Implementation of quantum and classical discrete fractional Fourier transforms

Area of Science:

• Optics and Photonics
• Quantum Information Science
• Applied Mathematics
• Signal Processing

Background:

• Fourier transforms are fundamental mathematical tools across science and engineering.
• The fractional Fourier transform (FrFT) generalizes the ordinary Fourier transform.
• Implementing discrete fractional Fourier transforms (dFFTs) has been a significant challenge.

Purpose of the Study:

• To develop and demonstrate classical and quantum optical realizations of the discrete fractional Fourier transform (dFFT).
• To explore the applicability of dFFT in both classical wave function manipulation and quantum optical systems.

Main Methods:

• Classical optical implementation of the dFFT for exemplary wave functions.
• Experimental demonstration of the shift theorem using the optical dFFT.
• Quantum optical implementation of dFFT for separable and entangled biphoton states.

Main Results:

• Successful realization of discrete fractional Fourier transforms using classical optical setups.
• Experimental validation of the shift theorem in the context of optical dFFTs.
• Demonstration of dFFT application to quantum states, including entangled biphotons.

Conclusions:

• The proposed optical approach provides a viable method for implementing discrete fractional Fourier transforms.
• This versatile technique bridges classical and quantum optics, with potential applications in diverse scientific fields.
• The work overcomes previous elusiveness in dFFT implementation, paving the way for new technological advancements.