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arXiv:1406.6239 · Physical Review A

Violation of Bell's inequality for phase-singular beams

Shashi Prabhakar, Salla Gangi Reddy, Abdul Rahim Aadhi, Chithrabhanu Perumangatt, G. K. Samanta, Ravindra Pratap Singh

Published Aug 14, 2015• 23 citations
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Abstract

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.

DOI
10.1103/physreva.92.023822

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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 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.
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.
Abstract Banaszek, Wódkiewicz and others (Banaszek and Wdkiewicz 1998 Phys. Rev . A 58 4345; Chen et al 2002 Phys. Rev. Lett . 88 040406; Chen and Zhang 2002 Phys. Rev . A 65 044102) made the surprising discovery that Einstein–Bell locality inequalities can be violated by the two mode squeezed vacuum by a factor , in spite of the fact that the state has a positive Wigner function. I use here the more general Gleason–Kochen–Specker assumption of non-contextuality (Gleason 1957 J. Math. Mech . 6 885) to express classicality. I then derive non-contextuality Bell inequalities for correlations of N pseudo spins embedded in an infinite dimensional continuous variable Hilbert space, and show that their maximum possible quantum violation is by a factor 2 ( N −1)/2 . I find quantum states for which this maximum violation is reached. I also show that the familiar displaced squeezed vacuum for a single optical mode, which has a positive Wigner function, can violate the inequality by a factor for odd . The arbitrarily large non-classicality means that realizations of the pseudo-spin measurements even in a single mode photon state might afford similar opportunities in quantum information tasks as entangled N qubit systems with large N .
There have been theoretical and experimental studies on quantum nonlocality for continuous variables, based on dichotomic observables. In particular, we are interested in two cases of dichotomic observables for the light field of continuous variables: One case is even and odd numbers of photons and the other case is no photon and the presence of photons. We analyze various observables to give the maximum violation of Bell's inequalities for continuous-variable states. We discuss an observable which gives the violation of Bell's inequality for any entangled pure continuous-variable state. However, it does not have to be a maximally entangled state to give the maximal violation of Bell's inequality. This is attributed to a generic problem of testing the quantum nonlocality of an infinite-dimensional state using a dichotomic observable.