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Multielectron High Harmonic Generation: simple man on a complex plane

From the famous classical "simple man" model to the recent multichannel model for polyatomic molecules, this tutorial will guide you through the several landmarks in our understanding of high harmonic generation and high harmonic spectroscopy. Our goal was to provide recipes and insights for modelling the harmonic response in various regimes, from the single active electron regime typical for noble gas atoms to the laser-driven attosecond hole dynamics in polyatomics. We have tried to pay particular attention to both simple recipes and their limitations, including the connection of real-valued classical and complex-valued quantum times and velocities. In addition to physical pictures, general approaches and their practical realizations, we have also tried to discuss some of the sticky technical issues, such as the possibility to factorize high harmonic response into the three steps of ionization, propagation and recombination and practical ways to treat strong field ionization, search for the saddle points of multi-dimensional integrals, deal with poles arising in the quantum-mechanical integrands, remove divergences appearing in approximate saddle point expressions, etc.