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arXiv:1611.00124 · Laser Physics Letters

Complementarity via error-free measurement in a two-path interferometer

Yanjun Liu, Jing Lu, Lan Zhou

Published Mar 28, 2017• 7 citations
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Abstract

Abstract We study both wave-like behavior and particle-like behavior in a general Mach–Zehnder interferometer with an asymmetric beam splitter. An error-free measurement in the detector is used to extract the which-path information. The fringe visibility V and the which-path information I path are derived, their complementary relation V + I p a t h ⩽ 1 is found, and the condition for the equality is also presented.

DOI
10.1088/1612-202x/aa6593

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Abstract We study both wave-like behavior and particle-like behavior in a general Mach–Zehnder interferometer with an asymmetric beam splitter. An error-free measurement in the detector is used to extract the which-path information. The fringe visibility V and the which-path information I path are derived, their complementary relation V + I p a t h ⩽ 1 is found, and the condition for the equality is also presented.
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 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.
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.
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.
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.
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.