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

Laurent Amour (1) , L. Jager (1) , J. Nourrigat (1)

1 URCA - Université de Reims Champagne-Ardenne

We are concerned in this paper with the connection between the dynamics of a model related to Nuclear Magnetic Resonance (NMR) in Quantum Field Theory (QFT) and 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, Physical Review, vol. 70, 460 (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 order 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. 1 Statement of the results. The aim of this work is to carry on the study of an infinite dimensional symbolic calculus begun in [4] and to apply it to the semiclassical limit of the evolution for a quantum field model in Nuclear Magnetic Resonance (NMR). This model is introduced in Section 4.11 in Reuse [50], (see also [20] and [39]). It is devoted to the interaction between a finite number of fixed (heavy) spin-1 2 particles, for instance atomic nuclei, with the quantized electromagnetic field together with a constant external magnetic field. This interaction model is closely related to the spin-boson model (for example, see [28], [29], [36], [11], [54] , [24]). The Hamiltonian of the system studied here is also similar to the more complicated Pauli Fierz Hamiltonian (see [23], [12]), where the terms concerning the spin particles motion are deleted. NMR can also be modelled by earlier equations due to F.Bloch [15] (1946). We prove here that they are the semiclassical limit of the QED model. In this early model, spins are viewed as vectors S λ (t) ∈ R 3 , λ = 1,. .. , P .

Mots clés: Semiclassical analysis; infinite dimensional analysis; composition of operators; Wick quantization; anti-Wick quantization; Wick symbol; Husimi function; Wiener spaces; Heat operator; symbolic calculus; QED; quantum electrodynamics; Maxwell-Bloch equations; Bloch equations; NMR; Nuclear Magnetic Resonance; photon emission; photon number

Domaines: Mathématiques générales [math.GM]; Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Laurent Amour, L. Jager, J. Nourrigat. Infinite dimensional semiclassical analysis and applications to a model in nuclear magnetic resonance. Journal of Mathematical Physics, 2019, 60 (7), pp.071503. ⟨10.1063/1.5094396⟩. ⟨hal-02177597⟩