# Simulating Maxwell-Schr\"odinger Equations by High-Order Symplectic FDTD Algorithm

## TL;DR

This paper introduces a high-order symplectic FDTD algorithm for accurately simulating light-matter interactions governed by Maxwell-Schr"odinger equations, demonstrating improved accuracy and efficiency over traditional methods.

## Contribution

The paper develops a fourth-order symplectic FDTD algorithm with Dirichlet boundary conditions for better long-term simulation of Maxwell-Schr"odinger systems.

## Key findings

- The SFDTD(4,4) outperforms FDTD(2,2) in accuracy and efficiency.
- Numerical validation through 3D Rabi oscillation simulations.
- Enhanced energy-conservation in complex nanoscale light-matter interactions.

## Abstract

A novel symplectic algorithm is proposed to solve the Maxwell-Schr\"odinger (M-S) system for investigating light-matter interaction. Using the fourth-order symplectic integration and fourth-order collocated differences, Maxwell-Schr\"odinger equations are discretized in temporal and spatial domains, respectively. The symplectic finite-difference time-domain (SFDTD) algorithm is developed for accurate and efficient study of coherent interaction between electromagnetic fields and artificial atoms. Particularly, the Dirichlet boundary condition is adopted for modeling the Rabi oscillation problems under the semi-classical framework. To implement the Dirichlet boundary condition, image theory is introduced, tailored to the high-order collocated differences. For validating the proposed SFDTD algorithm, three-dimensional numerical studies of the population inversion in the Rabi oscillation are presented. Numerical results show that the proposed high-order SFDTD(4,4) algorithm exhibits better numerical performance than the conventional FDTD(2,2) approach at the aspects of accuracy and efficiency for the long-term simulation. The proposed algorithm opens up a promising way towards a high-accurate energy-conservation modeling and simulation of complex dynamics in nanoscale light-matter interaction.

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Source: https://tomesphere.com/paper/1906.09589