bethe and
Salpeter in \cite{B-S}, such as proving the exponential decay in the Markovian
case assuming the Fermi Golden Rule (FGR) hypothesis (Theorem 1.17 or Theorem
1.15) and obtaining a FGR type approximation in the non Markovian case (Theorem
1.5). The second application, in the non Markovian case, includes the
derivation of Rabi cycles from QED (Theorem 1.7). All the results are
established under the following assumptions at some steps of the proofs: an
ultraviolet and an infrared regularization are imposed, the quadratic terms of
the Pauli-Fierz Hamiltonian are dropped, and the dipole approximation is
assumed but only to obtain optimal error controls.
