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European Journal of Physics

Evolution of superoscillations in the quantum mechanical harmonic oscillator

Max Bussell, P. Strange

Published Sep 8, 2015• 4 citations
Publisher

Abstract

Abstract A superoscillatory function is one in which the function oscillates faster than its fastest Fourier component. Superoscillations are now a well-established feature of certain types of wavepacket. Here we discuss quantum superoscillations in a wavepacket evolving according to the harmonic oscillator hamiltonian. The evolution of the wavepacket is investigated using an expansion in terms of eigenfunctions and in terms of the propagator using both an exact integration, and a saddle point approximation. The creation and decay of superoscillations as well as the barrier between them and normal oscillations is shown to depend on the behaviour of the saddle points. The work reported here is the result of a student project and we argue that super oscillations makes an excellent topic with which to introduce students to theoretical research.

DOI
10.1088/0143-0807/36/6/065028

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Abstract A superoscillatory function is one in which the function oscillates faster than its fastest Fourier component. Superoscillations are now a well-established feature of certain types of wavepacket. Here we discuss quantum superoscillations in a wavepacket evolving according to the harmonic oscillator hamiltonian. The evolution of the wavepacket is investigated using an expansion in terms of eigenfunctions and in terms of the propagator using both an exact integration, and a saddle point approximation. The creation and decay of superoscillations as well as the barrier between them and normal oscillations is shown to depend on the behaviour of the saddle points. The work reported here is the result of a student project and we argue that super oscillations makes an excellent topic with which to introduce students to theoretical research.
A superoscillatory function is one that oscillates faster than its fastest Fourier component - A phenomenon created by subtle interference between Fourier components. In this thesis, the work of Professor Sir Michael Berry [1, 2] and Professor Sandu Popescu on superoscillations in free space [2] has been applied to the case of the quantum harmonic oscillator. Superoscillations in free space are seen to persist in time longer than expected and are also seen to reform periodically after times much greater than their disappearance - for the quantum harmonic oscillator superoscillations reform periodically faster. The time of disappearance, td, depends upon a larger set of variables, this is due to the added complexity of the quantum harmonic oscillator. The evolution of the wavepacket is investigated using an expansion in terms of eigenfunctions and in terms of the propagator using both an exact integration and a saddle point approximation. The creation and disappearance of superoscillations is shown to depend on the behaviour of the saddle points. The frequency of the quantum harmonic oscillator, ! is found to be an extra control parameter which dictates the strength and duration of the superoscillations.
Abstract We analyse the evolution of superoscillations in a relativistic wavepacket. A simple superoscillating wavepacket is set up and is allowed to evolve freely according to both the Klein–Gordon equation and the Dirac equation. The superoscillations evolve anisotropically and decay after a time. Both the lifetime and anisotropy can be understood in terms of the interaction of contributions to the wavepacket from components with strongly differing complex wavenumber. The analysis is supported by numerical calculations and the results are compared with the non-relativistic analysis. A potential experiment in which the significance of relativistic effects on superoscillations could be measured is proposed.
Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. These functions appear in various fields of science and technology, in particular they were discovered in quantum mechanics in the context of weak values introduced by Y. Aharonov and collaborators. The evolution problem of superoscillatory functions as initial conditions for the Schrödinger equation is intensively studied nowadays and the supershift property of the solution of Schrödinger equation encodes the persistence of superoscillatory phenomenon during the evolution. In this paper, we prove that the evolution of a superoscillatory initial datum for spinning particles in a magnetic field has the supershift property. Our techniques are based on the exact propagator of spinning particles, the associated infinite order differential operators and their continuity on suitable spaces of entire functions with growth conditions.
A superoscillatory function—that is, a band-limited function f(x) oscillating faster than its fastest Fourier component—is taken to be the initial state of a freely-evolving quantum wavefunction ψ. The superoscillations persist for unexpectedly long times, but eventually disappear through the interaction of contributions to ψ with complex momenta that are exponentially disparate in magnitude; this is established by applying the asymptotics of integrals, supported by numerics. f(x) can alternatively be regarded as the wave generated by a diffraction grating, propagating paraxially and without evanescence as ψ in the space beyond. The persistence of superoscillations is then interpreted as the propagation of sub-wavelength structure farther into the field than the more familiar evanescent waves.
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.
Abstract A superoscillating function is defined by the property that it oscillates faster than its fastest Fourier components. This is mathematically possible because the coefficients of the linear combinations of the band limited components depend on the number of components. This phenomenon was discovered in the context of quantum physics, but it has important applications in a variety of areas, including metrology, antenna theory, and a new theory of superresolution in optics. In this paper we study the evolution of superoscillatory functions in uniform electric field by the Schrödinger equation where we assume that the Hamiltonian contains a even polynomial of the linear momentum p . This includes the classical case but also relativistic corrections of any order. Moreover, we extend our results to the case of several variables using the theory of superoscillating functions in several variables. We conclude by discussing a comparison of our work with the existing literature.