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The Weyl-Wigner formalism for the interpretation of parametric down-conversion experiments

Emilio Santos

Published May 18, 2019• 0 citations
OpenAlex

Abstract

We show that the Weyl-Wigner formalism in the Heisenberg picture may be used for the interpretation of experiments involving entangled photon pairs produced in nonlinear crystals via spontaneous parametric down-conversion. The calculations are usually no more involved than those with the Hilbert-space formalism. The WW formalism suggests a physical picture in terms of random variables and stochastic processes, but the picture is shown untenable in some instances.

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We show that the Weyl-Wigner formalism in the Heisenberg picture may be used for the interpretation of experiments involving entangled photon pairs produced in nonlinear crystals via spontaneous parametric down-conversion. The calculations are usually no more involved than those with the Hilbert-space formalism. The WW formalism suggests a physical picture in terms of random variables and stochastic processes, but the picture is shown untenable in some instances.
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.
In this paper, the theory of parametric down-conversion in the Wigner representation is applied to Ekert's quantum cryptography protocol. We analyse the relation between two-photon entanglement and (non-secure) quantum key distribution within the Wigner framework in the Heisenberg picture. Experiments using two-qubit polarization entanglement generated in nonlinear crystals are analysed in this formalism, along with the effects of eavesdropping attacks in the case of projective measurements.
We characterize the electromagnetic vacuum as a stochastic field. Some consequences, like the particle behaviour of light, are studied. The stochastic approach is connected with the standard Hilbert space formalism via the Weyl transform. Several experiments involving spontaneous parametric down conversion are studied comparing Hilbert space and Weyl–Wigner formalisms. This allows an intuitive picture of entanglement to be obtained as a correlation between field fluctuations in distant places, involving the vacuum fields. The analysis shows that the Bell definition of local realism is not general enough, whence the reported violation of Bell inequalities does not refute local realism.
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.
We develop a fully quantum-mechanical approach for the description of spontaneous parametric down-conversion in waveguides. Our method allows us to write an expression for the quantum state easily and directly, even when multiple photon pairs are generated in the process. The role of material dispersion in the normalization of the modes of the system, usually neglected, is properly treated here. The example of an AlyOx/Al0.5Ga0.5As{\text{Al}}_{y}{\text{O}}_{x}/{\text{Al}}_{0.5}{\text{Ga}}_{0.5}\text{As}Aly​Ox​/Al0.5​Ga0.5​As ridge waveguide is discussed to illustrate the application of the formalism.