Infinite dimensional semiclassical analysis and applications to a model in nuclear magnetic resonance

Abstract

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.

Compositions of states and observables in Fock spaces

“…It is proved in [3]) (formula (2.13) that, if A = Op AW h (F ), where the function F on H 2 must have a stochastic extension F , as we saw:…”

“…For the Weyl symbol, see [2] and [3]). Let us denote by σ h W eyl(A) and σ h W ick(A) the symbols of an operator A = Op AW h (F ), for a suitable given function F on H 2 .…”

Classical and quantum energies for interacting magnetic systems

“…The spin of one of several static interacting particles (fixed atomic nuclei) can be described in the quantum electrodynamic (QED) setting by a Hamiltonian operator given in [6] (see also [21]). Besides, it is common to represent these static 1 2 −spins as magnets able to turn around fixed points and interacting according to the classical physics laws. It our aim in this paper to make a link between these two points of view by studying the ground state energy of the QED Hamiltonian, recalled in Section 2 and having a non degenerate eigenvalue as the infimum of the spectrum (in the case of several spins).…”

“…First, we prove that E 2 (M ) is the smallest eigenvalue of some operator A M acting in the finite dimensional spin space H sp . The operator A M is defined in (2.9), quadratic of M = (M [1] , . .…”

“…It is however for specific sates X ∈ H sp , namely tensor products X = V [1] ⊗ • • • ⊗ V [P ] with the V [λ] in C 2 and normalized, also called product states or disentangled states, that the function < A M X, X > displays a very common form. To this end, first recall that, with any product state X ∈ H sp , the Hopf fibration defines S(X) = (S [1] (X), . .…”