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Journal of Modern Optics

Atomic Wehrl entropy for the Jaynes–Cummings model: explicit form and Bloch sphere radius

Faisal A. A. El-Orany

Published Dec 17, 2008• 24 citations
Publisher

Abstract

For the interaction between a two-level atom and the radiation field in the dipole and the rotating wave approximations, namely, the Jaynes–Cummings model, we obtain a closed form for the atomic Wehrl entropy in terms of the Bloch sphere radius. This explains analytically why the atomic Wehrl entropy has an identical behavior to that of the von Neumann entropy.

DOI
10.1080/09500340802517702

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For the interaction between a two-level atom and the radiation field in the dipole and the rotating wave approximations, namely, the Jaynes–Cummings model, we obtain a closed form for the atomic Wehrl entropy in terms of the Bloch sphere radius. This explains analytically why the atomic Wehrl entropy has an identical behavior to that of the von Neumann entropy.
The linear entropy, the von Neumann entropy and the atomic Wehrl entropy are frequently used to quantify entanglement in the quantum systems. These relations provide typical information on the entanglement in the Jaynes-Cummings model (JCM). In this Letter, we explain the origin of this analytically and derive a closed form for the atomic Wehrl entropy. Moreover, we show that it is more convenient to use the Bloch sphere radius for quantifying entanglement in the JCM instead of these entropic relations.
The concept of the classical-like Wehrl entropy is shown to be a very sensitive and informative measure describing the time evolution of a quantum system. Its applications to the Jaynes-Cummings model reveal relevant aspects of the field dynamics. The Wehrl entropy provides a compact one-parameter description giving, among other things, a clear signature for the splitting of the Q function. It gives also a clear indication of the phase randomization of the coherent electromagnetic field states during the interaction with a two-level atom.
In this paper, we present a structure for obtaining the exact eigenfunctions and eigenvalues of the Jaynes–Cummings model (JCM) without the rotating wave approximation (RWA). We study the evolution of the system in the strong coupling region using the time evolution operator without RWA. The entanglement of the system without RWA is investigated using the Von Neumann entropy as an entanglement measure. It is interesting that in the weak coupling regime, the population of the atomic levels and Von Neumann entropy without RWA model shows a good agreement with the RWA whereas in strong coupling domain, the results of these two models are quite different.
The behavior of atoms in the Jaynes–Cummings model has been discussed and analyzed extensively to date in terms of transition probability and entropy. In this paper, the Jaynes–Cummings model is described in terms of a quantum-mechanical channel expressing the state transition. For this channel, von Neumann entropy and quantum mutual entropy are derived. By means of these quantities, the variation of the atomic state and its nonreciprocity in this model are discussed. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 84(1): 95–103, 2001