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Physical Review C

Separation of two partially coherent beams from their interference pattern

R. da Silveira, Ch. Leclercq‐Willain

Published Jan 4, 2008• 0 citations
Publisher

Abstract

Two inequalities are derived which set upper and lower bounds on the intensity of two partially coherent interfering beams. These bounds are obtained from the observed interference pattern. Further, one establishes the connection between these inequalities and the ``duality relation'' obtained in the context of which-way information in two-path interferometry experiments.

DOI
10.1103/physrevc.77.014602

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Two inequalities are derived which set upper and lower bounds on the intensity of two partially coherent interfering beams. These bounds are obtained from the observed interference pattern. Further, one establishes the connection between these inequalities and the ``duality relation'' obtained in the context of which-way information in two-path interferometry experiments.
We propose for the experimental verification of an inequality concerning wave-particle duality by Englert [Phys. Rev. Lett. 77, 2154 (1996)] relating (or setting) an upper limit on distinguishability and visibility in a two-way interferometer. The inequality, quantifies the concept of wave-particle duality. The considered two-way interferometer is a Young's double-slit experiment involving two four-level atoms and is a slightly modified version of that of the recent experiment by Eichmann et al. [Phys. Rev. Lett. 70, 2359 (1993)]. The fringe visibility depends on the detected polarization direction of the scattered light and a read out of the internal state of one of the two atoms provides a partial which-way information.
The issue of interference and which-way information is addressed in the context of three-slit interference experiments. A new path distinguishability |DQ\mathcal {D}_{Q}DQ​| is introduced, based on unambiguous quantum state discrimination. An inequality connecting the interference visibility and path distinguishability, |V+2DQ3−DQ≤1\mathcal {V} + {2\mathcal {D}_Q\over 3- \mathcal {D}_Q} \le 1V+3−DQ​2DQ​​≤1|⁠, is derived which puts a bound on how much fringe visibility and which-way information can be simultaneously obtained. It is argued that this bound is tight. For two-slit interference, we derive a new duality relation which reduces to Englert's duality relation and the Greenberger–Yasin duality relation, in different limits.
We critically analyse the problem of formulating duality between fringe visibility and which-way information, in multibeam interference experiments. We show that the traditional notion of visibility is incompatible with any intuitive idea of complementarity, but for the two-beam case. We derive a number of new inequalities, not present in the two-beam case, one of them coinciding with a recently proposed multibeam generalization of the inequality found by Greenberger and YaSin. We show, by an explicit procedure of optimization in a three-beam case, that suggested generalizations of Englert's inequality do not convey, differently from the two-beam case, the idea of complementarity, according to which an increase of visibility is at the cost of a loss in path information, and vice versa.
It is well known that in a two-slit interference experiment, acquiring which-path information about the particle leads to a degrading of the interference. It is argued that path information has a meaning only when one can unambiguously tell which slit the particle went through. Using this idea, two duality relations are derived for the general case where the two paths may not be equally probable and the two slits may be of unequal widths. These duality relations, which are inequalities in general, saturate for all pure states. Earlier known results are recovered in suitable limit.
We set up an experiment with two independent laser beams that cross at a small angle, spatially separate, and end at detectors in front of each beam. The setup allows us to obtain maximum path or which-way information, K=1. However, when we scan a thin wire across the beam intersection the which-way information drops to K=0.985. The thin wire serves to verify the presence of an interference pattern with visibility, V=0.833, at the beam intersection. Our results appear to be in conflict with the complementarity inequality, V^2+K^2 smaller or equal to 1. We introduce a model that describes a virtual interference pattern. We resolve our paradoxical findings by proposing that the complementarity inequality does not apply to virtual interference patterns but only to physically real ones. We find that the interference pattern in our setup is virtual.
It is well known that in classical optics, the visibility of interference, in a two-beam light interference, is related to the optical coherence of the two beams. A wave-particle duality relation can be derived using this mutual coherence. The issue of wave-particle duality in classical optics is analyzed here, in the more general context of multipath interference. New definitions of interference visibility and path distinguishability have been introduced, which lead to a duality relation for multipath interference. The visibility is shown to be related to a new multipoint optical coherence function.