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—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.
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.
The principal differences between non-relativistic and relativistic quantum mechanics are: the existence of negative energy states, and the natural emergence of spin within a relativistic framework. We consider both these effects in the creation and evolution of superoscillating wavepackets. Superoscillations are a phenomena in which a function can oscillate faster than it’s fastest Fourier component. As a trade-off they suffer by being exponentially smaller than conventional oscillations. Previous studies of superoscillations within the construct of the Schrödinger equation have identified two key features: disappearance after a time,
td, and an asymmetrical evolution either side of a line in space (the wall effect). In this thesis, we expand this work into a relativistic framework by applying it to the Klein-Gordon and Dirac equations, to do this we require a thorough examination of 1+1 dimensional relativistic propagators. Although initially derived interms of Bessel functions, these have two key limits which allow for simpler calculation: the light-cone limit
(x→x0+ct) and the WKB limit
(h→0). Positive and negative energy superoscillations are best described in the WKBlimit. Through application of asymptotic integration, we find that both positiveand negative energy superoscillations possess a disappearance time and wall effect. For both Klein Gordon and Dirac equations,
td is equal. This implies, in terms of disappearance spin has no effect on the evolution of a superoscillatory, relativistic wavefunction of positive or negative energy. However, relativistic superoscillations disappear faster than non-relativistic superoscillations, and in each case, the disappearance time tends to a finite value as a superoscillatory parameter is increased. For non-relativistic superoscillations
td→1/2 compared to
td→0.3715 for the relativistic case. The wall effect is also noticeably different in relativistic and non-relativistic contexts. The walls appear at different points in space and, despite the relativistic wall effect being evident from
t=0, within the Schrödinger equation, it doesn’t appear until a time of
t=1/32. This contrast in the wall effects of both cases is the leading cause of the inequality of the disappearance times. The asymmetries between the wall effects are caused by differences in the behaviour of saddles (complex momenta) found when evaluating the wavefunction as an integral over the propagator. Again, there is no disparity between the walls of the Klein-Gordon or Dirac equation; spin does not affect the evolution of positive ornegative energy superoscillations. Where positive and negative energy wavefunctions appear as
h→0, mixed energy superoscillations appear at the light-cone. Mixed energy superoscillations do not exhibit the wall effect and neither do they exist in a non-relativistic description. However, they do have a disappearance time. In contrast to the positive and negative energy states,
td→0 as a superoscillatory parameter is increased. It is within a mixed energy construct that the effect of spin on the evolution of relativistic superoscillations appears; one of the components of the Dirac equation does not superoscillate. This is caused by this term existing at the WKB limit as opposed to the light-cone.
It is well known that one might describe a wave or a function in terms of it's Fourier components, therefore a function would be labelled as superoscillatory if it oscillates faster than the fastest Fourier component. This phenomena is known to be caused by the fine destructive interference between the Fourier components of a wave. There is a vast amount of literature on this special phenomena, however the main focus of this thesis will be studying and analysing the works of Sir Michael Berry, Sandu Popescu, Mark Dennis, and J.Lindberg [1][2][3][4][5], focusing mainly on naturally occurring superoscillations through the superposition of a large number of waves. In particular, investigating the effects of changing the band-width of the waves on the fractional superoscillatory area of the resultant wave as well as the effects of varying the number of waves (Fourier components) superimposing. In the papers by Mark Dennis et al [5] [4], an in depth description is given for superoscillations in speckle patterns, through the derivation of a joint probability density function of both intensity and phase gradient. Through this, a superoscillating fraction of 1 5 is found for a disk spectrum in addition an expression is found for an annular spectrum. The main work conducted for this thesis builds on this through a range of computational methods, specifically investigating the fraction of superoscillations obtained from the superposition of two dimensional non-monochromatic waves. An interesting result found shows that the fractional area which exhibit superoscillations is given to be much smaller than 1 5 for an annular spectrum as described in [5]. The calculations conducted here describes the effect of changing the bandwidth; in which the wavenumber of the individual waves are chosen from, has on the fraction of superoscillations. In addition the effect of varying the lowest wavenumber in the spectrum, as well as the number of sources, is provided. The study of all the papers mentioned earlier involves a vast amount of statistical analysis, ranging from Gaussian and speckle statistics to probability theory.
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.