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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)

Optical Fractional Fourier Transform - Experimental Implementation

Marcin Jastrzębski, Bartosz Niewelt, Stanisław Kurzyna, Jan Nowosielski, Wojciech Wasilewski, Mateusz Mazelanik, Michal P. Parniak

Published Jun 26, 2023• 0 citations
Publisher

Abstract

Light processing plays a central role in quantum communication, since optical quantum networks that will allow for secure quantum communication [1] and linking quantum computers [2] are of significant scientific and engineering interest [3], [4]. In this work, we present the experimental implementation of optical Fractional Fourier Transform. The protocol is realised in gradient echo memory. Hermite-Gaussian modes of light are encoded in rubidium atoms with the coupling impulse linearly chirped in frequency - this corresponds to the temporal lens [5]. Then the ac-Stark beam is applied to the atoms which leads to them acquiring the spectral phase corresponding to the propagation through a spectral lens. This operation results in the rotation of the Wigner function of the input state [6] and is exactly the Fractional Fourier Transform (FrFT) of the input function. The Hermite-Gaussian modes of light are the eigenfunctions of FrFT.

DOI
10.1109/cleo/europe-eqec57999.2023.10232517

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Light processing plays a central role in quantum communication, since optical quantum networks that will allow for secure quantum communication [1] and linking quantum computers [2] are of significant scientific and engineering interest [3], [4]. In this work, we present the experimental implementation of optical Fractional Fourier Transform. The protocol is realised in gradient echo memory. Hermite-Gaussian modes of light are encoded in rubidium atoms with the coupling impulse linearly chirped in frequency - this corresponds to the temporal lens [5]. Then the ac-Stark beam is applied to the atoms which leads to them acquiring the spectral phase corresponding to the propagation through a spectral lens. This operation results in the rotation of the Wigner function of the input state [6] and is exactly the Fractional Fourier Transform (FrFT) of the input function. The Hermite-Gaussian modes of light are the eigenfunctions of FrFT.
The fractional Fourier transform (FrFT), a fundamental operation in physics that corresponds to a rotation of phase space by any angle, is also an indispensable tool employed in digital signal processing for noise reduction. Processing of optical signals in their time-frequency degree of freedom bypasses the digitization step and presents an opportunity to enhance many protocols in quantum and classical communication, sensing, and computing. In this Letter, we present the experimental realization of the fractional Fourier transform in the time-frequency domain using an atomic quantum-optical memory system with processing capabilities. Our scheme performs the operation by imposing programmable interleaved spectral and temporal phases. We have verified the FrFT by analyses of chroncyclic Wigner functions measured via a shot-noise limited homodyne detector. Our results hold prospects for achieving temporal-mode sorting, processing, and superresolved parameter estimation.
Fractional Fourier transform (FrFT) is an important tool used in signal processing. It is usually utilized to filter the noise in acoustic signals and it is generalization of Fourier transform. In recent years it was also presented that it has a crucial role in optics, especially mode sorting [2]. One of the important interpretations of FrFT is rotation of the Wigner distribution in time-frequency domain called chronocyclic Wigner function. We have shown that it is possible to perform rotation of arbitrary angleφ\varphiφcoresponding to angle of FrFT. Despite its usefulness, it is still very difficult to perform FrFT in the time-frequency domain. Our experiment is based on the gradient echo memory built around cloud of ultracold rubidium 87 atoms trapped in the magneto-optical trap. The memory protocol uses two-photon Raman transition to store signal pulses in atomic coherence. By applying spatial lens with ac-Stark modulation and imposing two temporal lenses in specific order [1]. We have experimentally performed FrFT in spectro-temporal domain for the cat state.
A quantum memory for light is expected to play a crucial role in quantum communication protocols and distributed quantum computing. In addition to storage and buffering, a quantum memory can be used for manipulations of stored states to allow more complex quantum network operations. In this work, we demonstrate a quantum-compatible in-memory Fourier transform using a combination of two well-established quantum memory protocols: Gradient Echo Memory and Electromagnetically Induced Transparency. Our experiment is realized using an ensemble of rubidium atoms that are laser cooled in an elongated magneto-optic trap to maximize optical depth. The results of our time-frequency Fourier transform can be understood as a temporal double slit. We show that the interference between time-separated pulses depends on the relative phase and time between the pulses of light. The use of quantum memory protocols enables us to illuminate exactly where and how interference occurs between time-separated pulses. Time-frequency Fourier manipulation is a well-established technique in classical optical systems. Our combination of Fourier manipulation and quantum-compatible memory could be used to bring similar capability to quantum optical systems.