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Conference on Lasers and Electro-Optics Pacific Rim (CLEO-PR)

Fluctuations in a two-way interferometer

Jiehui Huang, Shi‐Yao Zhu

Published Aug 1, 2007• 0 citations
Publisher

Abstract

One Hermitian operator and one non-Hermitian operator are introduced for the predictability and visibility in a two-way interferometer. The sum of the variances of the predictability and visibility is used to measure the uncertainty, which is linked to the complementarity relation through the equation, (DeltaP)2+ (DeltaVf)2+ P2+ V2= 2.
DOI
10.1109/cleopr.2007.4391759

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In terms of operator, the two complementary quantities, the predictability and visibility, are reinvestigated in a two-way interferometer. One Hermitian operator and one non-Hermitian operator (composed of two Hermitian operators) are introduced for the predictability and visibility, respectively. The predictability and visibility can not be measured exactly simultaneously, due to the non-commutation between the two operators. The sum of the variances of the predictability and visibility (the total variance), is used to measure the uncertainty, which is linked to the complementarity relation through the equation, (δP)2+(δVf)2+P2+V2=2(δ_P)^2+(δ_Vf)^2+P^2+V^2=2(δP​)2+(δV​f)2+P2+V2=2 . This new description for the predictability and visibility connects the complementarity and the uncertainty relations, although neither of them can be derived directly from the other.
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 theoretical framework for the uncertainty relation of Hermitian operators is perfect and has been applied in many fields. At the same time, non-Hermitian operators are also widely used in some other fields. However, the uncertainty relation of non-Hermitian operators remains to be explored. K.W. Bong and his co-workers proposed the theory of unitary uncertainty relation and verified it in the experiment [Phys. Rev. Lett. 120, 230402 (2018)]. In this work, we generalized this unitary uncertainty relation theory and proposed uncertainty relations of non-Hermitian operators. Due to the difficulties in the direct measurement of non-Hermitian operators in the uncertainty relations, we simplified the uncertainty relation of two non-Hermitian operators with pure states and proposed a realizable experimental measurement scheme by using the Mach–Zehnder interferometer. When the two non-Hermitian operators are unitary, our result can reduce to Bong et al.’s result. Furthermore, for two non-Hermitian operators but not unitary, we obtained a generalized and analogous result of theirs.
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.
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, vi{\mathit{v}}_{\mathit{i}}vi​v12{\mathit{v}}_{12}v12​\ensuremath{\le}1/2. Some extensions are discussed.
Abstract The wave-particle duality, as a manifestation of Bohr’s complementarity, is usually quantified in terms of path predictability and interference visibility. Various characterizations of the wave-particle duality have been proposed from an operational perspective, most of them are in forms of inequalities, and some of them are expressed in forms of equalities by incorporating entanglement or coherence. In this work, we shed different insights into the nature of the wave-particle duality by casting it into a form of information conservation in a multi-path interferometer, with uncertainty as a unified theme. More specifically, by employing the simple yet fundamental concept of variance, we establish a resolution of unity, which can be interpreted as a complementarity relation among wave feature, particle feature, and mixedness of a quantum state. This refines or reinterprets some conventional approaches to wave-particle duality, and highlights informational aspects of the issue. The key idea of our approach lies in that a quantum state, as a Hermitian operator, can also be naturally regarded as an observable, with measurement uncertainty (in a state) and state uncertainty (in a measurement) being exploited to quantify particle feature and wave feature of a quantum state, respectively. These two kinds of uncertainties, although both are defined via variance, have fundamentally different properties and capture different features of a state. Together with the mixedness, which is a kind of uncertainty intrinsic to a quantum state, they add up to unity, and thus lead to a characterization of the wave-particle-mixedness complementarity. This triality relation is further illustrated by examples and compared with some popular wave-particle duality or triality relations.