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

Abstract

We continue the analysis of our previous articles which were devoted to type-I parametric down conversion, the extension to type-II being straightforward. We show that entanglement, in the Wigner representation, is just a correlation that involves both signals and vacuum fluctuations. An analysis of the detection process opens the way to a complete description of parametric down conversion in terms of pure Maxwell electromagnetic waves.The theory of parametric down conversion (PDC) was treated, in the Wigner formalism, in an earlier series of articles [1,2,3]. There we showed that, provided one considers the zeropoint fluctuations of the vacuum to be real, the description of radiation is fully within Maxwell electromagnetic theory. Effectively, because the Wigner function maintains its positivity, we can say that quantization is just the addition of a zeropoint radiation, and there is no need for any further quantization of the light field. In the present article we show that the same result extends, without any difficulty, from the type-I PDC case to the type-II situation.There seems to be a widespread reluctance to accept the reality of the vacuum fluctuations, in spite of the fact that they appear, quite naturally, in the Wigner function of the vacuum state. We remark that such fluctuations have been taken seriously, within a certain school of thought, throughout the entire history of the quantum theory, following the formulation of Max Planck, originating in 1911 [4]. Of course, it is true that, integrated over all frequencies, they give us a vacuum with infinite energy density; why then are all photographic plates not blackened instantaneously? But all photodetectors, including even our own eyes, are very selective, not only as regards the frequency, but even also the wave vectors, of the light components they analyze. This is especially the case with the detectors commonly used in PDC experiments. So, there is a noise to subtract, but it is not infinite! In our previous articles we indicated how the noise subtraction is made, according to the Wigner formalism, and showed how this subtraction is related to the standard calculating device, of normal ordering, used in the Hilbert-space formalism. Here we extend this analysis, in an informal manner, showing that, if we take into account the fact that all detectors integrate the light intensity over a large time window, the process of light detection, like that of light propagation, may also be described entirely in terms of real waves and positive probabilities. We are then able to see that, in terms of a purely wave description, the highly problematic concept of "entangledphoton" states of the field loses all its mystery. Entangled photons are just correlated waves! The only reason this description has taken so long to mature is that the word "classical", in reference to the light field, is restricted in its application to Glauber-classical states [5]. A discussion of the difference between classical and nonclassical effects has been given in Ref. [6]. The

Research on hidden variable theories: A review of recent progresses

“…A more ambitious program was started by a British-Spaniard group with the hope to build a real alternative to Quantum Optics [297,298,299,300,301,302,303]. The main idea was that the probability distribution for the hidden variable is given by the Wigner function, which is positive for photons experiments.…”

Simulation Results Related to Stochastic Electrodynamics

“…Marshall's experimental suggestions involves work he has advanced with colleagues, particularly E. Santos, in "stochastic optics," aiming in particular at a local model explanation of the Bell inequalities [60], [61]. At the end of the 1990s, Marshall developed with colleagues a local physical explanation of entanglement involving a nonlinear optical process often called spontaneous parametric down conversion [62], [63], [64], [65], [66], [67]. This work led them to the prediction of a new phenomena, still to be tested for, involving a natural explanation in terms of unquantized light and local quantities that they have termed spontaneous parametric up conversion [68], [69], [70], [71].…”

The Wigner Function as Distribution Function

“…It will be convenient for us later to refer to the above as the "time evolution" of S x under the "Hamiltonian" S z in Eq. (21): in this way we refer to the "rotation" angle, ϑ, as the time, t. Sticking to the geometric notation, the Bell operator (13) is (the superscripts refer to the channels, a, a † being channel 1 and b, b † channel 2) B = S 1 · n S 2 · m + S 1 · n S 2 · m + S 1 · n S 2 · m − S 1 · n S 2 · m (22) and the Bell inequality we study is…”