Studying Heisenberg-like Uncertainty Relation with Weak Values in One-dimensional Harmonic Oscillator

As one of the fundamental traits governing the operation of quantum world, the uncertainty relation, from the perspective of Heisenberg, rules the minimum deviation of two incompatible observations for arbitrary quantum states. Notwithstanding, the original measurements appeared in Heisenberg’s principle are strong such that they may disturb the quantum system itself. Hence an intriguing question is raised: What will happen if the mean values are replaced by weak values in Heisenberg’s uncertainty relation? In this work, we investigate the question in the case of measuring position and momentum in a simple harmonic oscillator via designating one of the eigenkets thereof to the pre-selected state. Astonishingly, the original Heisenberg limit is broken for some post-selected states, designed as a superposition of the pre-selected state and another eigenkets of harmonic oscillator. Moreover, if two distinct coherent states reside in the pre- and post-selected states respectively, the variance reaches the lower bound in common uncertainty principle all the while, which is in accord with the circumstance in Heisenberg’s primitive framework.

Integrated Communications but Disintegrating Society. Quo Vadis?

The goal of this article is to show the interconnection of the new quality of social communications defined by the term “digital integrated communications” and the social structure crisis. To do that, the author carries out the analysis and conceptual differentiation of the “communication”, “information” and “knowledge” notion; and brings forth the problem of harmful information – the information noise. Communications integration is analyzed in historical and modern contexts as a controversial factor of social and cultural dynamics that intensifies information exchange and social strife. The author reveals the social context of digital integrated communications: global urbanization and the strengthening of hedonistic individualism as mass worldview. The author arrives at the conclusion that integrated communications deepen anthropological crises at a time of weak values and institutes of social integration.

Holomorphic functions, relativistic sum, Blaschke products and superoscillations

Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. The notion of superoscillation is a particular case of that one of supershift. In the recent years, superoscillating functions, that appear for example in weak values in quantum mechanics, have become an interesting and independent field of research in complex analysis and in the theory of infinite order differential operators. The aim of this paper is to study some infinite order differential operators acting on entire functions which naturally arise in the study of superoscillating functions. Such operators are of particular interest because they are associated with the relativistic sum of the velocities and with the Blaschke products. To show that some sequences of functions preserve the superoscillatory behavior it is of crucial importance to prove that their associated infinite order differential operators act continuously on some spaces of entire functions with growth conditions.

Anomalous weak values via a single photon detection

Is it possible that a measurement of a spin component of a spin-1/2 particle yields the value 100? In 1988 Aharonov, Albert and Vaidman argued that upon pre- and postselection of particular spin states, weakening the coupling of a standard measurement procedure ensures this paradoxical result1. This theoretical prediction, called weak value, was realised in numerous experiments2–9, but its meaning remains very controversial10–19, since its “anomalous” nature, i.e., the possibility to exceed the eigenvalue spectrum, as well as its “quantumness” are debated20–22. We address these questions by presenting the first experiment measuring anomalous weak values with just a single click, without the need for statistical averaging. The measurement uncertainty is significantly smaller than the gap between the measured weak value and the nearest eigenvalue. Beyond clarifying the meaning of weak values, demonstrating their non-statistical, single-particle nature, this result represents a breakthrough in understanding the foundations of quantum measurement, showing unprecedented measurement capability for further applications of weak values to quantum photonics.

A new method to generate superoscillating functions and supershifts

Superoscillations are band-limited functions that can oscillate faster than their fastest Fourier component. These functions (or sequences) appear in weak values in quantum mechanics and in many fields of science and technology such as optics, signal processing and antenna theory. In this paper, we introduce a new method to generate superoscillatory functions that allows us to construct explicitly a very large class of superoscillatory functions.

Theory of neutrino detection: flavor oscillations and weak values

We revisit the theory of neutrino oscillations and describe it through the formalism of weak measurements with postselection. It is well understood that due to the large momentum uncertainty in detection, there is no collapse of the neutrino wavefunction in the momentum or energy basis, and the mass eigenstates are detected coherently. Here we show that postselection, which projects the system to a final flavor state, deforms the system wavefunction in such a way that the momentum detected is not the expectation value of the neutrino mass eigenstates momenta, but the corresponding weak value. We use the weak values to describe the intermediate state in the oscillation process, avoiding problems in defining probability currents for particle states with mass superposition.