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Transverse Entanglement of Biphotons

Felix Just, Andrea Cavanna, Maria V. Chekhova, Gerd Leuchs

Published Jan 1, 2013• 6 citations
Publisher

Abstract

We study the transverse entanglement of biphotons generated via parametric downconversion as they propagate from the near to the far field. We obtain the Fedorov ratio and the Schmidt number and compare the results.
DOI
10.1364/cleo_qels.2013.jth2a.98

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We measure the transverse entanglement of photon pairs on their propagation from the near to the far field of spontaneous parametric down-conversion (SPDC). The Fedorov ratio, depending on the widths of conditional and unconditional intensity measurements, is shown to be only able to characterize entanglement in the near and far field zones of the source. Therefore we also follow a different approach. By evaluating the first-order coherence of a subsystem of the state we can quantify its entanglement. Unlike previous measurements, which determine the Fedorov ratio via intensity correlations, our setup is sensitive to both phase and modulus of the biphoton state and thus always grants experimental access to the full transverse entanglement of the SPDC state. It is shown theoretically that this scheme represents a direct measurement of the Schmidt number.
The reduced density matrix ${\ensuremath{\rho}}_{r}(x,{x}^{\ensuremath{'}})$ of the transverse biphoton state beyond the near zone is found to be real on and only on diagonals in the plane of the photon's transverse coordinates $(x,{x}^{\ensuremath{'}})$. This remarkable feature is found to occur at any distances of photon propagation with diffraction spreading completely taken into account. The functions of the reduced density matrix at two orthogonal diagonals, ${\ensuremath{\rho}}_{r}(x,x)$ and ${\ensuremath{\rho}}_{r}(x,\ensuremath{-}x)$, are considered as the only two complementary single-particle distributions generated by the reduced density matrix. The ratio of widths of these two distributions is interpreted as the new entanglement parameter Rdiag{R}_{\mathrm{diag}}Rdiag​ which is found to be very close to and completely compatible with the Schmidt entanglement parameter KKK.
We show that, although the amount of mutual entanglement of photons propagating in free space is fixed, the type of correlations between the photons that determine the entanglement can dramatically change during propagation. We show that this amounts to a migration of entanglement in Hilbert space, rather than real space. For the case of spontaneous parametric down-conversion, the migration of entanglement in transverse coordinates takes place from modulus to phase of the biphoton state and back again. We propose an experiment to observe this migration in Hilbert space and to determine the full entanglement.