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

TL;DR

This paper analyzes the time evolution of a particle system interacting with the quantized electromagnetic field within non-relativistic QED, deriving approximations and error controls for Markovian and non-Markovian dynamics, including applications to decay and Rabi cycles.

Contribution

It provides new rigorous approximations of the time evolution in both Markovian and non-Markovian regimes, with detailed error estimates and applications to decay and Rabi oscillations.

Findings

Exponential decay proven under Fermi Golden Rule in Markovian case

FGR-type approximation established in non-Markovian case

Derivation of Rabi cycles from QED in non-Markovian setting

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 1.16) 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.