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Physical Review A

Influence of losses on the wave-particle duality

Aiai Jia, Jie‐Hui Huang, Tiancai Zhang, Shi‐Yao Zhu

Published Apr 8, 2014• 10 citations
Publisher

Abstract

We theoretically investigate the influence of losses on the wave-particle duality in a Mach-Zehnder interferometer. Balanced losses (equal loss on the two paths) have no effect on the visibility and predictability (also the duality relation), while unbalanced losses do have influence on them. If the unbalanced losses occur inside the interferometer, the losses have an effect on both the visibility and predictability, while the losses have no effect on the duality relation. If the unbalanced losses occur after the interferometer, the losses have no effect on the visibility, but do have an effect on the predictability (also the wave-particle duality), which can lead to a result that the duality relation exceeds 1, ${P}^{2}+{V}^{2}>1$. The influence of losses can be eliminated by exchanging the two detectors or the two inputs (one photon and the vacuum) of the interferometer and then averaging the two results. Consequently, we have the visibility, predictability, and duality relation ${P}^{2}+{V}^{2}\ensuremath{\le}1$ independent of the unbalanced losses. The obtained $P$ and $V$ for the unbalanced losses in the two paths do not represent the original predictability and visibility, and the result of ${P}_{D}^{2}+{V}_{D}^{2}>1$ does not mean the ``violation'' of the original duality.

DOI
10.1103/physreva.89.042103

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We theoretically investigate the influence of losses on the wave-particle duality in a Mach-Zehnder interferometer. Balanced losses (equal loss on the two paths) have no effect on the visibility and predictability (also the duality relation), while unbalanced losses do have influence on them. If the unbalanced losses occur inside the interferometer, the losses have an effect on both the visibility and predictability, while the losses have no effect on the duality relation. If the unbalanced losses occur after the interferometer, the losses have no effect on the visibility, but do have an effect on the predictability (also the wave-particle duality), which can lead to a result that the duality relation exceeds 1, P2+V2>1{P}^{2}+{V}^{2}>1P2+V2>1. The influence of losses can be eliminated by exchanging the two detectors or the two inputs (one photon and the vacuum) of the interferometer and then averaging the two results. Consequently, we have the visibility, predictability, and duality relation P2+V2\ensuremath≤1{P}^{2}+{V}^{2}\ensuremath{\le}1P2+V2\ensuremath≤1 independent of the unbalanced losses. The obtained PPP and VVV for the unbalanced losses in the two paths do not represent the original predictability and visibility, and the result of PD2+VD2>1{P}_{D}^{2}+{V}_{D}^{2}>1PD2​+VD2​>1 does not mean the ``violation'' of the original duality.
Wave-particle duality of photons with losses in the Mach-Zehnder interferometer (MZI) is investigated experimentally and theoretically. The experiment is done with the standard MZI with the beam splitter or the beam merger being continuously varied. The losses are deliberately introduced either inside the MZI (the two arms between the beam splitter and beam mergers) or outside the MZI (after the beam merger). It is proved that the unbalanced losses have great influence on the predictability PPP (particle nature) and visibility VVV (wave nature). For the former case the duality inequality holds while for the later the duality inequality is ``violated''. We get P2+V2>1P^2+V^2>1P2+V2>1. This ``violation'' could be eliminated in principle by switching the two paths and detectors and then averaging the results. The observed results can be exactly explained theoretically. The experiment is done with coherent beam, instead of single photons, and we have proved that they are exactly equivalent in duality experiment with MZI.
We present a study of the relation between wave-particle duality and quantum uncertainty in a two-path interferometer and derive equalities and inequalities involving the visibility (representing wave-like behavior), the predictability (representing particle-like behavior), and their variances. We experimentally demonstrate that, for a single photon in a Mach-Zehnder interferometer, these quantities are related via an equation that connects both duality and uncertainty. This relation holds for the single-photon source prepared either in a pure state or a mixed state.
The wave–particle duality relation derived by Englert sets an upper bound of the extractable information from wave and particle properties in a two-path interferometer. Surprisingly, previous studies demonstrated that the introduction of a quantum beamsplitter in the interferometer could break the limitation of this upper bound, due to interference between wave and particle states. Along the other line, a lot of efforts have been made to generalize this relation from the two-path setup to the N-path case. Thus, it is an interesting question that whether a quantum N-path beamsplitter can break the limitation as well. This paper systemically studies the model of a quantum N-path beamsplitter, and finds that the generalized wave–particle duality relation between interference visibility and path distinguishability is also broken in certain situations. We further study the maximal extractable information’s reliance on the interference between wave and particle properties, and derive a quantitative description. We then propose an experimental methodology to verify the break of the limitation. Our work reflects the effect of quantum superposition on wave–particle duality, and exhibits a new aspect of the relation between visibility and path distinguishability in N-path interference. Moreover, it implies the observer’s influence on wave–particle duality.
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.
We study the wave–particle duality in a general Mach–Zehnder interferometer with an asymmetric beam splitter from the viewpoint of quantum information theory. The correlations (including the classical correlation and the quantum correlation) between the particle and the which-path detector are derived when they are in pure state or mixed state at the output of Mach–Zehnder interferometer. It is found that the fringe visibility and the correlations are effected by the asymmetric beam splitter and the input state of the particle. The complementary relations between the fringe visibility and the correlations are also presented.