Hamiltonian systems and semiclassical dynamics for interacting spins in QED

In this article, we consider fixed spin-1/2 particles interacting through the quantized electromagnetic field in a constant magnetic field. We give approximate evolutions of coherent states. This uses spins-photon classical Hamiltonian mechanics. These approximations enable to derive that the approximate average fields and spins follow Maxwell-Bloch equations with a current density coming from spins. In addition, we obain a law concerning the evolution of the approximate average number of photons. Next, we provide stationary points of the spins-photon Hamiltonian when the spin particles belong to an orthogonal plane to the constant magnetic field. This allows for the construction of quasimodes with spins colinear to the constant magnetic field. Finally, a quasimode with an arbitrary high accuracy is built up and its first order radiative correction is computed.