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Optics Express

Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces

Jonathan Leach, Barry Jack, Jacquiline Romero, Monika Ritsch‐Marte, Robert W. Boyd, Anand K. Jha, Stephen Mark Barnett, Sonja Franke‐Arnold, Miles J. Padgett

Published Apr 30, 2009• 176 citations
Publisher

Abstract

We observe entanglement between photons in controlled super-position states of orbital angular momentum (OAM). By drawing a direct analogy between OAM and polarization states of light, we demonstrate the entangled nature of high order OAM states generated by spontaneous downconversion through violation of a suitable Clauser Horne Shimony Holt (CHSH)-Bell inequality. We demonstrate this violation in a number of two-dimensional subspaces of the higher dimensional OAM Hilbert space.

DOI
10.1364/oe.17.008287

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We observe entanglement between photons in controlled super-position states of orbital angular momentum (OAM). By drawing a direct analogy between OAM and polarization states of light, we demonstrate the entangled nature of high order OAM states generated by spontaneous downconversion through violation of a suitable Clauser Horne Shimony Holt (CHSH)-Bell inequality. We demonstrate this violation in a number of two-dimensional subspaces of the higher dimensional OAM Hilbert space.
Orbital angular momentum entangled photons emitted by a down-conversion source are shown to be in higher dimensional entangled states. This is experimentally confirmed by demonstrating a violation of Clauser-Horne-Shimony-Holt-type Bell inequalities. The experiment is performed in two steps. First, the two-photon two-dimensional entanglement is demonstrated for orbital angular momentum states by violating the usual CHSH Bell inequality. In a second step the setup is extended in order to show that the entanglement also holds in three dimensions. By using a generalized type of the CHSH Bell inequality, a violation of more than 18 standard deviations is achieved. Higher dimensional entangled states thus allow the realization of new types of quantum communication protocols. They also provide a more secure quantum cryptography scheme.
The symmetric orbital angular momentum (OAM) entanglement in the subspace of { ± ℓ} has been extensively exploited in quantum information science. Here we investigate instead the Bell inequalities for the two-dimensional subspace spanned by asymmetric OAM modes of ℓ 1 and ℓ 2 , usually corresponding to the quantum property of nonmaximal entanglement. We demonstrate, both theoretically and experimentally, the optimal violation of a suitable Clauser-Horne-Shimony-Holt Bell inequality for the asymmetric OAM modes, thus manifesting the entanglement for asymmetric OAM states space.
Orbital angular momentum entangled photons emitted by a down-conversion source are in higher dimensional entangled states. Here we report the experimental confirmation by demonstrating a violation of a generalized Clauser-Horne-Shimony-Holt-type Bell inequality in three dimensions by more than 18 standard deviations. Higher dimensional entangled states allow the realization of new types of quantum communication protocols. They also provide a more secure quantum cryptography scheme. Therefore our experimental results are likely to have applications in future quantum communication technology.
The Bell inequalities for azimuthal quantum number, i.e., orbital angular momentum, have been extensively reported to show quantum correlations between the spatial degree of freedom of entangled photon pairs. Here we investigate instead the Bell inequalities for radial quantum number. We demonstrate, both theoretically and experimentally, the violation of a suitable Clauser-Horne-Shimony-Holt Bell inequality for the radial index ppp of the Laguerre-Gaussian mode, thus manifesting the entanglement in the radial structure of two-photon wave functions.
We use spatial light modulators (SLMs) to measure correlations between arbitrary superpositions of orbital angular momentum (OAM) states generated by spontaneous parametric down-conversion. Our technique allows us to fully access a two-dimensional OAM subspace described by a Bloch sphere, within the higher-dimensional OAM Hilbert space. We quantify the entanglement through violations of a Bell-type inequality for pairs of modal superpositions that lie on equatorial, polar, and arbitrary great circles of the Bloch sphere. Our work shows that SLMs can be used to measure arbitrary spatial states with a fidelity sufficient for appropriate quantum information processing systems.
Based on spin-orbit coupling induced by q-plates, we present a feasible experimental proposal for preparing two-dimensional spatially inhomogeneous polarizations of light. We further investigate the quantum correlations between these inhomogeneous polarizations of photon pairs generated by spontaneous parametric downconversion, which in essence describe the so-called hypoentanglement that is established between composite spin-orbit variables of photons. The violation of the Clauser–Horne–Shimony–Holt–Bell inequality is predicted with S=22 to illustrate the entangled nature of the cylindrical symmetry of spatially inhomogeneous polarizations.