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arXiv:1401.7364 · International Journal of Quantum Information

Spatio–temporal entanglement of twin photons: An intuitive picture

A. Gatti, Lucia Caspani, T. Corti, E. Brambilla, Ottavia Jedrkiewicz

Published Mar 1, 2014• 6 citations
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Abstract

We draw an intuitive picture of the spatio–temporal properties of the entangled state of twin photons, where they are described as classical wave-packets. This picture predicts a precise relation between their temporal and transverse spatial separations at the crystal output. The space-time coupling described by classical arguments turns out to determine in a precise way the spatio–temporal structure of the quantum entanglement, analyzed by means of the biphotonic correlation and of the Schmidt dimensionality of the entanglement.

DOI
10.1142/s0219749914610164

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We draw an intuitive picture of the spatio–temporal properties of the entangled state of twin photons, where they are described as classical wave-packets. This picture predicts a precise relation between their temporal and transverse spatial separations at the crystal output. The space-time coupling described by classical arguments turns out to determine in a precise way the spatio–temporal structure of the quantum entanglement, analyzed by means of the biphotonic correlation and of the Schmidt dimensionality of the entanglement.
The entangled photon pairs (biphotons) produced from parametric down-conversion (PDC) in a χ(2)crystal are the key element for several quantum information and communication schemes. Recent theoretical investigations [1–2] have revealed the peculiar structure of the biphotonic correlation characterizing the entanglement of twin beams in the space-time domain. The biphotonic correlation in type I PDC is characterized in the near field by a X-shaped geometry, which turns out to be non-factorable in space and time, offering thus the relevant possibility of manipulating the temporal bandwidth of biphoton entanglement by acting on their spatial degrees of freedom. The name “X-entanglement” was coined to describe this geometry [1]. An other key feature which emerged is the extreme spatial and temporal localization of the biphotonic correlation, on the micrometer and femtosecond range, respectively.
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.
This work analyzes the temporal coherence and correlation of counterpropagating twin photons generated in a quasiphase matched nonlinear crystal by spontaneous parametric down-conversion. We find out different pictures depending on the pump pulse duration relative to two characteristic temporal scales, determined, respectively, by the temporal separation between the counterpropagating and the co-propagating wave packets. When the pump duration is intermediate between the two scales, we show a transition from a highly entangled state to an almost separable state, with strongly asymmetric spectral properties of the photons.