Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice

Abstract

We establish a link between the fractional Schrödinger equation (FSE) and light propagation in the honeycomb lattice (HCL) - the Dirac-Weyl equation (DWE). The fractional Laplacian in FSE causes a modulation of the dispersion relation of the system, which in the limiting case becomes linear. In the HCL, the dispersion relation is already linear around the Dirac point, suggesting a possible connection with the FSE. Here, we demonstrate this connection by describing light propagation in both FSE and HCL, using DWE. Thus, we propagate Gaussian beams according to FSE, HCL around the Dirac point, and DWE, to discover very similar behavior - the conical diffraction. However, if an additional potential is brought into the system, the link between FSE and HCL is broken, because the added potential serves as a perturbation, which breaks the translational periodicity of HCL and destroys Dirac cones in the dispersion relation.

6 pages, 4 figures. Comments and criticisms are welcome (Yiqi Zhang: [email protected])

References in corpus (8)

• The electronic properties of graphene
• Topological Photonics
• Fractional Quantum Mechanics
• Unveiling pseudospin and angular momentum in photonic graphene
• $\mathcal{P}\mathcal{T}$ symmetry in a fractional Schrödinger equation
• Diffraction-free beams in fractional Schrödinger equation
• Numerical study of fractional Nonlinear Schrödinger equations
• Transport properties in the photonic super-honeycomb lattice - a hybrid fermionic and bosonic system

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• Self-accelerating beam dynamics in the space fractional Schrödinger equation
• High-frequency electromagnetic emission from non-local wavefunctions
• Quantum droplets in two-dimensional optical lattices
• High-accuracy power series solutions with arbitrarily large radius of convergence for fractional nonlinear differential equations
• Vortex solitons in fractional nonlinear Schrödinger equation with the cubic-quintic nonlinearity