An intuitive and complete understanding of the underlying processes in high harmonic generation (HHG) in solids will enable the development and optimization of experimental techniques for attosecond measurement of dynamical and structural properties of solids. Here we introduce the Wannier quasi-classical (WQC) theory, which allows the characterization of HHG in terms of classical trajectories. The WQC approach completes the single-body picture for HHG in semiconductors, as it is in quantitative agreement with quantum calculations. The importance of WQC theory extends beyond HHG; it enables modeling of dynamic processes in solids with classical trajectories, such as for coherent control and transport processes, potentially providing better scalability and a more intuitive understanding.

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Keldysh ionization theory is one of the main pillars of strong field physics and attosecond science. It describes non-relativistic ionization rates of hydrogen-like atoms subjected to strong laser fields within the dipole approximation and the length gauge. According to this theory ionization can be described by two regimes: electronic tunneling through a laser-dressed atomic potential (tunnel ionization) and absorption of multiple photons at once (multi-photon ionization). There are many gaps in the mathematical steps and explanations in the original Keldysh paper. Therefore, the goal of this work is to give a detailed re-derivation of ionization rates following Keldysh’s formulation and to fill in the mathematical steps of this beautiful approach so that it is more accessible to a wider audience.