  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.
