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Physical Review A

Complementarity of one-particle and two-particle interference

Gregg S. Jaeger, Michael Horne, Abner Shimony

Published Aug 1, 1993• 153 citations
Publisher

Abstract

In a two-particle interferometer, one can study the variation of both single- and joint-detection probabilities as functions of the phase shifts of the beams. By combining the usual definition for one-particle fringe visibility ${\mathit{v}}_{\mathit{i}}$ (i=1,2) with a reasonable proposed definition for two-particle fringe visibility ${\mathit{v}}_{12}$, we show that ${\mathit{v}}_{\mathit{i}}^{2}$+${\mathit{v}}_{12}^{2}$\ensuremath{\le}1 or, equivalently, ${\mathit{v}}_{\mathit{i}}$${\mathit{v}}_{12}$\ensuremath{\le}1/2. Some extensions are discussed.

DOI
10.1103/physreva.48.1023

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Two interferometric complementarities

Gregg S. Jaeger, Abner Shimony, Lev Vaidman A

Two interferometric complementarities are formulated and demonstrated, and relations between them are analyzed. The first relates the distinguishability D(scrP) of the path of propagation of a particle (where scrP is the preparation of an ensemble) to the fringe visibility ${\mathit{v}}_{1}$ when amplitudes from two paths are combined. It is shown that [maxD(scrP)${]}^{2}$+${\mathit{v}}_{1}^{2}$=1, where the maximum is taken over all preparations compatible with a fixed density operator. The second complementarity relates the visibility of one-particle interference fringes to the visibility ${\mathit{v}}_{12}$ of two-particle fringes: ${\mathit{v}}_{12}^{2}$+${\mathit{v}}_{1}^{2}$\ensuremath{\le}1. With a suitable extension of operations on the pair of particles, this inequality is strengthened to an equality.

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WAVE-PARTICLE DUALITY IN MULTI-PATH INTERFEROMETERS: GENERAL CONCEPTS AND THREE-PATH INTERFEROMETERS

Berthold‐Georg Englert, Dagomir Kaszlikowski, L. C. Kwek +1 more

For two-path interferometers, the which-path predictability [Formula: see text] and the fringe visibility [Formula: see text] are familiar quantities that are much used to talk about wave-particle duality in a quantitative way. We discuss several candidates that suggest themselves as generalizations P of [Formula: see text] for multi-path interferometers, and treat the case of three paths in considerable detail. To each choice for the path knowledge P, the interference strength V — the corresponding generalization of [Formula: see text] — is found by a natural, operational procedure. In experimental terms, it amounts to finding those equal-weight superpositions of the path amplitudes which maximize P for the emerging intensities. Mathematically speaking, one needs to identify a certain optimum one among the Fourier transforms of the state of the interfering quantum object. Wave-particle duality is manifest, inasmuch as P = 1 implies V = 0 and V = 1 implies P = 0, whatever definition is chosen. The possible values of the pair (P,V) are restricted to an area with corners at (P,V) = (0,0), (P,V) = (1,0), and (P,V) = (0,1), with the shape of the border line from (1,0) to (0,1), depending on the particular choice for P and the induced definition of V.

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Comment on “Quantitative wave-particle duality in multibeam interferometers”

Giuseppe Bimonte, R. Musto

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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Distinguishability, contrast and complementarity in multimode two-particle interferences

Pedro Sancho

Multimode two-particle systems show interference effects in one-particle detections when both particles have common modes. We explore the possibility of extending the usual concepts of distinguishability and visibility to these types of systems. Distinguishability will refer now to the balance between common and different modes of a two-particle system, instead of the standard definition concerning available alternatives for a one-particle system. On the other hand, the usual concept of visibility is not suitable for our problem and must be replaced with that of contrast, measuring the ratio of detection probabilities with and without interference effects. Finally, we show that for the type of states considered in the paper there is a complementarity relation between distinguishability and contrast for two-boson states. In contrast, there is no two-fermion counterpart.

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Zeilinger et al. have recently proposed procedures for producing three-particle entanglements from two independent entangled pairs. Here I exhibit the quantum state behind one realization of this proposal and from this state calculate the visibility of the three-particle fringes.

