Physical Review Letters

Fringe Visibility and Which-Way Information: An Inequality

Berthold‐Georg Englert

Published Sep 9, 1996• 817 citations

Abstract

An inequality is derived according to which the fringe visibility in a two-way interferometer sets an absolute upper bound on the amount of which-way information that is potentially stored in a which-way detector. In some sense, this inequality can be regarded as quantifying the notion of wave-particle duality. The derivation of the inequality does not make use of Heisenberg's uncertainty relation in any form.

DOI: 10.1103/physrevlett.77.2154

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Physical Review Letters

Fringe Visibility and Which-Way Information: An Inequality

Berthold‐Georg Englert

Published Sep 9, 1996• 817 citations

Publisher

Abstract

DOI: 10.1103/physrevlett.77.2154

Related papers

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

Yonatan Abranyos, Matthias Jakob, János A. Bergou

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.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.

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Wave and particle properties of a quantum object cannot be observed simultaneously. In particular, the fringe visibility in an interferometer is limited by the amount of which-way information which can be obtained. This limit is set by the recently discovered duality relation. So far, all derivations of the duality relation are independent of Heisenberg’s uncertainty relation. Here we demonstrate that it is alternatively possible to derive the duality relation in the form of an uncertainty relation for some suitably chosen observables.Wave and particle properties of a quantum object cannot be observed simultaneously. In particular, the fringe visibility in an interferometer is limited by the amount of which-way information which can be obtained. This limit is set by the recently discovered duality relation. So far, all derivations of the duality relation are independent of Heisenberg’s uncertainty relation. Here we demonstrate that it is alternatively possible to derive the duality relation in the form of an uncertainty relation for some suitably chosen observables.

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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.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.

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Stephan Dürr, T. Nonn, Gerhard Rempe

We experimentally investigate the reduction of the fringe visibility in an atom interferometer due to the storage of which-way information. We focus on the case of incomplete which-way information and use the distinguishability $D$ to quantify how much information is stored. For a given value of $D$ , the fringe visibility $V$ is limited by the duality relation ${D}^{2}{+V}^{2}\ensuremath{\le}1$. We have measured $D$ and $V$ independently. Combining the results, we find good agreement with the duality relation.We experimentally investigate the reduction of the fringe visibility in an atom interferometer due to the storage of which-way information. We focus on the case of incomplete which-way information and use the distinguishability $D$ to quantify how much information is stored. For a given value of $D$, the fringe visibility $V$ is limited by the duality relation ${D}^{2}{+V}^{2}\ensuremath{\le}1$. We have measured $D$ and $V$ independently. Combining the results, we find good agreement with the duality relation.

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“Quality” of a which-way detector

Jesús Martínez-Linares, David A. Harmin

We introduce a measure $\mathcal{Q}$ of the ``quality'' of a quantum which-way detector, which characterizes its intrinsic ability to extract which-way information in an asymmetric two-way interferometer. The ``quality'' $\mathcal{Q}$ allows one to separate the contribution to the distinguishability of the ways arising from the quantum properties of the detector from the contribution stemming from a priori which-way knowledge available to the experimenter, which can be quantified by a predictability parameter $\mathcal{P}.$ We provide an inequality relating these two sources of which-way information to the value of the fringe visibility displayed by the interferometer. We show that this inequality is an expression of duality, allowing one to trace the loss of coherence to the two reservoirs of which-way information represented by $\mathcal{Q}$ and $\mathcal{P}.$ Finally, we illustrate the formalism with the use of a quantum logic gate: the symmetric quanton-detecton system (SQDS). The SQDS can be regarded as two qubits trying to acquire which-way information about each other. The SQDS will provide an illustrating example of the reciprocal effects induced by duality between system and which-way detector.We introduce a measure $\mathcal{Q}$ of the ``quality'' of a quantum which-way detector, which characterizes its intrinsic ability to extract which-way information in an asymmetric two-way interferometer. The ``quality'' $\mathcal{Q}$ allows one to separate the contribution to the distinguishability of the ways arising from the quantum properties of the detector from the contribution stemming from a priori which-way knowledge available to the experimenter, which can be quantified by a predictability parameter $\mathcal{P}.$ We provide an inequality relating these two sources of which-way information to the value of the fringe visibility displayed by the interferometer. We show that this inequality is an expression of duality, allowing one to trace the loss of coherence to the two reservoirs of which-way information represented by $\mathcal{Q}$ and $\mathcal{P}.$ Finally, we illustrate the formalism with the use of a quantum logic gate: the symmetric quanton-detecton system (SQDS). The SQDS can be regarded as two qubits trying to acquire which-way information about each other. The SQDS will provide an illustrating example of the reciprocal effects induced by duality between system and which-way detector.

arXiv:quant-ph/0306057Physical Review AJun 23, 20045 citations

We derive different relations quantifying duality in a generic two-way interferometer. These relations set different upper bounds to the visibility

\mathcal{V}

of the fringes measured at the output port of the interferometer. A hierarchy of inequalities is presented which exhibits the influence of the availability to the experimenter of different sources of which-way information contributing to the total distinguishability

\mathcal{D}

of the ways. For mixed states and unbalanced interferometers an inequality is derived, ${\mathcal{V}}^{2}+{\ensuremath{\Xi}}^{2}\ensuremath{\le}1$, which can be more stringent than the one associated with the distinguishability $({\mathcal{V}}^{2}+{\mathcal{D}}^{2}\ensuremath{\le}1)$.

D

to quantify how much information is stored. For a given value of

D

, the fringe visibility

V

is limited by the duality relation ${D}^{2}{+V}^{2}\ensuremath{\le}1$. We have measured

D

and

V

independently. Combining the results, we find good agreement with the duality relation.

We introduce a measure

\mathcal{Q}

of the ``quality'' of a quantum which-way detector, which characterizes its intrinsic ability to extract which-way information in an asymmetric two-way interferometer. The ``quality''

\mathcal{Q}

allows one to separate the contribution to the distinguishability of the ways arising from the quantum properties of the detector from the contribution stemming from a priori which-way knowledge available to the experimenter, which can be quantified by a predictability parameter

\mathcal{P}.

We provide an inequality relating these two sources of which-way information to the value of the fringe visibility displayed by the interferometer. We show that this inequality is an expression of duality, allowing one to trace the loss of coherence to the two reservoirs of which-way information represented by

\mathcal{Q}

and

\mathcal{P}.

Finally, we illustrate the formalism with the use of a quantum logic gate: the symmetric quanton-detecton system (SQDS). The SQDS can be regarded as two qubits trying to acquire which-way information about each other. The SQDS will provide an illustrating example of the reciprocal effects induced by duality between system and which-way detector.