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Physical Review E

Semiclassical corrections to the large- N limit of Dicke’s model

Mauricio Reis, Maria Carolina Nemes, J. G. Peixoto de Faria

Published Sep 22, 2008• 6 citations
Publisher

Abstract

We use a semiclassical expansion as an alternative derivation of the well-known, rigorous result obtained by Hepp and Lieb for the classical limit of the spin-boson model. We also explicitly derive correction terms to the classical limit previously obtained in the context of Heisenberg equations of motion. We analyze the size and shape of the N (number of atoms) vs t (time) domain whithin which the corrections so obtained are useful.

DOI
10.1103/physreve.78.036220

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We use a semiclassical expansion as an alternative derivation of the well-known, rigorous result obtained by Hepp and Lieb for the classical limit of the spin-boson model. We also explicitly derive correction terms to the classical limit previously obtained in the context of Heisenberg equations of motion. We analyze the size and shape of the N (number of atoms) vs t (time) domain whithin which the corrections so obtained are useful.
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 Tavis-Cummings model (the Dicke model treated in the rotating-wave approximation), describing many two-level systems coupled to a single bosonic mode, has been long known to show collective semiclassical oscillations when prepared in an inverted state, with all two-level systems excited and the bosonic mode is empty. This paper discusses how the quantum dynamics approaches this semiclassical result for large numbers of two-level systems, focusing on how the eigenvalues approach their semiclassical limit. The approach to the semiclassical result is found to be slow, scaling like a power of the logarithm of the system size. Considering also the effect of weak detuning between the two-level system and the bosonic field, quantum corrections are again found to decay slowly with system size, such that for a fixed detuning, the quantum effects of detuning are greater than the classical effect.
For quantum theories with a classical limit (which includes the large N limits of typical field theories), we derive a hierarchy of evolution equations for equal time correlators which systematically incorporate corrections to the limiting classical evolution. Explicit expressions are given for next-to-leading order, and next-to-next-to-leading order time evolution. The large N limit of N-component vector models, and the usual semiclassical limit of point particle quantum mechanics are used as concrete examples. Our formulation directly exploits the appropriate group structure which underlies the construction of suitable coherent states and generates the classical phase space. We discuss the growth of truncation error with time, and argue that truncations of the $\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\ensuremath{-}N$ evolution equations are generically expected to be useful only for times short compared to a ``decoherence'' time which scales like N1/2.{N}^{1/2}.N1/2.