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arXiv:1812.04348 · Chinese Physics B

Fringe visibility and distinguishability in two-path interferometer with an asymmetric beam splitter

Yanjun Liu, Jing Lu, Zhihui Peng, Lan Zhou, Dongning Zheng

Published Mar 1, 2019• 7 citations
arXivHTMLPDF

Abstract

We study the fringe visibility and the distinguishability of a general Mach–Zehnder interferometer with an asymmetric beam splitter. Both the fringe visibility V and the distinguishability D are affected by the input state of the particle characterized by the Bloch vector S = ( S x , S y , S z ) and the second asymmetric beam splitter characterized by the paramter β . For the total system is initially in a pure state, it is found that the fringe visibility reaches the upper bound and the distinguishability reaches the lower bound when cos β = − S x . The fringe visibility obtain the maximum only if S x = 0 and β = π /2 when the input particle is initially in a mixed state. The complementary relationship V 2 + D 2 ≤ 1 is proved in a general Mach–Zehnder interferometer with an asymmetric beam splitter, and the conditions for the equality are also presented.

DOI
10.1088/1674-1056/28/3/030303

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We study the fringe visibility and the which-path information (WPI) of a general Mach–Zehnder interferometer with an asymmetric beam splitter (BS). A minimum error measurement in the detector is used to extract the WPI. Both the fringe visibility V and the WPI Ipath are affected by the initial state of the photon and the second asymmetric BS. The condition in which the WPI takes the maximum is obtained. The complementarity relationship V 2 + I path 2 ≤ 1 is found, and the conditions for equality are also presented.
We investigate quantitatively the wave-particle duality in a general Mach-Zehnder interferometer setup with an asymmetric beam splitter. The asymmetric beam splitter introduces additional a priori which-path knowledge, and the additional knowledge is different for particles detected at one or the other output port. Accordingly, the interference patterns appearing at the two output ports are also different. Hence, in sharp contrast with the symmetric case, in the present case we should treat the two output ports separately. It turns out that two nonorthogonal unsharp observables are measured jointly in this setup. We apply the condition for joint measurability of these unsharp observables to obtain a trade-off relation between the fringe visibility of the interference pattern and the which-path distinguishability.
The wave--particle duality relation bounds visibility and path distinguishability but does not determine their attainable values in a fixed noisy interferometer. We characterize the complete attainable region for a two-path interferometer with a qubit marker, varying the marker input while fixing the source, preparation channel, path-dependent interaction, and subsequent marker channel. At each feasible fringe visibility, the Helstrom-optimal distinguishabilities form at most two closed intervals, which can be separated by a gap. The intervals arise from opposite axial marker polarizations with the same fringe-amplitude modulus. Fixing the complex fringe amplitude instead gives a connected interval; for balanced original-path discrimination, a gap requires nonunital marker preparation. For a balanced pure source, relative πππ rotations, and no subsequent marker noise, we obtain a necessary and sufficient geometric criterion for a gap under some rotation axis perpendicular to the preparation displacement. Explicit channels and a time-independent Lindblad realization show how the gap depends on preparation geometry and evolution, with exact onset and optimization formulas at fixed fractional visibility. We also give sufficient bounds for excluding distinguishability intervals under calibrated finite-resolution errors.
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