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Physical Review A

Spatial effects in two- and four-beam interference of partially entangled biphotons

Bahaa E. A. Saleh, Adel Joobeur, Malvin Carl Teich

Published May 1, 1998• 33 citations
Publisher

Abstract

Fourth-order interference patterns of biphotons generated by type-I spontaneous parametric downconversion are examined for beams with partial angular and spectral entanglement, both for degenerate and nondegenerate constituent photons. Two-beam configurations using a single beam-splitter interferometer and a Mach-Zehnder interferometer are explicitly studied, as are four-beam configurations using pairs of beam splitters and Mach-Zehnder interferometers. The interference pattern generally comprises a number of harmonic functions of the path-length difference with frequencies and visibilities that usually depend on direction, and may have a finite or infinite duration (fourth-order coherence length). The relation between the visibility and the spectral indistinguishability of the two beams is established. Certain components of the interference pattern are independent of direction so that they are not washed out by the use of apertures of finite size.

DOI
10.1103/physreva.57.3991

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Journal of Optics B Quantum and Semiclassical Optics•Feb 1, 2001•18 citations
Fourth-order interference patterns of biphotons generated by type-I spontaneous parametric downconversion are examined for beams with partial angular and spectral entanglement, both for degenerate and nondegenerate constituent photons. Two-beam configurations using a single beam-splitter interferometer and a Mach-Zehnder interferometer are explicitly studied, as are four-beam configurations using pairs of beam splitters and Mach-Zehnder interferometers. The interference pattern generally comprises a number of harmonic functions of the path-length difference with frequencies and visibilities that usually depend on direction, and may have a finite or infinite duration (fourth-order coherence length). The relation between the visibility and the spectral indistinguishability of the two beams is established. Certain components of the interference pattern are independent of direction so that they are not washed out by the use of apertures of finite size.
We examine the fourth-order interference properties of two photons at a lossless four-port optical device. The photon pair is represented by a joint single-photon wave-packet state with arbitrary spectral composition. We explicitly determine the coincidence probabilities at the output ports of a beam splitter and a Mach-Zehnder interferometer when the incident wave packet is described by a joint Gaussian spectral distribution and the two photons are incident one at each port of the device. For uncorrelated and distinguishable input photons, the fourth-order interference is readily understood in terms of the particle or wave behavior of each photon acting separately. On the other hand, highly correlated photon pairs, such as those generated from spontaneous parametric down-conversion, exhibit fourth-order interference effects at the output of the interferometer that depend expressly on the joint nature of the photon-pair wave packet, and cannot be described by the particle or wave behavior of either photon.
We report the results of two experiments performed with two-photon light, produced via collinear degenerate optical spontaneous parametric downconversion (SPDC), in which both second-order (one-photon) and fourth-order (two-photon) interferograms are recorded in a Mach-Zehnder interferometer (MZI). In the first experiment, high-visibility fringes are obtained for both the second- and fourth-order interferograms. In the second experiment, the MZI is modified by the removal of a mirror from one of its arms; this leaves the fourth-order interferogram unchanged, but extinguishes the second-order interferogram. A theoretical model that takes into consideration both the temporal and spatial degrees-of-freedom of the two-photon state successfully explains the results. While the temporal interference in the MZI is independent of the spatial coherence of the source, that of the modified MZI is not.
Theory and experiment of two-photon lquotlquotlquotdirectrquotrquotrquot interference for type-II spontaneous parametric down-conversion with an ultrafast pulse laser pump
Fourth-order spatial interference of entangled photon pairs generated in the process of spontaneous parametric down-conversion pumped by a femtosecond pulse laser has been performed for the first time. In theory, it takes into account the transverse correlation between the two photons and is used to calculate the dependence of the visibility of the interference pattern obtained in Young’s double-slit experiment. In this experiment, a short focal length lens and two narrow band interference filters were adopted to eliminate the effects of the broadband pump laser and improve the visibility of the interference pattern under the condition of nearly collinear light and degenerate phase matching.
Entangled-photon pairs (biphotons) generated by spontaneous optical parametric down-conversion exhibit a number of properties that are analogous to those of ordinary photons generated by incoherent sources. The spatial pump-field distribution and the two-particle wave function in the biphoton case play the respective roles of the source intensity distribution and the second-order coherence function in the incoherent case. The van Cittert--Zernike theorem, which is applicable for incoherent optical sources emitting independent photons, has a counterpart for biphotons. Likewise, the partial-coherence theory of image formation has an analogous counterpart for biphoton beams transporting spatial information. However, an underlying duality, rather than analogy, accompanies the mathematical similarity between incoherent and biphoton emissions if the comparison is made between the photon count rate in the incoherent case and the biphoton count rate in the entangled-photon case. The smaller the size of an incoherent source, the more separable is the coherence function and the more coherent is the field, and therefore the higher the visibility of ordinary interference fringes. In contrast, the narrower the size of a biphoton pump source, the more separable is the wave function and the less entangled is the field, and therefore the lower the visibility of biphoton interference fringes. This duality is similar to the complementarity between single and two-photon interference exhibited for biphotons.
We present theoretical and experimental second- and fourth-order interference patterns for entangled photons of different colors entering single and dual Mach-Zehnder interferometers in which dispersive elements have been deliberately placed. Although photon wave packets are generally broadened, as well as delayed, by passage through dispersive optical elements, coincidence measurements made with entangled photon pairs can be free of such broadening. This occurs for materials with particular dispersive behavior, as well as when the dispersion is balanced in both arms. This nonlocal behavior arises from the entanglement in the frequencies of the down-converted pair photons. We also show that nonlocal pump-frequency oscillations are present in the coincidence rate patterns for long path-length-difference times, confirming the robustness of this phenomenon in the presence of dispersion. However, the magnitude of these oscillations is reduced when an arbitrary dispersive material is used in an unbalanced configuration. Difference-frequency oscillations are also robust in the presence of dispersion, although they decay for large path-length-difference times.
In this paper we investigate the fourth-order coherence of biphoton beams from spontaneous parametric downconversion resulting from a nonlinear crystal as exemplified in a double-slit interference configuration where the signal and idler beams both pass through the same double slits. We find that the angle of the crystal optical axis and the crystal length are important factors, along with the double-slit separation and system bandwidth, in determining the nature of the fourth-order interference pattern obtained behind the slits. Only careful planning of system parameters and understanding of the tuning curves of the downconversion will illuminate the obtained coincidence patterns.