Semiconductor Wannier equations: A real-time, real-space approach to the nonlinear optical response in crystals

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We develop the semiconductor Wannier equations (SWEs), a real-time, real-space formulation of ultrafast light-matter dynamics in crystals, by deriving the equations of motion for the electronic reduced density matrix in a localized Wannier basis. Working in real space removes the structure-gauge ambiguities that hinder reciprocal-space semiconductor Bloch equations. Electron-electron interactions are included at the time-dependent Hartree plus static screened-exchange level. Decoherence is modeled with three complementary channels: pure dephasing, population relaxation, and distance-dependent real-space dephasing, providing physically grounded damping for strong-field phenomena such as high-harmonic generation. Conceptually, the SWEs bridge real-space semiclassical intuition with many-body solid-state optics, offering a numerically robust and gauge-clean alternative to reciprocal-space approaches for nonlinear optical response and attosecond spectroscopy in solids.