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arXiv:1505.06103 · New Journal of Physics

Experimental violation of a Bell-like inequality with optical vortex beams

B. Stoklasa, L. Motka, Jaroslav Řeháček, Z. Hradil, Luis Lorenzo Sánchez-Soto, G. S. Agarwal

Published Nov 19, 2015• 29 citations
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Abstract

Optical beams with topological singularities have a Schmidt decomposition.Hence, they display features typically associated with bipartite quantum systems; in particular, these classical beams can exhibit entanglement.This classical entanglement can be quantified by a Bell inequality formulated in terms of Wigner functions.We experimentally demonstrate the violation of this inequality for Laguerre-Gauss (LG) beams and confirm that the violation increases with increasing orbital angular momentum.Our experimental scheme, which is designed to give directly the parity of the Wigner function, yields negativity at the origin for LG 10 beams, whereas for LG 20 we always get a positive value.

DOI
10.1088/1367-2630/17/11/113046

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Optical beams with topological singularities have a Schmidt decomposition.Hence, they display features typically associated with bipartite quantum systems; in particular, these classical beams can exhibit entanglement.This classical entanglement can be quantified by a Bell inequality formulated in terms of Wigner functions.We experimentally demonstrate the violation of this inequality for Laguerre-Gauss (LG) beams and confirm that the violation increases with increasing orbital angular momentum.Our experimental scheme, which is designed to give directly the parity of the Wigner function, yields negativity at the origin for LG 10 beams, whereas for LG 20 we always get a positive value.
We consider optical beams with topological singularities which possess Schmidt decomposition and show that such classical beams share many features of two mode entanglement in quantum optics. We demonstrate the coherence properties of such beams through the violations of Bell inequality for continuous variables using the Wigner function. This violation is a consequence of correlations between the (x,px)(x,{p}_{x})(x,px​) and (y,py)(y,{p}_{y})(y,py​) spaces which mathematically play the same role as nonlocality in quantum mechanics. The Bell violation for the Laguerre-Gaussian beams is shown to increase with higher orbital angular momenta lll of the vortex beam. This increase is reminiscent of enhancement of nonlocality for many particle Greenberger-Horne-Zeilinger states or for higher spins. The states with large lll can be easily produced using spatial light modulators.
We have considered optical beams with phase singularity and experimentally verified that these beams, although classical, have properties of two-mode entanglement in quantum states. We have observed the violation of Bell's inequality for continuous variables using the Wigner distribution function (WDF) proposed by P. Chowdhury et al. [Phys. Rev. A 88, 013830 (2013)]. Our experiment establishes an alternate form of Bell's inequality in terms of the WDF which can be used for classical as well as quantum systems.
We perform Bell's measurement for the non-separable correlation between polarization and orbital angular momentum from the same classical vortex beam. The violation of Bell's inequality for such a non-separable classical correlation has been demonstrated experimentally. Based on the classical vortex beam and non-quantum entanglement between the polarization and the orbital angular momentum, the Hadamard gates and conditional phase gates have been designed. Furthermore, a quantum Fourier transform has been implemented experimentally.
We experimentally demonstrate a method for enhancing and flattening multiplexed entanglement in the four-wave mixing (FWM) process, which is implemented by replacing Laguerre-Gaussian (LG) modes with perfect vortex (PV) modes. For the topological charge l ranging from -5 to 5, the entanglement degrees of orbital angular momentum (OAM) multiplexed entanglement with PV modes are all larger than those of OAM multiplexed entanglement with LG modes. More importantly, for OAM multiplexed entanglement with PV modes, the degree of entanglement almost does not change with the topology value. In other words, we experimentally flatten the OAM multiplexed entanglement, which cannot be achieved in OAM multiplexed entanglement with LG modes based on the FWM process. In addition, we experimentally measure the entanglement with coherent superposition OAM modes. Our scheme provides a new, to the best of our knowledge, platform to construct an OAM multiplexed system and may find potential applications in realizing the parallel quantum information protocols.
Structured light has many applications in areas such as quantum information and quantum optics. The angular momentum of structured light makes it possible to access higher dimensional systems. Ince–Gaussian beams are the solution of the paraxial wave equation in elliptical coordinates, which are characterized by a parameter called ellipticity. In this work, we obtain expressions for the Wigner function and Bell inequality for Ince–Gauss modes. This is done through the corresponding generalization of the structured light modes that the Ince–Gauss modes represent and therefore of the Wigner function. Geometric representations of structured light modes are of great use here. The Wigner function is important for the purpose of determining the nonlocal properties of Ince–Gauss modes. In this way, we demonstrate the nonlocality of the Ince–Gauss modes through the violation of the Bell inequality. We also give a detailed analysis of the behavior of the Bell function and the violation of the Bell inequality of Laguerre–Gauss modes relevant for our description of Ince–Gauss modes, therefore characterizing their nonlocality.