© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1510.04353 · Journal of Mathematical Physics

Driving quantum systems with superoscillations

Achim Kempf, Angus Prain

Published Aug 1, 2017• 16 citations

Abstract

Superoscillations, i.e., the phenomenon that a bandlimited function can temporary oscillate faster than its highest Fourier component, are being much discussed for their potential for “superresolution” beyond the diffraction limit. Here, we consider systems that are driven with a time dependence that is off-resonance for the system, in the Fourier sense. We show that superoscillating sources can temporarily induce resonance during the period when the source is behaving superoscillatory. This observation poses the question as to how the system “undoes” the “false resonance” after the full source has acted and its band limitation is apparent. We discuss several examples of systems that might be capable of distilling the temporary excitation through some non-harmonic effects, such as dissipation or dispersion at high frequencies, opening up the possibility of low frequency detection of “fast” microphysics through superoscillations. We conclude that either superoscillations really can beat the bandlimit and achieve superresolution (“kinematic superresolution”) or the superoscillating high frequency is absorbed and we gain dynamical access to the physics of high frequency processes with low frequency signals (“dynamical superresolution”).

DOI: 10.1063/1.4996135

Related papers

Evolution of quantum superoscillations and optical superresolution without evanescent waves

Michael Berry, Sandu Popescu

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.

Journal of Physics A Mathematical and General•May 16, 2006•442 citations

Analysis of superoscillatory wave functions

Matt S. Calder, Achim Kempf

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.

arXiv:quant-ph/0405065•Journal of Mathematical Physics•Dec 22, 2004•26 citations

Analysis of Superoscillatory Wave Functions

Matt S. Calder, Achim Kempf

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.

•Jan 1, 2004•15 citations

Synthesizing superoscillations with just two frequencies

Denys I. Bondar, Diyar Talbayev

Superoscillations, band-limited signals that locally oscillate faster than their highest Fourier component, have recently enabled superspectroscopy and super-sensing. Previous experiments [Phys. Rev. Lett. 131, 153803 (2023), https://doi.org/10.1103/PhysRevLett.131.153803 and APL Photonics 10, 086107 (2025), https://doi.org/10.1063/5.0271556] relied on combining four quasi-sinusoidal harmonics to generate temporal superoscillations. Here, we show that just two harmonics are sufficient to produce superoscillations of comparable quality, as quantified by the local frequency. By reanalyzing prior experimental data with two harmonics (0.5 and 0.6 THz) and performing new experiments with 0.9 and 1 THz harmonics, we demonstrate a twofold enhancement in local frequency relative to the highest frequency component -- matching the enhancement previously achieved with four harmonics. We provide a simple analytical explanation for the bichromatic superoscillations based on near-complete destructive interference, and identify an optimal frequency separation of approximately 10% that balances the superoscillatory frequency enhancement against signal amplitude. This simplification substantially lowers the experimental barrier to implementing superoscillation-based technology.

physics.opticsCross-listedarXiv:2607.19542•Jul 23, 2026

Scaling properties of superoscillations and the extension to periodic signals

Eugene Tang, Lovneesh Garg, Achim Kempf

Abstract Superoscillatory wave forms, i.e., waves that locally oscillate faster than their highest Fourier component, possess unusual properties that make them of great interest from quantum mechanics to signal processing. However, the more pronounced the desired superoscillatory behavior is to be, the more difficult it becomes to produce, or even only calculate, such highly fine-tuned wave forms in practice. Here, we investigate how this sensitivity to preparation errors scales for a method for constructing superoscillatory functions which is optimal in the sense that it minimizes the energetic expense. We thereby also arrive at very accurate approximations of functions which are so highly superoscillatory that they cannot be calculated numerically. We then investigate to what extent the scaling and sensitivity results for superoscillatory functions on the real line extend to the experimentally important case of superoscillatory functions that are periodic.

arXiv:1512.00109•Journal of Physics A Mathematical and Theoretical•Jul 8, 2016•11 citations

New methods for creating superoscillations

Leilee Chojnacki, Achim Kempf

Abstract Superoscillating functions, i.e. functions that locally oscillate at a rate faster than their highest Fourier component, are of interest for applications from fundamental physics to engineering. Here, we develop a new method which allows one to construct superoscillations of arbitrarily high frequency and arbitrarily long duration in a computationally efficient way. We also present a method for constructing non-singular Schrödinger potentials whose ground state is a superoscillating wave function.

arXiv:1608.03121•Journal of Physics A Mathematical and Theoretical•Nov 23, 2016•17 citations

Superoscillations, i.e., the phenomenon that a bandlimited function can temporary oscillate faster than its highest Fourier component, are being much discussed for their potential for “superresolution” beyond the diffraction limit. Here, we consider systems that are driven with a time dependence that is off-resonance for the system, in the Fourier sense. We show that superoscillating sources can temporarily induce resonance during the period when the source is behaving superoscillatory. This observation poses the question as to how the system “undoes” the “false resonance” after the full source has acted and its band limitation is apparent. We discuss several examples of systems that might be capable of distilling the temporary excitation through some non-harmonic effects, such as dissipation or dispersion at high frequencies, opening up the possibility of low frequency detection of “fast” microphysics through superoscillations. We conclude that either superoscillations really can beat the bandlimit and achieve superresolution (“kinematic superresolution”) or the superoscillating high frequency is absorbed and we gain dynamical access to the physics of high frequency processes with low frequency signals (“dynamical superresolution”).

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.

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.

Superoscillations, band-limited signals that locally oscillate faster than their highest Fourier component, have recently enabled superspectroscopy and super-sensing. Previous experiments [Phys. Rev. Lett. 131, 153803 (2023), https://doi.org/10.1103/PhysRevLett.131.153803 and APL Photonics 10, 086107 (2025), https://doi.org/10.1063/5.0271556] relied on combining four quasi-sinusoidal harmonics to generate temporal superoscillations. Here, we show that just two harmonics are sufficient to produce superoscillations of comparable quality, as quantified by the local frequency. By reanalyzing prior experimental data with two harmonics (0.5 and 0.6 THz) and performing new experiments with 0.9 and 1 THz harmonics, we demonstrate a twofold enhancement in local frequency relative to the highest frequency component -- matching the enhancement previously achieved with four harmonics. We provide a simple analytical explanation for the bichromatic superoscillations based on near-complete destructive interference, and identify an optimal frequency separation of approximately 10% that balances the superoscillatory frequency enhancement against signal amplitude. This simplification substantially lowers the experimental barrier to implementing superoscillation-based technology.

Abstract Superoscillatory wave forms, i.e., waves that locally oscillate faster than their highest Fourier component, possess unusual properties that make them of great interest from quantum mechanics to signal processing. However, the more pronounced the desired superoscillatory behavior is to be, the more difficult it becomes to produce, or even only calculate, such highly fine-tuned wave forms in practice. Here, we investigate how this sensitivity to preparation errors scales for a method for constructing superoscillatory functions which is optimal in the sense that it minimizes the energetic expense. We thereby also arrive at very accurate approximations of functions which are so highly superoscillatory that they cannot be calculated numerically. We then investigate to what extent the scaling and sensitivity results for superoscillatory functions on the real line extend to the experimentally important case of superoscillatory functions that are periodic.

Abstract Superoscillating functions, i.e. functions that locally oscillate at a rate faster than their highest Fourier component, are of interest for applications from fundamental physics to engineering. Here, we develop a new method which allows one to construct superoscillations of arbitrarily high frequency and arbitrarily long duration in a computationally efficient way. We also present a method for constructing non-singular Schrödinger potentials whose ground state is a superoscillating wave function.