Experimental realisations of the fractional Schrödinger equation in the temporal domain

Abstract

The fractional Schrödinger equation (FSE) -- a natural extension of the standard Schrödinger 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{Lévy waveguide} and to reconstruct the amplitude and phase of the pulses. Varying the Lévy 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.

to be published in Nature Communications

References in corpus (4)

• Fractional Quantum Mechanics
• Propagation dynamics of the circular Airy Gaussian vortex beams in the fractional nonlinear Schrödinger equation
• Families of fundamental and multipole solitons in a cubic-quintic nonlinear lattice in fractional dimension
• Time-frequency Domain Analogues of Phase Space Sub-Planck Structures

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• One-dimensional L{é}vy Quasicrystal
• Two-dimensional fractional discrete NLS equations: dispersion relations, rogue waves, fundamental and vortex solitons
• Spectral and Dynamical Properties of the Fractional Nonlinear Schrödinger Equation under Harmonic Confinement
• Generalised fractional Rabi problem