Generalization of the Floquet theory to treat dynamics in pulse lasers

2 Generalization of the Floquet theory to treat dynamics in pulse lasers 

Laser-induced molecular dynamics we will examine below is driven by an oscillating electric field whose strength is comparable to the atomic Coulomb field. Solving for molecular dynamics in such strong optical field requires a non-perturbative approach. 

1 Expansion using Field-dressed periodic states 

Presumably the most straightforward approach to chemical dynamics in intense laser fields is to use the time-independent or time-dependent adiabatic states [352], which are the eigenstates of field-free or field-dependent Hamiltonian at given time points respectively, and solve the Schrodinger equation in a stepwise manner. However, when the laser field is approximately periodic, one can also use a set of field-dressed periodic states as an expansion basis. The set of quasi-static states in a periodic Hamiltonian is derived by a Floquet type analysis and is often referred to as the Floquet states [370]. Provided that the laser field is approximately periodic, advantages of using the latter basis set include (1) analysis and interpretation of the electron dynamics is clearer since the Floquet state population often vary slowly with the timescale of the pulse envelope and each Floquet state is characterized as a field-dressed quasi-stationary state, (2) under some moderate conditions, the nuclear dynamics can be approximated by mixed quantum-classical (MQC) nonadiabatic dynamics on the field-dressed PES. The latter point not only provides a powerful clue for interpretation of nuclear dynamics but also implies possible MQC formulation of intense field molecular dynamics. 

The Floquet theorem, when apphed to the quantum mechanics [370], implies the stationarity of Floquet states imder a perfectly periodic Hamiltonian. We define the electronic Floquet operator as 7ft = Hf — ihdt and the Floquet states as its periodic eigenstates which satisfy Tlt x t)) = A] A(f))- The above mentioned stationarity states that the solution of time-dependent Schrodinger equation ] t) can be expanded as 

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