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

Fractional Fourier Transform - Experimental Implementation for Spectro-Temporal Cat State

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

Published Jun 26, 2023• 0 citations

Abstract

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 angle$\varphi$coresponding 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.

DOI: 10.1109/cleo/europe-eqec57999.2023.10232484

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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 angle

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coresponding 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.

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.

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 (FRT) corresponds to an arbitrary-angle rotation in the phase space, e.g., the time-frequency (TF) space, and generalizes the fundamentally important Fourier Transform. FRT applications range from classical signal processing (e.g., time-correlated noise optimal filtering) to emerging quantum technologies (e.g., super-resolution TF sensing) which rely on or benefit from coherent low-noise TF operations. Here a versatile low-noise single-photon-compatible implementation of the FRT is presented. Optical TF FRT can be synthesized as a series of a spectral disperser, a time-lens, and another spectral disperser. Relying on the state-of-the-art electro-optic modulators (EOM) for the time-lens, our method avoids added noise inherent to the alternatives based on non-linear optical interactions (such as wave-mixing, cross-phase modulation, or parametric processes). Precise control of the EOM-driving radio-frequency signal enables fast all-electronic control of the FRT angle. In the experiment, we demonstrate FRT angles of up to 1.63 rad for pairs of coherent temporally separated 11.5 ps-wide pulses in the near-infrared (800 nm). We observe a good agreement between the simulated and measured output spectra in the bright-light and single-photon-level regimes, and for a range of pulse separations (20 ps to 26.7 ps). Furthermore, a tradeoff is established between the maximal FRT angle and optical bandwidth, with the current setup accommodating up to 248 GHz of bandwidth. With the ongoing progress in EOM on-chip integration, we envisage excellent scalability and vast applications in all-optical TF processing both in the classical and quantum regimes.