Experimental realisations of the fractional Schrodinger equation in the temporal domain

The fractional Schrodinger equation (FSE)-a natural extension of the standard Schrodinger equation-is the basis of fractional quantum mechanics. It can be obtained by replacing the kinetic-energy operator with a fractional derivative. Here, we report the experimental realisation of an optical FSE for femtosecond laser pulses in the temporal domain. Programmable holograms and the single-shot measurement technique are respectively used to emulate a Levy waveguide and to reconstruct the amplitude and phase of the pulses. Varying the Levy index of the FSE and the initial pulse, the temporal dynamics is observed in diverse forms, including solitary, splitting and merging pulses, double Airy modes, and "rain-like" multi-pulse patterns. Furthermore, the transmission of input pulses carrying a fractional phase exhibits a "fractional-phase protection" effect through a regular (non-fractional) material. The experimentally generated fractional time-domain pulses offer the potential for designing optical signal-processing schemes. Studies on the fractional Schrodinger equation (FSE) remain mostly theoretical, due to the lack of materials supporting fractional dispersion or diffraction. Here, the authors indirectly realized the FSE using two programmable holograms acting as an optical Levy waveguide.