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arXiv:1106.4805 · Advances in Mathematical Physics

Partial Bell‐State Analysis with Parametric down Conversion in the Wigner Function Formalism

A. Casado, S. Guerra, J. Plácido

Published Jan 1, 2010• 8 citations
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Abstract

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.

DOI
10.1155/2010/501521

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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.
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 of parametric down conversion in the Heisenberg picture is applied to the study of the Rome teleportation experiment. We investigate the physical meaning of the zeropoint inputs at the different areas of the experimental set-up. In particular, we establish a quantitative relationship between the zeropoint sets of modes that are needed for the preparation of the quantum state to be teleported, the idle channels inside the one-photon polarization-momentum Bell-state analyser, and the possibility of performing teleportation of a polarization state with certainty.