Simulation of the Fractional Schrödinger Equation in Materials with Sub- and Super-Diffusive Transport*

Fractional derivatives (FDs) allow us to effectively describe transport in bumpy, wrinkled, or jagged materials in nature. In quantum systems that have these rough characteristics, such as in materials with sub- and super-diffusive transport, it is described by the fractional Schrödinger equation (FSE); as such, simulating the FSE is important to understanding quantum phenomena in complex geometries. We constructed an algorithm that solves the FSE in space using an integer derivative series approximation to the FD and evolves it in time using Volterra-like integral equations of first kind. Using this algorithm, we were able to simulate the FSE in materials that exhibit a nonlocal or fractional geometry. We were also able to quantitatively describe, in general, what the effects are that the fractional geometry has on the solution to the FSE. This applies to quantum information processing with connected systems because complex networks and complexity science can be related to the same source of space and time non-locality as found in the FSE. Using this simulation of the FSE, we want to construct materials from ground up with prescribed transport properties where we may control the movement of charge and spin through materials in quantum computing.