  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 imaging)
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
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
26.67 ps). Furthermore, a tradeoff is established between the maximal FRT angle
and 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.
