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Journal of Optics B Quantum and Semiclassical Optics

Probing entanglement in phase space: signature of GHZ states in the Wigner function

Marcelo França Santos, L. G. Lutterbach, L. Davidovich

Published Feb 1, 2001• 11 citations
Publisher

Abstract

We show that the value at the origin of phase space of the Wigner function of entangled GHZ states, involving three modes of the electromagnetic field either in the same cavity or in different ones, is a signature of the quantum character of those states. This multimode Wigner function can be directly measured by a simple extension of a method previously proposed by two of the present authors. A single measurement on an atom which crosses the three modes is enough to distinguish between quantum mechanics and local hidden variable theories.

DOI
10.1088/1464-4266/3/1/360

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We show that the value at the origin of phase space of the Wigner function of entangled GHZ states, involving three modes of the electromagnetic field either in the same cavity or in different ones, is a signature of the quantum character of those states. This multimode Wigner function can be directly measured by a simple extension of a method previously proposed by two of the present authors. A single measurement on an atom which crosses the three modes is enough to distinguish between quantum mechanics and local hidden variable theories.
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We propose a scheme for the reconstruction of a cavity field state. In the scheme the cavity field is first displaced by a microwave source, and then couples with a three-level Λ-type atom in a Raman manner. After an appropriate interaction time the measurement of the difference between the probabilities of the atom being in two ground states directly yields the Wigner function for the field state. The scheme can be easily generalized to measure the Wigner function of an entangled state of two spatially separated single-mode cavities.
We propose a scheme for measuring the Wigner function of a two-mode cavity field. The scheme bases on the interaction between the two-mode cavity field and three-level atoms. We find a simple relation between the Wigner function and the atomic population. One can obtain the Wigner function by measuring the atomic population with a micromaser-like experiment and doing a numerical integral. By using the two-mode Wigner function one can obtain the Clauser–Horne combination and test the Bell's inequalities. We test our equations with a two-mode entanglement state and the results are rather good.
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