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Frontiers in Optics

Multiphoton discrete fractional Fourier operations in waveguide beam splitters

Konrad Tschernig, Roberto de J. León‐Montiel, Omar S. Magaña‐Loaiza, Alexander Szameit, Kurt Busch, Armando Pérez-Leija

Published Jan 1, 2018• 0 citations
Publisher

Abstract

We show that exciting a waveguide beam-splitter with N indistinguishable photons, give rise to lattice-like structures in the photon number space that are formally equivalent to coupled systems that perform discrete fractional Fourier transforms.
DOI
10.1364/fio.2018.jtu2a.67

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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.
We demonstrate that when a waveguide beam splitter (BS) is excited by N indistinguishable photons, the arising multiphoton states evolve in a way as if they were coupled to each other with coupling strengths that are identical to the ones exhibited by a discrete fractional Fourier system. Based on the properties of the fractional Fourier transform, we then derive a multiphoton suppression law for 50/50 BSs, thereby generalizing the Hong–Ou–Mandel effect. Furthermore, we examine the possibility of performing simultaneous multiphoton quantum random walks by using a single waveguide BS in combination with photon-number-resolving detectors. We anticipate that the multiphoton lattice-like structures unveiled in this work will be useful to identify new effects and applications of high-dimensional multiphoton states.