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Physical Review A

Fourth-order interference in the Wigner representation for parametric down-conversion experiments

Alberto Casado, A. Fernández-Rueda, Trevor W. Marshall, Ramón Risco, Emilio Santos

Published May 1, 1997• 38 citations
Publisher

Abstract

In the Wigner formalism, after giving a general description of a light beam, the theory of parametric down-conversion is developed to second-order in the coupling constant. We then describe the detection process by calculating the auto-correlation and cross-correlations of the signal and idler beams. Four recent experiments are analyzed in detail: interference on a screen, Franson's experiments [Phys. Rev. Lett. 62, 2205 (1989)], Rarity and Tapster's experiments [Phys. Rev. Lett. 64, 2495 (1990)], and induced coherence without induced emission [X. Y. Zou, L. J. Wang, and L. Mandel, Phys. Rev. Lett. 67, 318 (1991)].

DOI
10.1103/physreva.55.3879

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In the Wigner formalism, after giving a general description of a light beam, the theory of parametric down-conversion is developed to second-order in the coupling constant. We then describe the detection process by calculating the auto-correlation and cross-correlations of the signal and idler beams. Four recent experiments are analyzed in detail: interference on a screen, Franson's experiments [Phys. Rev. Lett. 62, 2205 (1989)], Rarity and Tapster's experiments [Phys. Rev. Lett. 64, 2495 (1990)], and induced coherence without induced emission [X. Y. Zou, L. J. Wang, and L. Mandel, Phys. Rev. Lett. 67, 318 (1991)].
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.
The interference of signal and idler waves produced by optical parametric down-conversion was the object of a classic paper by Ghosh and Mandel [Phys. Rev. Lett. 1987, 59, 1903–1905] many years ago. In this paper we investigate the problem in the continuous variable regime, with the aim of identifying quantum signatures that arise from the entanglement of signal and idler waves. By using a model which allows for analytical calculations we show first of all that no quantum features appear for the interference in the fourth-order correlation function in the high-gain regime of parametric down-conversion. The quantum signatures emerge instead from the spatio-temporal correlation function of a generic quadrature component, and we demonstrate that they are clearly linked to the signal/idler entanglement. Relevant and useful links with the theory of quantum images complete the picture.
Normally, the signal and idler photons produced in the process of parametric down- conversion exhibit no second-order interference. However, in the presence of an external laser beam that follows the path of the signal photon and induces stimulated down-conversions, one would expect coherence to develop between the down-converted signal and idler beams. We have demonstrated the existence of this phenomenon in an experiment with two down-converters pumped by similar pump beams, in which two mutually coherent laser beams followed the paths of the two down-converted signal photons and induced coherence between the two down- converted idler beams. The induced coherence gives rise to observable interference effects, which are presented.
The quantum theory of fourth-order interference of light is presented in a general format and compared with classical wave theory. The conditions under which nonclassical phenomena occur are discussed. In particular, the interference between the quantum field and classical field may give rise to a nonclassical effect. For some special states of light, the interference pattern does not disappear even though one field is much stronger than the other, for which no classical analog exists. Fourth-order effects in the interference between two independent fields are analyzed in detail. It is pointed out that the fourth-order interference between independent fields will not disappear when the integration time of detection is of the order of the reciprocal bandwidth of the two light fields as long as the spectra of the two fields are symmetric around the same center frequency, and for some correlated fields, the interference does not vanish even if the detection time is much larger than the reciprocal bandwidth of the fields. A new type of fourth-order interference experiment involving a beam splitter is proposed in which local realism of the Einstein-Podolsky-Rosen form is violated for quantum mechanics. This general argument is then applied to the interference between two photons generated in the parametric down-conversion process. The possibility of violations of Bell's inequalities in interference experiments is investigated.