Wave and particle properties of a quantum object cannot be observed simultaneously. In particular, the fringe visibility in an interferometer is limited by the amount of which-way information which can be obtained. This limit is set by the recently discovered duality relation. So far, all derivations of the duality relation are independent of Heisenberg’s uncertainty relation. Here we demonstrate that it is alternatively possible to derive the duality relation in the form of an uncertainty relation for some suitably chosen observables.
Wave and particle properties of a quantum object cannot be observed simultaneously. In particular, the fringe visibility in an interferometer is limited by the amount of which-way information which can be obtained. This limit is set by the recently discovered duality relation. So far, all derivations of the duality relation are independent of Heisenberg’s uncertainty relation. Here we demonstrate that it is alternatively possible to derive the duality relation in the form of an uncertainty relation for some suitably chosen observables.
Publisher Summary Wave-particle duality means that a quantum object can exhibit either wave or particle properties, depending on the experimental situation. The wave nature gives rise to interference phenomena, whereas knowledge about the path taken by the object testifies to its particle nature. It is impossible to observe wave and particle properties simultaneously. Atom interferometry addresses fundamental aspects of the quantum theory of measurement in real experiments. This chapter discusses the loss of interference fringes in a which-way experiment need not be due to the Heisenberg's uncertainty relation. It demonstrates how interference can be restored by erasing the which-way information. The chapter investigated a new regime, where only incomplete which-way information is available. This allowed studying the continuous transition between wave and particle pictures and to test wave-particle duality in a quantitative manner. Future experimental investigations in this field focuses on the investigation of decoherence effects to study the transition between quantum and classical physics, leading to a more detailed understanding of the processes underlying the collapse of the wave function.
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.
Updating the wave-particle duality Maria C. Bosca e-mail: bosca@ugr.es Departamento de Fisica Atomica, Molecular y Nuclear Universidad de Granada E-18071. Granada Spain The wave-particle-duality, the fundamental component of the new quantum formalism in Bohr’s opinion, must be reformulated by incorporating the results of some experiments accomplished in the last decades of twentieth century. The Bohr´s complementarity principle stated the mutual exclusiveness and joint full completeness of the two (classical) descriptions of quantum systems; after Einstein-Podolsky-Rosen’paper, the wave-particle duality, or wave-particle complementarity, could be expressed by stating that it is impossible to build up an experimental arrangement in which we observe at the same time both corpuscular and wave aspects. In a two-slit experiment, they would correspond, respectively, to the which-way knowledge and the observation of interference pattern. Bohr showed this mutual exclusivity in numerous examples , and linked it to the unavoidable disturbance inherent in any measurement event. In quantum mechanical formalism, the complementarity principle has a clear mathematical expression: two observables are complementary if precise knowledge of one of them implies that all possible outcomes of the other are equally probable; their extension to classical concepts (as wave and particle) is not concerned. In 1991 Scully et al published a variant of the two-slit experiment that incorporates two micromasers cavities and a laser beam to provide which-path information without net momentum transferred during the interaction ; the impossibility of know which slit an atom went through and still observe the interference fringes is preserved by the establishing of quantum correlations between the measuring apparatus and the system being observed. They claimed that complementarity, of which wave-particle duality would be to them a manifestation, is more fundamental than the uncertainty principle . In 1996 B-G. Englert, following an approach originally due to Wooters and Zurek , derived , without making use of Heisenberg’s uncertainty relation, an inequality that quantifies the mutual compatibility relation between fringe visibility and which-way information. The inequality, that they denominated as “interferometric duality”, has the expression D^2 + V^2 ≤ 1, where D stands for the distinguishability of the ways and V for the fringe visibility; both of them are mathematical expressions that can be measured to check experimentally the inequality . Today, it is clear that intermediate particle-wave behaviours exist and, in addition to that, there are single experiments in which both classical wave-like and particle-like behaviours are showed total and simultaneously on an individual system. For instance, in the Bose’s double-prism experiment , tunnelling and perfect anticoincidence were observed in single photon states. Consequently, the meaning of the wave-particle duality must incorporate the simultaneous use of the two classical descriptions in the interpretation of experiments, loosing their original mutual exclusivity , which is incorporated as an extreme case in the new interferometric duality, a continuous quantum concept . The wave-particle duality, therefore, must be formulated as an interpretative addition to quantum mechanics, to which is possible to renounce if any pretension of visualize quantum phenomena in terms of classical concepts and intuitions is abandoned.
An interferometer - no matter how clever the design - cannot allow the experimenter to see both the wave and particle behavior of a quantum system. This fundamental idea has been captured quantitatively by inequalities, so-called wave-particle duality relations (WPDRs), that upper bound the sum of the path distinguishability (particle behavior) and fringe visibility (wave behavior). Another fundamental quantum concept is Heisenberg's uncertainty principle, stating that some pairs of observables cannot be known simultaneously. The distinction between these two concepts has been debated in the literature. Here we provide closure to the debate by showing that WPDRs correspond to a modern formulation of the uncertainty principle, namely, the uncertainty relation for the min- and max-entropies, which is used in quantum cryptography. At the technical level, our unification relies on a novel operational meaning for the max-entropy in terms of the guessing probability. Furthermore, our unification provides a framework for solving an outstanding problem of how to rigorously prove WPDRs for interferometers with more than two paths, and we employ this framework to derive some novel WPDRs.
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.