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Journal of the Optical Society of America B

Nonlocality and quantum correlations in Ince–Gauss structured light modes

María A. González-Domínguez, A Elena Piceno-Martínez, L. Rosales-Zárate

Published Feb 27, 2023• 7 citations
Publisher

Abstract

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.

DOI
10.1364/josab.482580

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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.
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 present a quantum-mechanical analysis of the orbital angular momentum of a class of recently discovered elliptically symmetric stable light fields---the so-called Ince-Gauss modes. We study, in a fully quantum formalism, how the orbital angular momentum of these beams varies with their ellipticity, and we discover several compelling features, including nonmonotonic behavior, stable beams with real continuous (noninteger) orbital angular momenta, and orthogonal modes with the same orbital angular momenta. We explore, and explain in detail, the reasons for this behavior. These features may have applications in quantum key distribution, atom trapping, and quantum informatics in general---as the ellipticity opens up an alternative way of navigating the spatial photonic Hilbert space.
Fields generated in a large number of nonlinear-optical processes (including those with losses) have a Wigner distribution that is Gaussian centered around the mean value of the field. We show that the Bell inequality for an optical-correlation experiment with two coupled modes generated in a nonlinear process can be expressed as an inequality relating to the parameters of the underlying Wigner distribution function. We consider examples of parametric down-conversion and a two-mode squeezed laser, which can be used in experiments to demonstrate quantum nonlocality.
Photons with complex spatial mode structures open up possibilities for new fundamental high-dimensional quantum experiments and for novel quantum information tasks. Here we show entanglement of photons with complex vortex and singularity patterns called Ince-Gauss modes. In these modes, the position and number of singularities vary depending on the mode parameters. We verify two-dimensional and three-dimensional entanglement of Ince-Gauss modes. By measuring one photon and thereby defining its singularity pattern, we nonlocally steer the singularity structure of its entangled partner, while the initial singularity structure of the photons is undefined. In addition we measure an Ince-Gauss specific quantum-correlation function with possible use in future quantum communication protocols.
In this work, we present an extensive analysis on the nature and performance of Ince-Gauss beams, elliptical solutions of the paraxial wave equation that have orbital angular momentum, as information carriers in turbulent atmosphere. We perform numerical simulations of the propagation of these beams, and focus on the effects that the order, degree and ellipticity parameters have on the robustness of the beams. We find that the choice of basis in which a mode is constructed does not greatly influence the mode performance and it is instead strongly affected by the combination of order and degree values.
We propose an experimental scheme for testing nonlocality of a correlated two-mode quantum state of light. We show that the correlation functions violating the Bell inequalities in the proposed experiment are equal to the joint two-mode QQQ function and the Wigner function. This assigns a novel operational meaning to these two quasidistribution functions in tests of quantum nonlocality and also establishes a direct relationship between two intriguing aspects of quantum mechanics: the nonlocality of entangled states and the noncommutativity of quantum observables, which underlies the nonclassical structure of the phase-space quasidistribution functions.