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

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