Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering

We give a general overview of the high-frequency regime in periodically\ndriven systems and identify three distinct classes of driving protocols in\nwhich the infinite-frequency Floquet Hamiltonian is not equal to the\ntime-averaged Hamiltonian. These classes cover systems, such as the Kapitza\npendulum, the Harper-Hofstadter model of neutral atoms in a magnetic field, the\nHaldane Floquet Chern insulator and others. In all setups considered, we\ndiscuss both the infinite-frequency limit and the leading finite-frequency\ncorrections to the Floquet Hamiltonian. We provide a short overview of Floquet\ntheory focusing on the gauge structure associated with the choice of\nstroboscopic frame and the differences between stroboscopic and\nnon-stroboscopic dynamics. In the latter case one has to work with dressed\noperators representing observables and a dressed density matrix. We also\ncomment on the application of Floquet Theory to systems described by static\nHamiltonians with well-separated energy scales and, in particular, discuss\nparallels between the inverse-frequency expansion and the Schrieffer-Wolff\ntransformation extending the latter to driven systems.\n