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arXiv:1311.0017 · Nature Communications

Quantum mechanical which-way experiment with an internal degree of freedom

Konrad Banaszek, Paweł Horodecki, Michał Karpiński, Czesław Radzewicz

Published Oct 28, 2013• 24 citations

Abstract

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.

DOI: 10.1038/ncomms3594

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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.

Quantum-mechanical wave-particle duality is quantified in terms of a trade-off relation between the fringe visibility and the which-way distinguishability in an interference experiment. This relation was recently generalized by Banaszek et al. [Nat. Commun. 4, 2594 (2013)] when the particle is equipped with an internal degree of freedom such as spin. Here, we extend the visibility-distinguishability trade-off relation to quantum interference of a particle possessing an internal degree of freedom, when the which-way detector state is entangled with an ancillary system. We introduce an extended which-way distinguishability

{\mathcal{D}}_{E}

and the associated extended fringe visibility

{\mathcal{V}}_{E}

, satisfying the inequality ${\mathcal{D}}_{E}^{2}+{\mathcal{V}}_{E}^{2}\ensuremath{\le}1$ in this scenario. We illustrate, with the help of three specific examples, that while the which-way information inferred solely from the detector state (without ancilla) vanishes, the extended distinguishability retrievable via measurements on the detector-ancilla entangled state is nonzero. Furthermore, in all the three examples, the extended visibility and the generalized visibility (which was introduced by Banaszek et al.) match identically with each other.

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.

Complementarity lies at the heart of conceptual foundation of orthodox quantum mechanics. The wave-particle duality makes it impossible to tell which slit each particle passes through and still observe an interference pattern in a Young's double-slit experiment. In this paper, this duality is appraised under the standard formulation of quantum mechanics for atom interferometers. It is found that the internal freedoms like electronic states of an atom can be used to detect the which-path information of each atom while uphold the interference pattern of atoms when the center-of-mass motion of atoms is detected.

The notion of wave-particle duality may be quantified by the inequality

{V}^{2}{+K}^{2}<~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.

The wave—particle duality of a single particle with an n-dimensional internal degree of freedom is re-examined theoretically in a Mach—Zehnder interferometer. The famous duality relation D 2 + V 2 ≤ 1 is always valid in this situation, where D is the distinguishability and V is the visibility. However, the sum of the particle information and the wave information, can be smaller than one for the input of a pure state if this initial pure state includes the internal degree of freedom of the particle, while the quantity D 2 + V 2 is always equal to one when the internal degree of freedom of the particle is excluded.

Young’s double-slit experiment played a crucial role in establishing the wave nature of light. In this chapter, the shocking result that incident electrons yield a similar interference pattern as that formed by light waves is described. It is shown that the only way the experimental results could be explained is via a wave function description of electrons. It is also shown that, in the same experiment, the interference fringes disappear if the which-path information becomes available. This is the essence of wave–particle duality. The first of the Einstein–Bohr debates on wave-particle duality and Bohr’s principle of complementarity in the double-slit experiment is also discussed. Also presented are the counterintuitive notions of delayed choice and quantum eraser effects showing how the availability or erasure of information generated in the future can affect how the data in the present can be interpreted.