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The Fractional Fourier Transform (FRT) corresponds to an arbitrary-anglerotation in the phase space, e.g. the time-frequency (TF) space, andgeneralizes the fundamentally important Fourier Transform. FRT applicationsrange from classical signal processing (e.g. time-correlated noise optimalfiltering) to emerging quantum technologies (e.g. super-resolution TF imaging)which rely on or benefit from coherent low-noise TF operations. Here aversatile low-noise single-photon-compatible implementation of the FRT ispresented. Optical TF FRT can be synthesized as a series of a spectraldisperser, a time-lens, and another spectral disperser. Relying on thestate-of-the-art electro-optic modulators (EOM) for the time-lens, our methodavoids added noise inherent to the alternatives based on non-linearinteractions (such as wave-mixing, cross-phase modulation, or parametricprocesses). Precise control of the EOM-driving radio-frequency signal enablesfast all-electronic control of the FRT angle. In the experiment, we demonstrateFRT angles of up to 1.63 rad for pairs of coherent temporally separated 11.5ps-wide pulses in the near-infrared (800 nm). We observe a good agreementbetween the simulated and measured output spectra in the bright-light andsingle-photon-level regimes, and for a range of pulse separations (20 ps to26.67 ps). Furthermore, a tradeoff is established between the maximal FRT angleand bandwidth, with the current setup accommodating up to 248 GHz of bandwidth.With the ongoing progress in EOM on-chip integration, we envisage excellentscalability and vast applications in all-optical TF processing both in theclassical and quantum regimes.

Ultrafast electro-optic Time-Frequency Fractional Fourier Imaging at the Single-Photon Level