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Journal of Physics A Mathematical and General

The semiclassical limit of the spin boson model

M. B. Cibils, Y. Cuche, Valdecir Marvulle, Walter Felipe Wreszinski, J.-P. Amiet, H Beck

Published Apr 7, 1991• 10 citations
Publisher

Abstract

The authors study the semiclassical limit of the spin boson model, where the spin remains quantized and the field is expanded around the classical limit of a large number of photons, using a phase representation due to Bialynicki-Birula and Bialynicki-Birula (1976). They derive the semiclassical Hamiltonian and prove that in the semiclassical expansion there always arises a back reaction of the atom on the field. The nearest-neighbour spacing distribution of levels is studied in the same limit and shown to be, in general, qualitatively different from the distribution for the original model. However, the comparison between the Dyson-Mehta statistics is also studied and no qualitative difference is visible for the function Delta 3 .

DOI
10.1088/0305-4470/24/7/035

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The authors study the semiclassical limit of the spin boson model, where the spin remains quantized and the field is expanded around the classical limit of a large number of photons, using a phase representation due to Bialynicki-Birula and Bialynicki-Birula (1976). They derive the semiclassical Hamiltonian and prove that in the semiclassical expansion there always arises a back reaction of the atom on the field. The nearest-neighbour spacing distribution of levels is studied in the same limit and shown to be, in general, qualitatively different from the distribution for the original model. However, the comparison between the Dyson-Mehta statistics is also studied and no qualitative difference is visible for the function Delta 3 .
The Jaynes-Cummings model (JCM) of a two-level atom interacting with a quantized radiation field has been investigated theoretically. Of particular interest is the transition of a quantized electromagnetic field of the model to a classical one in the limit of very large average photon number. In this thesis, two aspects of the JCM are examined. Firstly, it is shown that, in contrast to the general case, for specic initial atom-field states, there is exact correspondence between the atomic inversion predicted by the JCM and that predicted by the semiclassical model in which the field is treated classically from the outset. Secondly, the degree of quantum entanglement of the JCM is investigated quantitatively and it is shown that, at any nite absolute temperature greater than zero, the atom and field are never separable and the degree of quantum entanglement approaches zero in the high and low temperature limits
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.
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We write the density matrix of a two-level atom interacting with a single-mode radiation field in terms of the average photon number and the degree of offdiagonality. The latter represents quantum correlations, which are transformed to a phase variable analogous to the phase of the classical field. Using a consistent expansion in the inverse powers of the square root of the average photon number we obtain the ordinary classical limit. The surprising feature which emerges is that the zero point energy contributes to the classical field parameter as efficiently as the photon occupation number; this result agrees with one derived differently by Bialynicki-Birula and Bialynicka-Birula.