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Physical Review A

Violation of a Bell inequality in two-dimensional state spaces for radial quantum number

Dongkai Zhang, Xiaodong Qiu, Wuhong Zhang, Lixiang Chen

Published Oct 29, 2018• 22 citations
Publisher

Abstract

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 $p$ of the Laguerre-Gaussian mode, thus manifesting the entanglement in the radial structure of two-photon wave functions.

DOI
10.1103/physreva.98.042134

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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.
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 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.
Orbital angular momentum (OAM) entanglement is investigated in the Bessel-Gaussian (BG) basis. Having a readily adjustable radial scale, BG modes provide an alternative basis for OAM entanglement over Laguerre-Gaussian modes. We show that the OAM bandwidth in terms of BG modes can be increased by selection of particular radial wavevectors and leads to a flattening of the spectrum, which allows for higher dimensionality in the entangled state. We demonstrate entanglement in terms of BG modes by performing a Bell-type experiment and showing a violation of the Clauser-Horne-Shimony-Holt inequality for the ℓ = ±1 subspace. In addition, we use quantum state tomography to indicate higher-dimensional entanglement in terms of BG modes.
We show that nonlocal correlation experiments on the two spatially separated modes of a maximally path-entangled number state may be performed. They lead to a violation of a Clauser-Horne Bell inequality for any finite photon number NNN. We also present an analytical expression for the two-mode Wigner function of a maximally path-entangled number state and investigate a Clauser-Horne-Shimony-Holt Bell inequality for such a state. We test other Bell-type inequalities. Some are violated by a constant amount for any NNN.
Single photons emerging from Pancharatnam-Berry phase optical elements (q plates) exhibit entanglement in the degrees of freedom of spin and orbital angular momentum. We put forward an experimental scheme for probing the spin-orbit correlations of single photons. It is found that the Clauser-Horne-Shimony-Holt parameter S for the single-photon spin-orbit entangled state could be as high as 2√2, evidently violating the Bell-like inequality and thus invalidating the noncontextual hidden variable theories.