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arXiv:1003.0180 · Physical Review A

Spatial multipartite entanglement and localization of entanglement

D. Daems, Nicolas J. Cerf

Published Sep 9, 2010• 12 citations
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Abstract

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 $N$ 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 $\sqrt{N}$.

DOI
10.1103/physreva.82.032303

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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​.
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.
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.
Quantum multipartite entanglement is a striking phenomenon predicted by quantum mechanics when several parts of a physical system share the same quantum state that cannot be factorized into the states of individual subsystems. The Gaussian quantum states are usually characterized by the covariance matrix of the quadrature components. A powerful formalism for treating the Gaussian states is that of the symplectic eigenvalues. In particular, a quantitative measure of multipartite entanglement is the so-called logarithmic negativity, related to the symplectic eigenvalues of the partially transposed covariance matrix. Considering only global variances of the field quadratures one completely neglects the spatiotemporal properties of the electromagnetic field. We propose, following the spirit of quantum imaging, to generalize the theory of multipartite entanglement for the continuous variable Gaussian states by considering the local correlation matrix at two different spatiotemporal points [see manuscript for characters] and [see manuscript for characters] with [see manuscript for characters] being the transverse coordinate. For stationary and homogeneous systems one can also introduce the spatiotemporal Fourier components of the correlation matrix. The formalism of the global symplectic eigenvalues can be straightforwardly generalized to the frequency-dependent symplectic eigenvalues. This generalized theory allows, in particular, to introduce the characteristic spatial area and time of the multipartite entanglement, which in general depend on the number of "parties" in the system. As an example we consider multipartite entanglement in spontaneous parametric down-conversion with spatially-structured pump. We investigate spatial properties of such entanglement and calculate its characteristic spatial length.
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.
By making use of the spatial shape of paired photons, parametric down-conversion allows the generation of two-photon entanglement in a multidimensional Hilbert space. How much entanglement can be generated in this way? In principle, the infinite-dimensional nature of the spatial degree of freedom renders unbounded the amount of entanglement available. However, in practice, the specific configuration used, namely, its geometry, the length of the nonlinear crystal, and the size of the pump beam, can severely limit the value that could be achieved. Here we show that the use of quasi-phase-matching engineering allows one to increase the amount of entanglement generated, reaching values of tens of ebits of entropy of entanglement under different conditions. Our work thus opens a way to fulfill the promise of generating massive spatial entanglement under a diverse variety of circumstances, some more favorable for its experimental implementation.