Introduction to the standard Floquet analysis

We first consider an analysis of physical systems in periodic external fields using the Floquet theorem [134, 168]. As we shall see below, the theorem provides theoretical basis for the existence of field-dressed quasi-stationary state which expand the propagator. Earlier application of the Floquet theorem or related ideas to physical problems includes Refs. [224, 370], whereas the progress in this field is recently reviewed in Ref. [90]. Although it will later be extended to allow small non-periodic modulations, discussions in this section assumes perfect periodicity. The Schrodinger equation is given as 

Proof of the Floquet theorem Readers not familiar with the Floquet theorem are recommended to refer to textbooks on ordinary differential equations such as Ref. [168], where one would find a formal proof of the theorem different from the following discussions. Here we try to derive an intuitive but physically appealing explanation of the Floquet theorem based on Fourier analysis. We start with the Fourier transformation of the time dependent Schrodinger equation (8.1),  

Calculation of the Floquet states We now solve the eigenvalue equation (8.5) for Floquet states By virtue of the periodicity, each Floquet state ua(t)) can be expanded in a Fourier series. We take an arbitrary time-independent orthonormal basis set that expands the Hilbert space of our interest, then the Floquet state can be expanded using the basis 

The Fourier representation of the Floquet operator thus obtained is an infinite dimensional matrix in general. Fortunately, however, it often takes a simple form when the time dependence of the Hamiltonian is monochromatic, or composed of a single mode with frequency w in such case have nonzero values only for n — m = 0, 1. 

For any Floquet state ( )), its modulation, e Mc(f)), with an arbitrary integer n, is also a Floquet state with quasi-energy — hnw. 

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