Experimental realisations of the fractional Schr\"{o}dinger equation in the temporal domain

Abstract

The fractional Schr\"{o}dinger equation (FSE) -- a natural extension of the standard Schr\"{o}dinger 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 \textit{L\'evy waveguide} and to reconstruct the amplitude and phase of the pulses. Varying the L\'evy 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.