Optical Fractional Fourier Transform in the time-frequency domain based on quantum-memory.

Fractional Fourier Transform (FrFT) has a number of applications ranging from noise reduction to radar science and mode sorting of light. It has an intuitive meaning when we represent it as a rotation of chronocyclic Wigner function in time-frequency space. This can be achieved by applying specific time and frequency quadratic phases to the input signal by means of linear modulation of frequency and AC-Stark shift applied to light stored in the form of atomic coherence in the Gradient Echo Memory. Previous experiments show that quantum memories allow for versatile processing of quantum states of light including super-resolved spectroscopy and Fourier transform. We expand that idea and demonstrate implementation of FrFT in GEM. We benchmark the protocol by showing transformation of two-pulse "Schroedinger cat" states and Hermite-Gauss modes–eigenfunctions of FrFT--proving its possible application in mode sorting. We are the first to implement the FrFT in the optical time-frequency domain. This achievement opens up new avenues in optical signal processing. In particular, allowing tailored noise reduction protocols to be implemented purely in the optical domain. Our setup allows for manipulation of signals with bandwidth reaching 1 MHz and duration of 25 μs, allowing operation with ultra narrow band light compatible with atomic and optomechanical systems.