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Journal of Modern Optics

Complementarity of interferometers interacting with the dynamical model of the quantum measurement process

Masashi Ban, Fumiaki Shibata, Sachiko Kitajima

Published Dec 17, 2008• 6 citations
Publisher

Abstract

Complementarity of wave-like and particle-like properties in the Mach–Zehnder and Michelson interferometers is investigated by means of the exactly solvable model for a quantum measurement process. When the error-minimum measurement and unambiguous measurement are used for extracting the which-path information, the fringe visibility and which-path information are derived and their complementary relation is examined. It is shown that there is a possibility of the recovery of the quantum coherence in the Michelson interferometer since the which-path information can be erased by the propagating particle itself.

DOI
10.1080/09500340802517694

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Complementarity of wave-like and particle-like properties in the Mach–Zehnder and Michelson interferometers is investigated by means of the exactly solvable model for a quantum measurement process. When the error-minimum measurement and unambiguous measurement are used for extracting the which-path information, the fringe visibility and which-path information are derived and their complementary relation is examined. It is shown that there is a possibility of the recovery of the quantum coherence in the Michelson interferometer since the which-path information can be erased by the propagating particle itself.
The complementarity for wave-like and particle-like properties of a quantum mechanical particle in the Mach–Zehnder interferometer is investigated by means of the dynamical approach. Measurement devices are placed on the paths of the interferometer to obtain the information of a propagating particle. The which-path information of the particle and the fringe visibility of the interferometer are derived by considering the dynamics of the particle and devices. When each device consists of a single qubit, the maximum amount of the which-path information is obtained and the complementary relation is examined.
Which-path information of a quantum particle in interferometers is the key to infer the past of the quantum particle. It has led to many extensive discussions, including discussions of quantum complementarity and the path-visibility relation. The basics of these discussions are the description, detection, and control of which-path information. In this paper, we focus on the investigation of multidimensional which-path information in a nested Mach-Zehnder interferometer. A general expression of which-path information is given and can be partially extracted by different detection methods. Further analysis shows that the which-path information can be controlled by the phase differences and beam-splitting ratios between the arms of the nested Mach-Zehnder interferometer. Moreover, an alternative which-path information elimination phenomenon has been predicted and demonstrated experimentally. Our work can help us to understand the physics of quantum particles, with potential applications for quantum information processing and quantum metrology.
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.
Complementarity, or more specifically, the wave-particle duality, and quantum coherence are two fundamental concepts in quantum mechanics. Recently, motivated by the progress of the quantification of quantum coherence, the complementary relation between coherence and path information is investigated by many authors, and various duality relations between them are established. Such relations not only provide insights into fundamental problems of quantum mechanics, such as the understanding of quantum coherence and wave-particle duality; but also are important in applications of quantum technologies. In this paper, based on the Bures distance and unambiguous quantum state discrimination, systematic analysis of the complementarity between the quantum coherence and path information in two path interferometers is carried out. Similarly as other related works, the wave aspect, or the visibility of the interferometer, is quantified by the l1-norm measure of quantum coherence, and the path information is considered via unambiguous quantum state discrimination. In this way, a novel duality relation in two path interferometers is obtained. Compared with known results, our work considers mixed states as well as pure states; considers the path predictability resulting from the intrinsic path asymmetry of the quantum state, as well as the path distinguishability resulting from the use of a which-path detector. Therefore, our work systematically generalizes known results in two path interferometers by removing all the unnecessary restrictions. Specifically, the most general form of quantum states in two path interferometers is considered and the duality relation between quantum coherence and path information is proved based on the positivity property of density matrices. The cases of path predictability and path distinguishability are considered separately. For path predictability, the proof is straightforward; whereas some advanced mathematical techniques, such as the Schur-Weyl inequality, properties of the fidelity and properties of positive matrices, are required in order to give a rigorous proof of the duality relation between coherence and path distinguishability. Concrete examples are provided to illustrate the abstract method and results. Our work concerns about two path interferometers exclusively and depends heavily that the dimensionality is two, therefore it would be an interesting task to generalize the results in this paper to n-path interferometers.
We present an analysis of a quantum nondemolition (QND) measurement, which determines which of the two possible paths a photon follows through a Mach-Zehnder interferometer. The measurement is effected by the nonlinear Kerr effect. If a photon traverses the arm of the interferometer containing the Kerr medium, it shifts the phase of a probe beam and then continues through to the interferometer output. The measurement does not destroy the photon but does disturb its phase. Using various probe states, we show how such phase disturbances reduce fringe visibility and enforce a principle of complementarity; the more accurately the QND measurement determines the path of the photon, the lower is the fringe visibility. We determine the appropriate parameters to optimize the observation of the two complementary quantities.