Searcharxiv⌕ Search

arXiv subjects

J. Tollaksen

Publications and source records attributed to J. Tollaksen.

10 recordsLinked to original sources
arXiv2021

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.

math-ph↗
arXiv2016

A new approach to ultrasensitive gravitational wave detection

We suggest here a method to detect gravitational waves (GW) different from the interferometric approach. It is based on two critical steps: conversion of the GW action into rotational motion and subsequent conversion into electric current. The ability to detect extremely tiny currents empowers this suggested approach in which the gravitational wave moves charges around closed loops. This new method may allow detection of gravitational waves with strain sensitivities beyond the reach of the interferometric approaches.

gr-qc↗
arXiv2015

The mathematics of superoscillations

In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations. Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics. From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum. Purpose of this work is twofold: on one hand we provide a self-contained survey of the existing literature, in order to offer a systematic mathematical approach to superoscillations; on the other hand, we obtain some new and unexpected results, by showing that superoscillating sequences can be seen of as solutions to a large class of convolution equations and can therefore be treated within the theory of Analytically Uniform spaces. In particular, we will also discuss the persistence of the superoscillatory behavior when superoscillating sequences are taken as initial values of the Schrödinger equation and other equations.

math-ph↗
arXiv2015

Superconducting Antenna Concept for Gravitational Wave Radiation

We present an idea for detecting gravitational waves (GWs) by measuring the current induced in a superconducting loop by the rotation of a frame to which it is attached. The frame experiences a torque caused by a GW propagating perpendicular to it because it is made of materials of different density in adjacent quadrants. Superconducting Cooper pair system responds symmetrically to the wave and stays at rest, while the ions of the superconductor are being accelerated by the moving frame. This generates an electric current in the loop which can be measured precisely by sensitive electronics. For that task the antenna consists of two superconducting loops parallel to each other. One of them, the primary loop, rotationally oscillates because of the described reasons, while the second stays at rest. In the oscillating loop the current creates an oscillating magnetic flux. This flux should be compensated due to fluxoid quantization, by the oppositely directed current in the non-moving, secondary loop, in which the ions are at rest, and the Cooper pairs move. We estimate the resulting signal/noise ratio and discuss the signal detectability. Several designs are envisioned, both for terrestrial and for orbital arrangements of the antenna.

cond-mat.supr-con↗
arXiv2014

The quantum pigeonhole principle and the nature of quantum correlations

The pigeonhole principle: "If you put three pigeons in two pigeonholes at least two of the pigeons end up in the same hole" is an obvious yet fundamental principle of Nature as it captures the very essence of counting. Here however we show that in quantum mechanics this is not true! We find instances when three quantum particles are put in two boxes, yet no two particles are in the same box. Furthermore, we show that the above "quantum pigeonhole principle" is only one of a host of related quantum effects, and points to a very interesting structure of quantum mechanics that was hitherto unnoticed. Our results shed new light on the very notions of separability and correlations in quantum mechanics and on the nature of interactions. It also presents a new role for entanglement, complementary to the usual one. Finally, interferometric experiments that illustrate our effects are proposed.

quant-ph↗
arXiv2011

Dynamical features of interference phenomena in the presence of entanglement

A "strongly" interacting, and entangling, heavy, non recoiling, external particle effects a significant change of the environment. Described locally, the corresponding entanglement event is a generalized electric Aharonov Bohm effect, that differs from the original one in a crucial way. We propose a gedanken interference experiment. The predicted shift of the interference pattern is due to a self induced or "private" potential difference experienced while the particle is in vacuum. We show that all non trivial Born Oppenheimer potentials are "private" potentials. We apply the Born Oppenheimer approximation to interference states. Using our approach we calculate the relative phase of the external heavy particle as well as its uncertainty throughout an interference experiment /entanglement event. We thus complement the Born Oppenheimer approximation for interference states.

quant-ph↗
arXiv2007

Multiple-time states and multiple-time measurements in quantum mechanics

We discuss experimental situations that consist of multiple preparation and measurement stages. This leads us to a new approach to quantum mechanics. In particular, we introduce the idea of multi-time quantum states which are the appropriate tools for describing these experimental situations. We also describe multi-time measurements and discuss their relation to multi-time states. A consequence of our new formalism is to put states and operators on an equal footing. Finally we discuss the implications of our new approach to quantum mechanics for the problem of the flow of time.

quant-ph↗
arXiv1996

Adiabatic Measurements on Metastable Systems

In several situations, most notably when describing metastable states, a system can evolve according to an effective non hermitian Hamiltonian. To each eigenvalue of a non hermitian Hamiltonian is associated an eigenstate $\vertϕ\rangle$ which evolves forward in time and an eigenstate $\langleψ\vert$ which evolves backward in time. Quantum measurements on such systems are analyzed in detail with particular emphasis on adiabatic measurements in which the measuring device is coupled weakly to the system. It is shown that in this case the outcome of the measurement of an observable $A$ is the weak value $\langleψ\vert A\vertϕ\rangle / \langleψ\vertϕ\rangle $ associated to the two-state vector $\langleψ\vert$ $\vertϕ\rangle$ corresponding to one of the eigenvalues of the non hermitian Hamiltonian. The possibility of performing such measurements in a laboratory is discussed.

quant-ph↗