Implementation of Quantum and Classical Discrete Fractional Fourier Transforms

TL;DR

This paper presents classical and quantum optical implementations of the discrete fractional Fourier transform, demonstrating experimental results and potential applications across multiple scientific fields.

Contribution

It introduces a novel integrated approach to realize discrete fractional Fourier transforms in both classical and quantum optics, enabling observation of all transformation orders simultaneously.

Findings

Successful implementation of classical discrete fractional Fourier transforms.

Experimental demonstration of the shift theorem in classical optics.

Quantum transformations of entangled biphoton wave functions.

Abstract

Fourier transforms are ubiquitous mathematical tools in basic and applied sciences. We here report classical and quantum optical realizations of the discrete fractional Fourier transform, a generalization of the Fourier transform. In the integrated configuration used in our experiments, the order of the transform is mapped onto the longitudinal coordinate, thus opening up the prospect of simultaneously observing all Transformation orders. In the context of classical optics, we implement discrete fractional Fourier transforms, both integer and fractional, of exemplary wave functions and experimentally demonstrate the shift theorem. Moreover, we apply this approach in the quantum realm to transform separable and highly entangled biphoton wave functions. The proposed approach is versatile and could find applications in various fields where Fourier transforms are essential tools, such as quantum chemistry and biology, physics and mathematics.