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Single- and two-particle interference

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Abstract The reconciliation of wave-like optical interference phenomena with the particle-like properties of the quantized radiation field has intrigued physicists since the early years of the present century. Two main topics of discussion have been the effect of very low light levels-of the order of one photon at a time in the interferometer-on the interference fringes, and the possibility of determining which of the two optical paths through the interferometer were taken by the photons that produce the fringes. The first experiment with very low light levels was performed by Taylor (1909) at the instigation of J. J. Thomson. He found that Young’s interference fringes persisted when the low light levels were compensated by long photographic exposure times.

•Mar 21, 1991•3 citations
In a two-particle interferometer, one can study the variation of both single- and joint-detection probabilities as functions of the phase shifts of the beams. By combining the usual definition for one-particle fringe visibility vi{\mathit{v}}_{\mathit{i}}vi​ (i=1,2) with a reasonable proposed definition for two-particle fringe visibility v12{\mathit{v}}_{12}v12​, we show that vi2{\mathit{v}}_{\mathit{i}}^{2}vi2​+v122{\mathit{v}}_{12}^{2}v122​\ensuremath{\le}1 or, equivalently, {\mathit{v}}_{\mathit{i}}$${\mathit{v}}_{12}\ensuremath{\le}1/2. Some extensions are discussed.
Two interferometric complementarities are formulated and demonstrated, and relations between them are analyzed. The first relates the distinguishability D(scrP) of the path of propagation of a particle (where scrP is the preparation of an ensemble) to the fringe visibility v1{\mathit{v}}_{1}v1​ when amplitudes from two paths are combined. It is shown that [maxD(scrP)]2{]}^{2}]2+v12{\mathit{v}}_{1}^{2}v12​=1, where the maximum is taken over all preparations compatible with a fixed density operator. The second complementarity relates the visibility of one-particle interference fringes to the visibility v12{\mathit{v}}_{12}v12​ of two-particle fringes: v122{\mathit{v}}_{12}^{2}v122​+v12{\mathit{v}}_{1}^{2}v12​\ensuremath{\le}1. With a suitable extension of operations on the pair of particles, this inequality is strengthened to an equality.
For two-path interferometers, the which-path predictability [Formula: see text] and the fringe visibility [Formula: see text] are familiar quantities that are much used to talk about wave-particle duality in a quantitative way. We discuss several candidates that suggest themselves as generalizations P of [Formula: see text] for multi-path interferometers, and treat the case of three paths in considerable detail. To each choice for the path knowledge P, the interference strength V — the corresponding generalization of [Formula: see text] — is found by a natural, operational procedure. In experimental terms, it amounts to finding those equal-weight superpositions of the path amplitudes which maximize P for the emerging intensities. Mathematically speaking, one needs to identify a certain optimum one among the Fourier transforms of the state of the interfering quantum object. Wave-particle duality is manifest, inasmuch as P = 1 implies V = 0 and V = 1 implies P = 0, whatever definition is chosen. The possible values of the pair (P,V) are restricted to an area with corners at (P,V) = (0,0), (P,V) = (1,0), and (P,V) = (0,1), with the shape of the border line from (1,0) to (0,1), depending on the particular choice for P and the induced definition of V.
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.
Multimode two-particle systems show interference effects in one-particle detections when both particles have common modes. We explore the possibility of extending the usual concepts of distinguishability and visibility to these types of systems. Distinguishability will refer now to the balance between common and different modes of a two-particle system, instead of the standard definition concerning available alternatives for a one-particle system. On the other hand, the usual concept of visibility is not suitable for our problem and must be replaced with that of contrast, measuring the ratio of detection probabilities with and without interference effects. Finally, we show that for the type of states considered in the paper there is a complementarity relation between distinguishability and contrast for two-boson states. In contrast, there is no two-fermion counterpart.
Abstract The reconciliation of wave-like optical interference phenomena with the particle-like properties of the quantized radiation field has intrigued physicists since the early years of the present century. Two main topics of discussion have been the effect of very low light levels-of the order of one photon at a time in the interferometer-on the interference fringes, and the possibility of determining which of the two optical paths through the interferometer were taken by the photons that produce the fringes. The first experiment with very low light levels was performed by Taylor (1909) at the instigation of J. J. Thomson. He found that Young’s interference fringes persisted when the low light levels were compensated by long photographic exposure times.