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IMA Journal of Applied Mathematics

An alternative point of view to the theory of fractional Fourier transform

G. Dattoli

Published Jun 1, 1998• 24 citations
Publisher

Abstract

The concept of the fractional Fourier transform is framed within the context of quantum evolution operators. This point of view yields an extension of the above concept and greatly simplifies the underlying operational algebra. It is also proved that a multidimensional extension can be performed by using a biorthogonal multiindex harmonic oscillator basis. It is finally shown that most of the proposed physical interpretations of the fractional Fourier transform are just trivial consequences of the analysis developed in this paper.

DOI
10.1093/imamat/60.3.215

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The concept of the fractional Fourier transform is framed within the context of quantum evolution operators. This point of view yields an extension of the above concept and greatly simplifies the underlying operational algebra. It is also proved that a multidimensional extension can be performed by using a biorthogonal multiindex harmonic oscillator basis. It is finally shown that most of the proposed physical interpretations of the fractional Fourier transform are just trivial consequences of the analysis developed in this paper.
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.
Fourier and fractional-Fourier transformations are widely used in theoretical physics. In this paper we make quantum perspectives and generalization for the fractional Fourier transformation (FrFT). By virtue of quantum mechanical representation transformation and the method of integration within normal ordered product (IWOP) of operators, we find the key point for composing FrFT, and reveal the structure of FrFT. Following this procedure, a full family of generalized fractional transformations are discovered with the usual FrFT as one special case. The eigen-functions of arbitrary GFrT are derived explicitly.
Fourier transform has taken place in different areas and applications, in this paper has been revised an important new form of application in the paradigm of quantum computing. Quantum Fourier transforms have gained increased attention with the rise of quantum walks, boson sampling, and quantum metrology [2]. In the Shor’s Algorithm it is used for find discrete logarithms on a quantum computer with two modular exponentiations and two quantum Fourier transforms [1]. Our propose consist in show a new quantum gate that can perform the Fractional Fourier Transform defined by Namias as a tool to solve the differential equation of the quantum mechanical oscillator [3], which it can satisfy the condition of to be unitary. We apply the quantum gate in different states through of a real quantum computer offered by IBM for check that it carry out the transform successfully, it results were compared with results of Quantum Fourier Transform for can understand its application. In the geometric point of view showed in the figure (2a) and (2c) through of Bloch Sphere, we can see the difference between the application of Quantum Fourier Transform (QFT) and Quantum Fractional Fourier Transform (with our quantum gate) to the basis state |0〉, while in the figure (2b) and (2d) is possible see with a theoretical result that probability density is equal for both.