Floquet representation

A quite different approach from all other presented in this review has been recently proposed by Coffey [41]. This approach allows both the MFPT and the integral relaxation time to be exactly calculated irrespective of the number of degrees of freedom from the differential recurrence relations generated by the Floquet representation of the FPE. 

In order to illustrate how the Floquet representation of the FPE, Eq. (5.44), may be used to calculate first passage times, we first take the Laplace transform of... 

Fourier and Floquet space. The definitions of the different block-elements are given in a for the Fourier representation and in b for the Floquet representation. The Hamiltonians retain the same general form, but the time-dependent elements in a become time-independent in b with the addition of the number operator elements on the diagonal... 

Relation Between Eigenvectors and Diagonalization Transformations Appendix B Coherent States in the Floquet Representation... 

Since in the Floquet representation the Hamiltonian K defined on the enlarged Hilbert space is time-independent, the analysis of the effect of perturbations (like, e.g., transition probabilities) can be done by stationary perturbation theory, instead of the usual time-dependent one. Here we will present a formulation of stationary perturbation theory based on the iteration of unitary transformations (called contact transformations or KAM transformations) constructed such that the form of the Hamiltonian gets simplified. It is referred to as the KAM technique. The results are not very different from the ones of Rayleigh-Schrodinger perturbation theory, but conceptually and in terms of speed of convergence they have some advantages. 

Rigorously, the above Floquet representation is valid only when the field is periodic. Imagine now that the field amplitude Eo in Eq. (12) carries a time dependence that denotes a slow modulation of the cosine field which oscillates with a frequency in the UV-Vis spectral range. Without the amplitude modulation, the dynamics under the UV-Vis field is well captured by the Floquet representation. If the amplitude modulation is slow. 

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