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Annual Meeting Optical Society of America

Multimode analysis of single photon nonlocality in parametric down-conversion processes

Z. Y. Ou, Harry J. Kimble

Published Jan 1, 1991• 0 citations
Publisher

Abstract

With regard to the system of parametric downconversion, we analyze a recently proposed experiment for demonstrating the nonlocal property of a single photon.1 In the original proposal by Tan et al., a light field in a single photon-number state is split into two beams, which are then homodyned with two spatially separated weak local oscillators. The two-photon detection probability with the two homodyned beams provides evidence for nonlocality. It is shown that this phenomenon is in essence a phase-sensitive fourth-order interference effect. A single photonnumber state can be achieved from the pair of photons produced in parametric downconversion processes.2 A multimode model for the downconversion process is used to carry out the calculation of three-photon detection probability that is associated with nonlocality of a single photon.

DOI
10.1364/oam.1991.mbb6

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With regard to the system of parametric downconversion, we analyze a recently proposed experiment for demonstrating the nonlocal property of a single photon.1 In the original proposal by Tan et al., a light field in a single photon-number state is split into two beams, which are then homodyned with two spatially separated weak local oscillators. The two-photon detection probability with the two homodyned beams provides evidence for nonlocality. It is shown that this phenomenon is in essence a phase-sensitive fourth-order interference effect. A single photonnumber state can be achieved from the pair of photons produced in parametric downconversion processes.2 A multimode model for the downconversion process is used to carry out the calculation of three-photon detection probability that is associated with nonlocality of a single photon.
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.
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 theoretical treatment is given of the process in which the two photons produced simultaneously in the parametric frequency splitting of light are allowed to interfere. It is shown that, while there is no interference in the usual sense involving quantities that are of the second order in the field, fourth-order interference effects are present. These may be revealed by measuring the joint probability of detecting two photons at two points x,x' in the interference plane with photoelectric detectors as a function of the separation x-x'. The probability exhibits a cosine modulation with x-x', with visibility that can approach 100%, even though the integration time in the experiment may greatly exceed the reciprocal bandwidth of the photons. The interference effect has a nonclassical origin and implies a violation of local realism in the highly correlated two-photon state.
A number of multiphoton interference schemes involving independent fields from parametric down-conversion are analyzed. They include fourth-order (two-photon) interference between two fields from two independent parametric down-conversion processes, interference with two independent single photon Fock states (produced by gated parametric down-conversion), and quantum state teleportation and entanglement swapping. It is shown that the visibility in these interference experiments is related to a quantity that is responsible for the photon bunching effect in one of the fields (signal or idler) from the parametric down-conversion process. Besides the two-photon interference schemes, a four-photon interference scheme is discussed where we consider four-photon partition at a beamsplitter. It is found that probability distribution of the partition is dramatically changed by four-photon interference of six paths and the four-photon effect also depends on the same quantity responsible for photon bunching. We will discuss the difference between two-photon interference and four-photon interference.
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.
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.