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arXiv:1002.1798 · Journal of the Optical Society of America B

Tripartite entanglement in parametric down-conversion with spatially structured pump

D. Daems, F. Bernard, Nicolas J. Cerf, Mikhail I. Kolobov

Published Feb 16, 2010• 35 citations
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Abstract

Most investigations of multipartite entanglement have been concerned with temporal modes of the electromagnetic field and have neglected its spatial structure. We present a simple model which allows us to generate tripartite entanglement between spatial modes by parametric down-conversion with two symmetrically tilted plane waves serving as a pump. The characteristics of this entanglement are investigated. We also discuss the generalization of our scheme to 2N+1 partite entanglement using 2N symmetrically tilted plane pump waves. Another interesting feature is the possibility of entanglement localization in just two spatial modes.

DOI
10.1364/josab.27.000447

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Most investigations of multipartite entanglement have been concerned with temporal modes of the electromagnetic field and have neglected its spatial structure. We present a simple model which allows us to generate tripartite entanglement between spatial modes by parametric down-conversion with two symmetrically tilted plane waves serving as a pump. The characteristics of this entanglement are investigated. We also discuss the generalization of our scheme to 2N+1 partite entanglement using 2N symmetrically tilted plane pump waves. Another interesting feature is the possibility of entanglement localization in just two spatial modes.
We present a simple model together with its physical implementation which allows one to generate multipartite entanglement between several spatial modes of the electromagnetic field. It is based on parametric down-conversion with NNN pairs of symmetrically tilted plane waves serving as a pump. The characteristics of this spatial entanglement are investigated in the cases of zero as well as nonzero phase mismatch. Furthermore, the phenomenon of entanglement localization in just two spatial modes is studied in detail and shown to result in an enhancement of the entanglement by a factor N\sqrt{N}N​.
In this Rapid Communication, we propose, following the spirit of quantum imaging, to generalize the theory of multipartite entanglement for the continuous-variable Gaussian states by considering, instead of the global covariance matrix, the local correlation matrix at two different spatiotemporal points $(\mathrm{\ensuremath{\rho}P\vec},t)$ and $(\mathrm{\ensuremath{\rho}P\vec} {}^{'},{t}^{'})$, with $\mathrm{\ensuremath{\rho}P\vec}$ being the transverse coordinate. Our approach makes it possible to introduce the characteristic spatial length and the characteristic time of the multipartite entanglement, which in general depend on the number of ``parties'' in the system. As an example, we consider tripartite entanglement in spontaneous parametric down-conversion with a spatially structured pump. We investigate spatiotemporal properties of such entanglement and calculate its characteristic spatial length and time.
It is known that the entanglement present in the transverse spatial properties of photon pairs generated by spontaneous parametric down-conversion depends explicitly on the characteristics of the pump laser beam. Here we show that, for pump beams described by a Hermite-Gaussian mode, the entanglement depends on the beam width as well as the order of the beam. The amount of entanglement generally increases with the order of the beam and is less dependent on the beam width for higher orders. We propose a method to measure the spatial entanglement directly, independent of the dimension of the subsystems.
We use a versatile model to evaluate the multipartite entanglement and the nonclassical light generation in optical parametric oscillators, exploring the differences between doubly and triply resonant cavity configurations. We demonstrate the entanglement of the pump mode with converted fields in both situations, and the fundamental differences of oscillators using parametric down conversion and four wave mixing processes as the intracavity amplification technique. The strong correlations involving the sidebands of the pump and converted fields give the signatures of a rich dynamic of multipartite entanglement.
We show that the transfer of the plane-wave spectrum of the pump beam to the fourth-order transverse spatial correlation function of the two-photon field generated by spontaneous parametric down-conversion leads to the conservation and entanglement of orbital angular momentum of light. By means of a simple experimental setup based on fourth-order (or two-photon) interferometry, we show that our theoretical model provides a good description for down-converted fields carrying orbital angular momentum.