  We are interested in this paper with the connection between the dynamics of a
model related to Nuclear Magnetic Resonance (NMR) in Quantum Field Theory (QFT)
with its classical counterpart known as the Maxwell-Bloch equations. The model
in QFT is a model of Quantum Electrodynamics (QED) considering fixed spins
interacting with the quantized electromagnetic field in an external constant
magnetic field. This model is close to the common spin-boson model. The
classical model goes back to F. Bloch [15] in 1946. Our goal is not only to
study the derivation of the Maxwell-Bloch equations but to also establish a
semiclassical asymptotic expansion of arbitrary high orders with control of the
error terms of this standard nonlinear classical motion equations. This
provides therefore quantum corrections of any order in powers of the
semiclassical parameter of the Bloch equations. Besides, the asymptotic
expansion for the photon number is also analyzed and a law describing the
photon number time evolution is written down involving the radiation field
polarization. Since the quantum photon state Hilbert space (radiation field) is
infinite dimensional we are thus concerned in this article with the issue of
semiclassical calculus in an infinite dimensional setting. In this regard, we
are studying standard notions as Wick and anti-Wick quantizations, heat
operator, Beals characterization theorem and compositions of symbols in the
infinite dimensional context which can have their own interest.
