We are concerned in this paper with the connection between the dynamics of a model related to nuclear magnetic resonance 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 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 Bloch [Phys. Rev. 70, 460 (1946)]. Our goal is not only to study the derivation of the Maxwell-Bloch equations but also to establish a semiclassical asymptotic expansion of arbitrary high order with control of the error terms of these 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.
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Infinite dimensional semiclassical analysis and applications to a model in nuclear magnetic resonance
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July 2019
Research Article|
July 08 2019
Infinite dimensional semiclassical analysis and applications to a model in nuclear magnetic resonance
L. Amour;
L. Amour
a)
CNRS, LMR FRE 2011, Université de Reims Champagne-Ardenne
, Moulin de la Housse, BP 1039, 51687 Reims Cedex 2, France
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L. Jager
;
L. Jager
b)
CNRS, LMR FRE 2011, Université de Reims Champagne-Ardenne
, Moulin de la Housse, BP 1039, 51687 Reims Cedex 2, France
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J. Nourrigat
J. Nourrigat
c)
CNRS, LMR FRE 2011, Université de Reims Champagne-Ardenne
, Moulin de la Housse, BP 1039, 51687 Reims Cedex 2, France
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J. Math. Phys. 60, 071503 (2019)
Article history
Received:
March 01 2019
Accepted:
May 29 2019
Citation
L. Amour, L. Jager, J. Nourrigat; Infinite dimensional semiclassical analysis and applications to a model in nuclear magnetic resonance. J. Math. Phys. 1 July 2019; 60 (7): 071503. https://doi.org/10.1063/1.5094396
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