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Physical Review A

Wigner-function description of quantum-mechanical nonlocality

Anu Venugopalan, Rupamanjari Ghosh

Published Nov 1, 1991• 8 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.44.6109

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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.
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.
We discuss violation of Bell inequalities by the regularized Einstein-Podolsky-Rosen (EPR) state, which can be produced in a quantum optical parametric down-conversion process. We propose an experimental photodetection scheme to probe nonlocal quantum correlations exhibited by this state. Furthermore, we show that the correlation functions measured in two versions of the experiment are given directly by the Wigner function and the Q function of the EPR state. Thus, the measurement of these two quasidistribution functions yields a novel scheme for testing quantum nonlocality.
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.
Nonclassical fields with coherent components generated in a large number of optical processes including losses can be characterized by a Gaussian Wigner function centered around the mean value of the field. The photon-number distributions for such fields are calculated and the numerical results are presented for a range of parameters. As a special case we give the number distributions for photons produced in down-conversion problems. We also give the analytical expression for the photoelectron distributions.
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