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arXiv:1207.6932 · Physical Review A

Dimensionality of the spatiotemporal entanglement of parametric down-conversion photon pairs

A. Gatti, T. Corti, E. Brambilla, D. B. Horoshko

Published Nov 5, 2012• 60 citations
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Abstract

In this work, the Schmidt number of the two-photon state generated by parametric down-conversion (PDC) is evaluated in the framework of a fully spatiotemporal model for PDC. A comparison with the results obtained in either purely spatial or purely temporal models shows that the degree of entanglement of the PDC state cannot be trivially reduced to the product of the Schmidt numbers obtained in models with lower dimensionality, unless the detected bandwidth is very narrow. This result is a consequence of the nonfactorability of the state in the spatial and temporal degrees of freedoms of twin photons. In the limit of a broad pump beam, we provide a geometrical interpretation of the Schmidt number as the ratio between the volume of the phase-matching region and of a correlation volume.

DOI
10.1103/physreva.86.053803

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In this work, the Schmidt number of the two-photon state generated by parametric down-conversion (PDC) is evaluated in the framework of a fully spatiotemporal model for PDC. A comparison with the results obtained in either purely spatial or purely temporal models shows that the degree of entanglement of the PDC state cannot be trivially reduced to the product of the Schmidt numbers obtained in models with lower dimensionality, unless the detected bandwidth is very narrow. This result is a consequence of the nonfactorability of the state in the spatial and temporal degrees of freedoms of twin photons. In the limit of a broad pump beam, we provide a geometrical interpretation of the Schmidt number as the ratio between the volume of the phase-matching region and of a correlation volume.
We investigate the spatiotemporal structure of the biphoton entanglement in parametric down-conversion (PDC) and we demonstrate its nonfactorable X-shaped geometry. Such a structure gives access to the ultrabroad bandwidth of PDC, and can be exploited to achieve a biphoton temporal localization in the femtosecond range. This extreme localization is connected to our ability to resolve the photon positions in the source near field. The nonfactorability opens the possibility of tailoring the temporal entanglement by acting on the spatial degrees of freedom of twin photons.
A simple analytic decomposition of the spatially entangled two-photon field allows us to generalize the earlier results of Law and Eberly [Phys. Rev. Lett. 92, 127903 (2004)] to more realistic experimental geometries of spontaneous parametric down-conversion. We quantify analytically how spatial filtering reduces the Schmidt number or dimensionality of the two-photon entanglement from a ``generated'' value to a generally much lower ``usable or detected'' value, for both collinear and noncollinear phase matching. We also discuss the intimate relation between the (two-photon) Schmidt number and the classical (one-photon) concept of etendue or geometric extent.
We investigate the spatio-temporal structure of the biphoton entanglement in Parametric Down Conversion (PDC). In particular we study the biphoton amplitude at the output face of the nonlinear crystal (near-field) and we demonstrate its X-shaped geometry in the space-time dimensions, i.e. the non-factorability of the state with respect to spatial and temporal variables. Our analysis provides a precise and quantitative characterization of this structure in various regimes and types of phase matching of PDC. The key elements of novelty emerging from our analysis are the non-factorability of the state with respect to spatial and temporal variables, and the extreme relative localization of the entangled photons, both in space (few microns) and time (few femtoseconds). This extreme localization is connected to our ability to resolve the photon positions in the source near-field. The non factorability opens the possibility of tailoring the temporal entanglement by acting on the spatial degrees of freedom of twin photons.
We investigate the spatio-temporal structure of the bi-photon entanglement in parametric down-conversion (PDC) and we demonstrate its non-factorable X-shaped geometry. Such a structure gives access to the ultra-broad bandwidth of PDC, and can be exploited to achieve a bi-photon temporal localization in the femtosecond range. This extreme localization is connected to our ability to resolve the photon positions in the source near-field. The non factorability opens the possibility of tailoring the temporal entanglement by acting on the spatial degrees of freedom of twin photons.
In this work we propose and analyze a scheme where the full spatiotemporal correlation of twin photons or beams generated by parametric down-conversion is detected by using its inverse process, i.e., sum frequency generation. Our main result is that, by imposing independently a temporal delay $\ensuremath{\Delta}t$ and a transverse spatial shift $\ensuremath{\Delta}x$ between two twin components of PDC light, the up-converted light intensity provides information on the correlation of the PDC light in the full spatiotemporal domain and should enable the reconstruction of the peculiar X-shaped structure of the correlation predicted previously [A. Gatti et al. Phys. Rev. Lett. 102, 223601 (2009); L. Caspani et al. Phys. Rev. A 81, 033808 (2010); E. Brambilla et al. Phys. Rev. A 82, 013835 (2010)]. Through both a semianalytical and a numerical modeling of the proposed optical system, we analyze the feasibility of the experiment and identify the best conditions to implement it. In particular, the tolerance of the phase-sensitive measurement against the presence of dispersive elements, imperfect imaging conditions, and possible misalignments of the two crystals is evaluated.