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Journal of Physics Conference Series

Novel relationships between superoscillations, weak values, and modular variables

Jeff Tollaksen

Published May 1, 2007• 15 citations
Publisher

Abstract

We present several novel, unexpected relationships between superoscillations, weak values and modular variables. For example, we show how an uncertain phase, which characterizes the process of projecting a particle onto a superoscillatory region, can create the high-momentum associated with the super-oscillation. If an uncertain phase can localize the particle, then a definite phase can also localize it. This introduction of a relative phase corresponds to a nonlocal exchange of modular variables. We also present a new way to measure the nonlocality in the equation of motion for modular variables by using weak measurements.

DOI
10.1088/1742-6596/70/1/012016

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We present several novel, unexpected relationships between superoscillations, weak values and modular variables. For example, we show how an uncertain phase, which characterizes the process of projecting a particle onto a superoscillatory region, can create the high-momentum associated with the super-oscillation. If an uncertain phase can localize the particle, then a definite phase can also localize it. This introduction of a relative phase corresponds to a nonlocal exchange of modular variables. We also present a new way to measure the nonlocality in the equation of motion for modular variables by using weak measurements.
We present several novel, unexpected relationships between superoscillations, weak values and modular variables. For example, we show how an uncertain phase, which characterizes the process of projecting a particle onto a superoscillatory region, can create the high-momentum associated with the super-oscillation. If an uncertain phase can localize the particle, then a definite phase can also localize it. This introduction of a relative phase corresponds to a nonlocal exchange of modular variables. We also present a new way to measure the nonlocality in the equation of motion for modular variables by using weak measurements.
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.
Weak measurements and the theory of weak values have a very deep meaning in quantum mechanics, and new phenomena associated with them has recently been observed experimentally. This theory has also directly led to the notion of superoscillating sequences of functions. In this paper we consider Cauchy problems with superoscillatory initial conditions (in particular, the Cauchy problem for the Schrödinger equation and some of its variations), and we give conditions under which the superoscillations persist in time. Our work is based on results from the theory of formal solutions to Cauchy problems and on the study of the specific growth of superoscillatory sequences, when regarded as functions of a complex variable. There are two main aims of this paper: one is to explain the mathematical tools that are necessary to study superoscillations, also repeating a few results that we have already proved in other papers in order to clarify the strategy. The second aim is to show that our technique applies to a large class of problems, showing under which conditions the superoscillatory phenomenon persists. Finally, we point out that our strategy can be applied also to non-constant coefficients differential equations as the quantum harmonic oscillator.
The aim of this paper is to study generating functions for the coefficients of the classical superoscillatory function associated with weak measurements. We also establish some new relations between the superoscillatory coefficients and many well-known families of special polynomials, numbers, and functions such as Bernstein basis functions, the Hermite polynomials, the Stirling numbers of second kind, and also the confluent hypergeometric functions. Moreover, by using generating functions, we are able to develop a recurrence relation and a derivative formula for the superoscillatory coefficients.
Surprisingly, differentiable functions are able to oscillate arbitrarily faster than their highest Fourier component would suggest. The phenomenon is called superoscillation. Recently, a practical method for calculating superoscillatory functions was presented and it was shown that superoscillatory quantum mechanical wave functions should exhibit a number of counter-intuitive physical effects. Following up on this work, we here present more general methods which allow the calculation of superoscillatory wave functions with custom-designed physical properties. We give concrete examples and we prove results about the limits to superoscillatory behavior. We also give a simple and intuitive new explanation for the exponential computational cost of superoscillations.
Abstract. Surprisingly, differentiable functions are able to oscillate arbitrarily faster than their highest Fourier component would suggest. The phenomenon is called superoscillation. Recently, a practical method for calculating superoscillatory functions was presented and it was shown that superoscillatory quantum mechanical wave functions should exhibit a number of counter-intuitive physical effects. Following up on this work, we here present more general methods which allow the calculation of superoscillatory wave functions with custom-designed physical properties. We give concrete examples and we prove results about the limits to superoscillatory behavior. We also give a simple and intuitive new explanation for the exponential computational cost of superoscillations. 1 2 1.