The transverse spatial effects observed in photon pairs produced by parametric down-conversion provide a robust and fertile testing ground for studies of quantum mechanics, non-classical states of light, correlated imaging and quantum information. Over the last 20 years there has been much progress in this area, ranging from technical advances and applications such as quantum imaging to investigations of fundamental aspects of quantum physics such as complementarity relations, Bell's inequality violation and entanglement. The field has grown immensely: a quick search shows that there are hundreds of papers published in this field, some with hundreds of citations. The objective of this article is to review the building blocks and major theoretical and experimental advances in the field, along with some possible technical applications and connections to other research areas.

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We report a quantum eraser experiment which actually uses a Young double-slit to create interference. The experiment can be considered an optical analogy of an experiment proposed by Scully, Englert and Walther(SEW) [Nature 351, 111 (1991)]. One photon of an entangled pair is incident on a Young double-slit of appropriate dimensions to create an interference pattern in a distant detection region. Quarter-wave plates, oriented so that their fast axes are orthogonal, are placed in front of each slit to serve as which-path markers. The quarter-wave plates mark the polarization of the interfering photon and thus destroy the interference pattern. To recover interference, we measure the polarization of the other entangled photon. In addition, we perform the experiment under "delayed erasure" circumstances.42.50. Ar, 42.25.Kb

We report an experiment to generate entangled states of D-dimensional quantum systems, qudits, by using transverse spatial correlations of two parametric down-converted photons. Apertures with D slits in the arms of the twin photons define the qudit space. By manipulating the pump beam correctly, the twin photons will pass only by symmetrically opposite slits, generating entangled states between these different paths. Experimental results for qudits with D = 4 and 8 are shown. We demonstrate that the generated states are entangled states.

A fourth-order Young interference experiment was done to demonstrate a practical way to measure the de Broglie wavelength of a two-photon wave packet. A two-photon collinear beam is generated by type-II spontaneous parametric down-conversion. By modifying the transverse field profile of the pump laser beam that generates the two-photon beam we demonstrate that it is possible to measure the de Broglie wavelength of the single-photon constituents of the two-photon wave packet, the de Broglie wavelength of the two-photon wave packet as a whole and an ill defined intermediate de Broglie wavelength between the two cases. [S0031-9007 (99)08826-2] PACS numbers: 42.50.Ar, 03.65.BzIn a recent article, Jacobson et al.[1] have shown theoretically that the measured de Broglie wavelength of an object is dependent on its internal structure as well as on the detection system. Motivated by a recent experiment [2,3] with molecules they proposed an idealized interferometer that is capable of measuring the de Broglie wavelength of an incident multiphoton wave packet as a whole. For an ensemble of photons with average number n and wavelength l 0 , the de Broglie wavelength is l 0 n . Their proposed Mach-Zehnder interferometer has an "effective" beam splitter (BS) that is a function of a parameter x that varies from 0 (BS does not divide the composite system in constituent quanta) to 1 (BS divides the composite system in its constituent quanta). A calculation done with an incident state jc 0 ͘ j2, 0͘ (two photons incident in one of the ports and zero photons in the other) shows that the oscillation period of the interferometer varies from l 0 ͑x 1͒, corresponding to single-photon interference, to l 0 2 ͑x 0͒, corresponding to two-photon interference. It is interesting to notice that for intermediate values of x the de Broglie wavelength is not well defined, even though the state energy is well defined.In this article, we demonstrate that we can measure the de Broglie wavelength of a two-photon wave packet (biphoton) with a Young double-slit experiment. The incident two-photon wave packet is generated collinearly from a nonlinear crystal by the process of spontaneous parametric down-conversion. The photons transmitted by the double slit form a fourth-order pattern which is a superposition of two Young interference patterns with different periodicity

We propose a simple scheme for complete Bell-state measurement of photons using hyperentangled states -entangled in multiple degrees of freedom. In addition to hyperentanglement, our scheme requires only linear optics and single photon detectors, and is realizable with current technology. At the cost of additional classical communication, our Bell-state measurement can be implemented nonlocally. We discuss the possible application of these results to quantum dense coding and quantum teleportation.

We consider multimode two-photon interference at a beam splitter by photons created by spontaneous parametric down-conversion. The resulting interference pattern is shown to depend upon the transverse spatial symmetry of the pump beam. In an experiment, we employ the first-order Hermite-Gaussian modes in order to show that, by manipulating the pump beam, one can control the resulting two-photon interference behavior. We expect these results to play an important role in the engineering of quantum states of light for use in quantum information processing and quantum imaging.PACS numbers: 03.65Bz, 42.50.ArEntangled photons have proven to be a great tool in the study of quantum phenomena and promise to play an important role in quantum information processing [1] as well as quantum imaging [2,3]. The most common source of entangled photons is spontaneous parametric down-conversion (SPDC), in which the interaction of a pump photon with a birefringent nonlinear crystal creates two daughter photons. Under certain experimental conditions, the down-converted photons may be entangled in momentum [4], energy [5], polarization [6,7] and/or angular momentum [8].Two-photon interference at a beam splitter was first demonstrated by Hong, Ou and Mandel (HOM) [9]. It has since been utilized in quantum tests of nonlocality [10] as well as many optical implementations of quantum information protocol such as Bell-state measurements [11,12] and may be used to construct quantum optical logic gates [13,14]. To date, however, most experiments utilizing HOM-type interference consider an ideal monomode situation. In this paper, we consider multimode two-photon interference of photon pairs created by SPDC. We show how the transverse amplitude profile of the pump beam in SPDC determines whether the down-converted fields interfere constructively or destructively. We present our experiment and conclude by noting the relevance of these results to quantum optical information processing.

We characterize entanglement in two-qubit pure states encoded in transverse momenta of twin photons obtained from spontaneous parametric down-conversion. Two alternate methods are employed: ͑i͒ measurement of conditional interference patterns and ͑ii͒ measurement of the marginal probability that yields the single-photon interference pattern. Both methods are local with classical communication and rely on Schmidt decomposition of the quantum state, which is generated by letting the photons propagate through an appropriate lens system. In both cases we can obtain the concurrence either through the Schmidt coefficients in the first method or directly relating the concurrence to the visibility of the interference pattern in the second one.