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

Abstract

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 1.14) 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 (Bethe and Salpeter, 1957), such as proving the exponential decay in the Markovian case assuming the Fermi Golden Rule (FGR) hypothesis (Theorem 1.15 or Theorem 1.13) 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.