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Journal of Physics A Mathematical and General

On the classical limits in the spin-boson model

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

Published Feb 21, 1990• 16 citations
Publisher

Abstract

The authors show that, while the classical equations of motion of Belobrov, Zaslavski and Tartakovski (1976) are rigorously obtained from the thermodynamic limit of the Heisenberg equations of motion of the many-atom version of the spin-boson model in suitable states, those of Milonni, Ackerhalt and Galbraith (1983) are derivable from a quantum Hamiltonian only if s)1/2, where s is the spin quantum number. Both equations are known to display chaotic behaviour for large coupling constants.

DOI
10.1088/0305-4470/23/4/022

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The authors show that, while the classical equations of motion of Belobrov, Zaslavski and Tartakovski (1976) are rigorously obtained from the thermodynamic limit of the Heisenberg equations of motion of the many-atom version of the spin-boson model in suitable states, those of Milonni, Ackerhalt and Galbraith (1983) are derivable from a quantum Hamiltonian only if s)1/2, where s is the spin quantum number. Both equations are known to display chaotic behaviour for large coupling constants.
We study the quantum-classical correspondence for systems with interacting spin particles that are strongly chaotic in the classical limit. This is done in the presence of constants of motion associated with the fixed angular momenta of individual spins. Our analysis of the Lyapunov spectra reveals that the largest Lyapunov exponent agrees with the Lyapunov exponent that determines the local instability of each individual spin moving under the influence of all other spins. Within this picture, we introduce a rigorous and simple test of ergodicity for the spin motion, and use it to identify when classical chaos is both strong and global in phase space. In the quantum domain, our analysis of the Hamiltonian matrix in a proper representation allows us to obtain the conditions for the onset of quantum chaos as a function of the model parameters. From the comparison between the quantum and classical domains, we demonstrate that quantum quantities, such as the local density of states (LDoS) and the shape of the chaotic eigenfunctions written in the noninteracting many-body basis, have well-defined classical counterparts. We also find a relationship between the Kolmogorov-Sinai entropy and the width of the LDoS, which is useful for studies of many-body dynamics.
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 .
We study a model system composed of interacting quantum spins, with every spin coupled to every other, in the limit where the number of spins K and the angular momentum j of each spin are both large, aiming to explore the effect of large system size on the breakdown of the classical limit of quantum mechanics. We obtain the exact spectrum of the Hamiltonian, and hence the trace of the quantum-mechanical time-evolution operator. We examine the time dependence of , finding a simple approximation which is valid when j and K are large. At a time proportional to , this approximation breaks down, and the long-time behaviour is extremely complex. We use a renormalization scheme to investigate this complexity. The scheme is based upon a generalization of the Gauss continued-fraction map to the complex plane.