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Physical Review A

Two-color photons and nonlocality in fourth-order interference

J. G. Rarity, P. R. Tapster

Published May 1, 1990• 90 citations
Publisher

Abstract

Angle-separated pair photons emitted in spontaneous parametric fluorescence are superimposed at a beamsplitter and detected by coincidence techniques. The coincidence disappears when the photon path lengths to the beamsplitter are equal to within their inverse bandwidth due to fourth-order interference. The photon bandwidth is set by wide apertures placed to satisfy phase-matching conditions at the nonlinear crystal. The resulting broadband photons are short (39-fs full width at half maximum) and show an oscillatory tail resulting from the hard-edge apertures. Two-color photons are created by blocking the central portion of the apertures leading to strong oscillations of the coincidence rate as a function of path-length difference. Both effects are predicted by the theory of fourth-order interference coupled with the detailed phase-matching conditions in the nonlinear crystal used as a source. The two colors overlap at different positions on the beamsplitter. We show how this could be used as a test of local realistic theories by formulating a Bell inequality based on relative phases between the colors.

DOI
10.1103/physreva.41.5139

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Angle-separated pair photons emitted in spontaneous parametric fluorescence are superimposed at a beamsplitter and detected by coincidence techniques. The coincidence disappears when the photon path lengths to the beamsplitter are equal to within their inverse bandwidth due to fourth-order interference. The photon bandwidth is set by wide apertures placed to satisfy phase-matching conditions at the nonlinear crystal. The resulting broadband photons are short (39-fs full width at half maximum) and show an oscillatory tail resulting from the hard-edge apertures. Two-color photons are created by blocking the central portion of the apertures leading to strong oscillations of the coincidence rate as a function of path-length difference. Both effects are predicted by the theory of fourth-order interference coupled with the detailed phase-matching conditions in the nonlinear crystal used as a source. The two colors overlap at different positions on the beamsplitter. We show how this could be used as a test of local realistic theories by formulating a Bell inequality based on relative phases between the colors.
Quantum mechanical models of beam splitters1 predict interesting fourth-order interference effects when nonclassical light is used. If indistinguishable one-photon states are simultaneously input into each port of a 50/50 beam splitter, the theory predicts that each output will contain either two photons or none. A naive (particle) theory would predict a mixture of two’s and one’s. The required |1,1〉 photon state can be selected in nondegenerate parametric downconversion experiments by photon-counting coincidence detection. The disappearance of this coincidence after a beam splitter has recently been studied,1-2 and the range of path length differences over which the effect can be seen is related by Fourier transform to the photon bandwidth. In our work the bandwidth is not limited by external filters, and we have developed a theory which relates the measured photon length to the geometry of the apparatus and the detailed phase matching conditions at the parametric crystal. This theory of phase matching viewed through a beam splitter produces some unexpected results when the beam splitter is tilted and when large aperture detectors are considered.
A two-photon state can be selected by coincidence detection of photon pairs created in parametric downconversion. We demonstrate a reduction in the coincidence-count rate when pairs of photons are combined in a beam splitter. This reduction occurs only when photon wave functions overlap and can permit accurate (subpicosecond) relative timing measurements. We relate the width of the coincidence reduction to the bandwidth of the downconverted light and thence to the detailed phase-matching conditions in the parametric downconversion crystal. We also demonstrate that the loss of coincidences is associated with an increased probability of detection of photon pairs in each output of the beam splitter.
An interference experiment with photon pairs produced by degenerate parametric fluorescence is performed. The pairs are split and directed to two spatially separated Michelson interferometers. The interferometers are set to path differences which cause time delays far exceeding the coherence time of the light. We investigate photon coincidences between the output arms of the interferometers and observe nonlocal fourth-order interferences in the absence of second-order interferences. Maximum visibilities up to 86% are obtained, leading to a violation of Bell's inequality for energy and time by several standard deviations.
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.
A theoretical treatment is given of the situation in which two photons produced in the parametric frequency splitting of light in a nonlinear medium are allowed to interfere. It is shown that there is no interference pattern in the usual sense of quantities that are of the second order in the field, but that fourth-order interference effects are present. These could be demonstrated by placing two photodetectors at positions x,x′ in the interference plane and measuring the joint probability of two detections as a function of x – x′. The probability is predicted to exhibit a cosine modulation with separation x – x′ of close to 100% visibility, even when the integration time in the experiment greatly exceeds the reciprocal bandwidth of the accepted photons. As a result two photons cannot be detected at two points separated by a distance corresponding to an odd number of half-interference fringes. This is an interference effect that violates the laws of classical probability.