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arXiv:quant-ph/9711042 · Journal of the Optical Society of America B

Type II parametric downconversion in the Wigner-function formalism: entanglement and Bell's inequalities

Alberto Casado, Trevor W. Marshall, Emilio Santos

Published May 1, 1998• 28 citations
arXivHTMLPDF

Abstract

We continue the analysis of our previous papers that were devoted to type I parametric downconversion, the extension to type II being straightforward. We show that entanglement, in the Wigner representation, is merely a correlation that involves both signals and vacuum fluctuations. An analysis of the detection process opens the way to a complete description of parametric downconversion in terms of pure Maxwell electromagnetic waves.

DOI
10.1364/josab.15.001572

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Wigner-PDC description of photon entanglement can still be made completely local

David Rodríguez

Regardless of past proposals (already disproved in experimental work by Brida et al), the Wigner picture of Parametric Down Conversion (works by Casado et al) can be interpreted as a local-realistic formalism, without the need to depart from quantum mechanical predictions at any step, at least for the relevant subset of QED-states. This involves reinterpreting the expressions for the detection probabilities, by means of an additional mathematical manipulation; though such manipulation seemingly provides enough freedom to guarantee consistency with the expectable, experimentally testable behavior of detectors, this is, in any case, irrelevant in relation to our main result, of a purely mathematical nature. Besides, the Wigner-PDC approach opens clear room to explain the elusive nature of a loop-hole free proof of the incompatibility of nature with that local-realism, without the need to assume hypothetical technological inefficiencies yet to overcome after several decades of experimental work. In relation to this, we also address the consequences of this framework in relation to typical Bell experiments. Additionally, we also propose an extremely simple interpretation on that apparently awkward subtraction of the average ZPF intensity at the Wigner image of the detection process.

•Nov 16, 2011•0 citations
We continue the analysis of our previous papers that were devoted to type I parametric downconversion, the extension to type II being straightforward. We show that entanglement, in the Wigner representation, is merely a correlation that involves both signals and vacuum fluctuations. An analysis of the detection process opens the way to a complete description of parametric downconversion in terms of pure Maxwell electromagnetic waves.
We apply the Wigner formalism of quantum optics to study the role of the zeropoint field fluctuations in entanglement swapping produced via parametric down conversion. It is shown that the generation of mode entanglement between two initially non interacting photons is related to the quadruple correlation properties of the electromagnetic field, through the stochastic properties of the vacuum. The relationship between the process of transferring entanglement and the different zeropoint inputs at the nonlinear crystal and the Bell-state analyser is emphasized.
We apply the Wigner formalism of quantum optics in the Heisenberg picture to study the role of the zeropoint field fluctuations in entanglement swapping produced via parametric down conversion. It is shown that the generation of mode entanglement between two initially non-interacting photons is related to the quadruple correlation properties of the electromagnetic field, through the stochastic properties of the vacuum. The relationship between the process of transferring entanglement and the different zeropoint inputs at the non-linear crystal and the Bell-state analyser is emphasized.
In this paper, the nexus between the Bell-state measurement and extracting phase information from the zeropoint field is investigated. For this purpose, the Wigner representation in the Heisenberg picture is applied in a Bell-type experiment in which the polarisation-entangled photon pairs generated in a type-II parametric down-conversion do not overlap. The signal intensities at the detectors are calculated in a four-mode approximation, being expressed as functions of the modules and phases of the four zeropoint amplitudes entering the crystal. A general criterion for identifying the correlated detectors is proposed based on the equality of the signal intensities, and without involving the calculation of the joint detection probabilities. In addition, from the analyses in the rectilinear and diagonal basis, it is shown that the distinguishability of the polarisation Bell states, which is in direct correspondence with the joint detection events in each experiment, can be related to the knowledge of the phases of the vacuum field entering the entanglement source, and giving rise to correlated detections. To this purpose, it is conjectured that a detection event is associated with a maximum value of the signal intensity averaged in the modules of the zeropoint amplitudes, as a function of the vacuum phases.
The Wigner representation is developed in the Heisenberg picture for the study of experiments involving photon pairs created in the process of parametric down conversion. After the general description of a light beam and the theory of detection (restricted here to single count probabilities in the Wigner formalism) the theory of parametric downconversion is developed to the point of calculating the autocorrelations of the signal and idler beams. Two recent experiments are analyzed in detail: frustrated two-photon creation by interference, and induced coherence and indistinguishability.
We apply the Wigner function formalism to partial Bell‐state analysis using polarization entanglement produced in parametric down conversion. Two‐photon statistics at a beam‐splitter are reproduced by a wave‐like description with zeropoint fluctuations of the electromagnetic field. In particular, the fermionic behaviour of two photons in the singlet state is explained from the invariance on the correlation properties of two light beams going through a balanced beam‐splitter. Moreover, we show that a Bell‐state measurement introduces some fundamental noise at the idle channels of the analyzers. As a consequence, the consideration of more independent sets of vacuum modes entering the crystal appears as a need for a complete Bell‐state analysis.
Regardless of past proposals (already disproved in experimental work by Brida et al), the Wigner picture of Parametric Down Conversion (works by Casado et al) can be interpreted as a local-realistic formalism, without the need to depart from quantum mechanical predictions at any step, at least for the relevant subset of QED-states. This involves reinterpreting the expressions for the detection probabilities, by means of an additional mathematical manipulation; though such manipulation seemingly provides enough freedom to guarantee consistency with the expectable, experimentally testable behavior of detectors, this is, in any case, irrelevant in relation to our main result, of a purely mathematical nature. Besides, the Wigner-PDC approach opens clear room to explain the elusive nature of a loop-hole free proof of the incompatibility of nature with that local-realism, without the need to assume hypothetical technological inefficiencies yet to overcome after several decades of experimental work. In relation to this, we also address the consequences of this framework in relation to typical Bell experiments. Additionally, we also propose an extremely simple interpretation on that apparently awkward subtraction of the average ZPF intensity at the Wigner image of the detection process.