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Physical Review A

Interference and partial which-way information: A quantitative test of duality in two-atom resonance

Yonatan Abranyos, Matthias Jakob, János A. Bergou

Published Dec 8, 1999• 8 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.61.013804

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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.
In a recent paper, D\"urr [Phys. Rev. A 64, 042113 (2001)] proposed an interesting multibeam generalization of the quantitative formulation of interferometric wave-particle duality, discovered by Englert for two-beam interferometers. The proposed generalization is an inequality that relates a generalized measure of the fringe visibility to certain measures of the maximum amount of which-way knowledge that can be stored in a which-way detector. We construct an explicit example where, with three beams in a pure state, the scheme proposed by D\"urr leads to the possibility of an ideal which-way detector that can achieve a better path discrimination at the same time as a better fringe visibility. In our opinion, this seems to be in contrast with the intuitive idea of complementarity, as it is implemented in the two-beams case, where an increase in path discrimination always implies a decrease of fringe visibility, if the beams and the detector are in pure states.
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.
Young's double slit experiment has often been used to illustrate the concept of complementarity in quantum mechanics. If information can in principle be obtained about the path of the photon, then the visibility of the interference fringes is reduced or even destroyed. This Gedanken experiment discussed by Bohr and Einstein can be realized when the slit is replaced by individual atoms sensitive to the transferred recoil momentum of a photon which "passes through the slit". Early pioneering experiments were done with trapped ions and atom pairs created via photo-dissociation. Recently, it became possible to perform interference experiments with single neutral atoms cooled to the absolute ground state of a harmonic oscillator potential. The slits are now single atoms representing a two-level system, and the excitation in the harmonic oscillator potential is the which-way marker. In this note, we analyze and generalize two recent experiments performed with single atoms and emphasize the different ways they record which-way information.
The notion of wave-particle duality may be quantified by the inequality V2+K2< 1,{V}^{2}{+K}^{2}<~1,V2+K2< 1, relating interference fringe visibility V, and path knowledge K. With a single-photon interferometer in which polarization is used to label the paths, we have investigated the relation for various situations, including pure, mixed, and partially mixed input states. A quantum-eraser scheme has been realized that recovers interference fringes even when no which way information is available to erase.
For a particle travelling through an interferometer, the trade-off between the available which-way information and the interference visibility provides a lucid manifestation of the quantum mechanical wave–particle duality. Here we analyse this relation for a particle possessing an internal degree of freedom such as spin. We quantify the trade-off with a general inequality that paints an unexpectedly intricate picture of wave–particle duality when internal states are involved. Strikingly, in some instances which-way information becomes erased by introducing classical uncertainty in the internal degree of freedom. Furthermore, even imperfect interference visibility measured for a suitable set of spin preparations can be sufficient to infer absence of which-way information. General results are illustrated with a proof-of-principle single-photon experiment. Quantum mechanics dictates that the interference pattern cast by particles after passing through a double slit depends on how much information it is possible to know about which slit they went through. Banaszek et al. show how this behaviour extends to a system’s internal degrees of freedom.