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Implementation of quantum and classical discrete fractional Fourier transforms

Author

Listed:
  • Steffen Weimann

    (Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena)

  • Armando Perez-Leija

    (Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena)

  • Maxime Lebugle

    (Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena)

  • Robert Keil

    (Institut für Experimentalphysik, Universität Innsbruck)

  • Malte Tichy

    (University of Aarhus)

  • Markus Gräfe

    (Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena)

  • René Heilmann

    (Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena)

  • Stefan Nolte

    (Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena)

  • Hector Moya-Cessa

    (INAOE, Coordinacion de Optica)

  • Gregor Weihs

    (Institut für Experimentalphysik, Universität Innsbruck)

  • Demetrios N. Christodoulides

    (CREOL, The College of Optics & Photonics, University of Central Florida)

  • Alexander Szameit

    (Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena)

Abstract

Fourier transforms, integer and fractional, are ubiquitous mathematical tools in basic and applied science. Certainly, since the ordinary Fourier transform is merely a particular case of a continuous set of fractional Fourier domains, every property and application of the ordinary Fourier transform becomes a special case of the fractional Fourier transform. Despite the great practical importance of the discrete Fourier transform, implementation of fractional orders of the corresponding discrete operation has been elusive. Here we report classical and quantum optical realizations of the discrete fractional Fourier transform. In the context of classical optics, we implement discrete fractional Fourier transforms of exemplary wave functions and experimentally demonstrate the shift theorem. Moreover, we apply this approach in the quantum realm to Fourier transform separable and path-entangled biphoton wave functions. The proposed approach is versatile and could find applications in various fields where Fourier transforms are essential tools.

Suggested Citation

  • Steffen Weimann & Armando Perez-Leija & Maxime Lebugle & Robert Keil & Malte Tichy & Markus Gräfe & René Heilmann & Stefan Nolte & Hector Moya-Cessa & Gregor Weihs & Demetrios N. Christodoulides & Ale, 2016. "Implementation of quantum and classical discrete fractional Fourier transforms," Nature Communications, Nature, vol. 7(1), pages 1-8, April.
    • Handle: RePEc:nat:natcom:v:7:y:2016:i:1:d:10.1038_ncomms11027
    DOI: 10.1038/ncomms11027
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    Cited by:

    1. Urzúa, Alejandro R. & Ramos-Prieto, Irán & Soto-Eguibar, Francisco & Moya-Cessa, Héctor, 2020. "Dynamical analysis of mass–spring models using Lie algebraic methods," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).

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