Canonical symplectic structure and structure-preserving geometric algorithms for Schr\"odinger-Maxwell systems

TL;DR

This paper develops structure-preserving geometric algorithms for Schr"odinger-Maxwell systems that maintain symplectic structure, wavefunction unitarity, and energy bounds, enabling accurate long-term simulations of photon-matter interactions.

Contribution

It introduces novel numerical algorithms that preserve key geometric properties of Schr"odinger-Maxwell systems, enhancing simulation accuracy and stability.

Findings

Algorithms preserve symplectic structure and wavefunction unitarity.

Energy errors remain bounded over long simulations.

Enables accurate first-principle simulations of photon-matter interactions.

Abstract

An infinite dimensional canonical symplectic structure and structure-preserving geometric algorithms are developed for the photon-matter interactions described by the Schr\"odinger-Maxwell equations. The algorithms preserve the symplectic structure of the system and the unitary nature of the wavefunctions, and bound the energy error of the simulation for all time-steps. This new numerical capability enables us to carry out first-principle based simulation study of important photon-matter interactions, such as the high harmonic generation and stabilization of ionization, with long-term accuracy and fidelity.