We present a short review article of recent new developments of Floquet formulations of time-dependent density functional theory (TDDFT) for ab initio nonperturbative treatment of multiphoton and high-order nonlinear optical processes of many-electron quantum systems in intense laser fields. In the presence of monochromatic or polychromatic (multi-color) time-dependent fields, it is shown that the time-dependent Kohn-Sham equation can be exactly reformulated as a time-independent generalized Floquet matrix eigenvalue problem. The theory has been extended to include the time-dependent current density functional theory (TDCDFT) as well. In the case of bound-free transition processes such as multiphoton ionization or dissociation, we show that the complex scaling transformation can be implemented into the Floquet formulation of TDDFT and TDCDFT, and the field-induced complex quasienergy (dressed) states of many-electron quantum systems can be determined by means of the solution of a rime-independent non-Hermitian Floquet matrix eigenvalue problem. To facilitate the exploration of the constraints satisfied by the universal but unknown time-dependent exchange-correlation energy functionals, we develop several exact relations and theorems for the quasienergy functionals. Finally we present some initial applications of the Floquet-TDDFT formalism for the study of multiphoton ionization of neutral atoms and electron angular distribution in multiphoton detachment of negative ions in intense laser fields. The Floquet-TDDFT/TDCDFT formalism provides a powerful new theoretical foundation and feasible procedures for the exploration of many-electron atomic and molecular physics in strong fields in the future.
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