Generalized Bloch Equation, Semiclassical Expansion

Laurent Amour, Lisette Jager, Jean Nourrigat

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.

Maurice A. de Gosson

A. M. Escobar-Ruiz, Edward Shuryak, Alexander V. Turbiner

This is the second paper on the semiclassical approach based on the density matrix given by the Euclidean time path integral with fixed coinciding end points. The classical path, interpolating between this point and the classical vacuum (called a ``flucton''), as well as systematic one- and two-loop corrections were calculated in the first paper [M. A. Escobar-Ruiz, E. Shuryak, and A. V. Turbiner, Phys. Rev. D 93, 105039 (2016)] for a double-well potential. Here, we extend them for a number of quantum-mechanical problems, such as an anharmonic oscillator and the sine-Gordon potential. The method is based on a systematic expansion in Feynman diagrams and thus can be extended to quantum field theories (QFTs). We show that the loop expansion in quantum mechanics resembles the leading-log approximations in QFT. In this sequel, we present a complete set of results obtained using this method in a unified way. Alternatively, starting from the Schr\"odinger equation we derive a generalized Bloch equation whose semiclassical-like, iterative solution generates the loop expansion. We rederive the two-loop expansions for all three of the above potentials and extend them to three loops, which has not yet been done via Feynman diagrams. All results for both methods are fully consistent with each other. An asymmetric (tilted) double-well potential (nondegenerate minima) is also studied using the second method.

Matthias Zens, Dmitry O. Krimer, Stefan Rotter

We numerically study the dynamics and stationary states of a spin ensemble strongly coupled to a single-mode resonator subjected to loss and external driving. Employing a generalized cumulant expansion approach we analyze finite-size corrections to a semiclassical description of amplitude bistability, which is a paradigm example of a driven-dissipative phase transition. Our theoretical model allows us to include inhomogeneous broadening of the spin ensemble and to capture in which way the quantum corrections approach the semiclassical limit for increasing ensemble size $N$. We set up a criterion for the validity of the Maxwell-Bloch equations and show that close to the critical point of amplitude bistability even very large spin ensembles consisting of up to ${10}^{4}$ spins feature significant deviations from the semiclassical theory.

P. A. Horváthy, L. Martina