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Acta Physica Hungarica A) Heavy Ion Physics

Wigner functions and quantum interference in optics

I. Földesi, József Janszky, Петер Адам, S. Szabó, I. Tarján, M. S. Kim

Published Aug 1, 1996• 0 citations
Publisher
DOI
10.1007/bf03053671

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Einstein–Podolsky–Rosen (EPR) correlations produced for two-photon entangled wave packets are analyzed by use of optical Wigner functions. In particular, type-II optical parametric downconversion systems are used, and the effects of the length of the crystal and the filter bandwidth on the Wigner functions are studied. The relation between the optical Wigner functions and the interference experiment with a beam splitter is studied. We find that, whereas in first-order correlation experiments (typified, for example, by Young’s interference experiment) the “photon interferes with itself” [ P. A. M. Dirac , The Principles of Quantum Mechanics (Clarendon, Oxford, UK, 1958), p. 9], in the EPR correlations the two-photon amplitude function interferes with itself. Such interference is analyzed by use of the optical Wigner functions.
The Gouy phase is essential for accurately describing various wave phenomena, ranging from classical electromagnetic waves to matter waves and quantum optics. In this work, we employ phase-space methods based on the cross-Wigner transformation to analyze spatial and temporal interference in the evolution of matter waves characterized initially by a correlated Gaussian wave packet. First, we consider the cross-Wigner of the initial wave function with its free evolution, and second for the evolution through a double-slit arrangement. Different from the wave function which acquires a global Gouy phase, we find that the cross-Wigner acquires a Gouy phase difference due to different evolution times. The results suggest that temporal like-Gouy phase difference is important for an accurate description of temporal interference. Furthermore, we propose a technique based on the Wigner function to reconstruct the cross-Wigner from the spatial intensity interference term in a double-slit experiment with matter waves.
Wigner-positive quantum states have the peculiarity to admit a Wigner function that is a genuine probability distribution over phase space. The Shannon differential entropy of the Wigner function of such states—called Wigner entropy for brevity—emerges as a fundamental information-theoretic measure in phase space and is subject to a conjectured lower bound, reflecting the uncertainty principle. In this work, we prove that this Wigner entropy conjecture holds true for a broad class of Wigner-positive states known as beam-splitter states, which are obtained by evolving a separable state through a balanced beam splitter and then discarding one mode. Our proof relies on known bounds on the p -norms of cross Wigner functions and on the interference formula, which relates the convolution of Wigner functions to the squared modulus of a cross Wigner function. Originally discussed in the context of signal analysis, the interference formula is not commonly used in quantum optics although it unveils a strong symmetry under convolution exhibited by Wigner functions of pure states. We provide here a simple proof of the formula and highlight some of its implications. Finally, we prove an extended conjecture on the Wigner-Rényi entropy of beam-splitter states, albeit in a restricted range for the Rényi parameter α ≥ 1 / 2 .