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Journal of Modern Optics

Wehrl's Entropy Dynamics in a Kerr-like Medium

Igor Jex, Arkadiusz Orłowski

Published Dec 1, 1994• 40 citations
Publisher

Abstract

We report on the time evolution of the Wehrl entropy in a Kerr-like medium. We show that the Wehrl entropy gives a clear signature for the formation of finite superpositions of coherent states (cat-like states). In addition, the actual value of the Wehrl entropy at the time of a superposition formation gives the number of coherent components taking part in the superposition.

DOI
10.1080/09500349414552151

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We report on the time evolution of the Wehrl entropy in a Kerr-like medium. We show that the Wehrl entropy gives a clear signature for the formation of finite superpositions of coherent states (cat-like states). In addition, the actual value of the Wehrl entropy at the time of a superposition formation gives the number of coherent components taking part in the superposition.
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The Wehrl information entropy and its phase density, the so-called Wehrl phase distribution, are applied to describe Schrödinger cat and cat-like (kitten) states. The advantages of the Wehrl phase distribution over the Wehrl entropy in a description of the superposition principle are presented. The entropic measures are compared with a conventional phase distribution from the Husimi Q -function. Compact-form formulae for the entropic measures are found for superpositions of well separated states. Examples of Schrödinger cats (including even, odd and Yurke-Stoler coherent states), as well as the cat-like states generated in the Kerr medium, are analysed in detail. It is shown that, in contrast to the Wehrl entropy, the Wehrl phase distribution properly distinguishes between different superpositions of unequally weighted states with respect to their number and phase-space configuration.
The phase properties of an initially coherent field propagating through a Kerr medium are examined using two different measures of phase uncertainty: (i) variance of periodic functions and (ii) the entropy. The variance of a periodic function of phase (with period 2\ensuremath{\pi}/m) is found to be minimized only at the times when the field evolves into a discrete superposition of n distinguishable coherent states (Schr\"odinger-cat states) where n is a factor of m. The phase entropy is found to be more sensitive, as it is minimized at the times when the field evolves into a superposition of an arbitrary number of distinguishable coherent states. The sum of the photon number and phase entropies is found to display behavior very similar to the Wehrl entropy for the evolving field, i.e., the sum is minimal at the times the cat states occur, and at these times it has a value that indicates the number of component coherent states in the cat states.
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.