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Perturbation theory for quasienergy (Floquet) solutions in the low-frequency regime of the oscillating electric field

For a simple illustrative model Hamiltonian for Xenon in low frequency linearly polarized laser field we obtain a remarkable agreement between the zero-order energy as well as amplitude and phase of the zero-order Floquet states and the exact eigenvalues and eigenfunctions of the Floquet operator. Here we use as a zero-order Hamiltonian the adiabatic Hamiltonian where time is used as an instantaneous parameter. Moreover, for a variety of low laser frequencies, $ω$, the deviation of the zero-order solutions from the exact quasi-energy (QE) Floquet solutions approaches zero at the time the oscillating laser field is maximal. This remarkable result gives a further justification to the validity of the first step in the simple man model. It should be stressed that the numerical calculations of the exact QE (Floquet) solutions become extremely difficult when $ω$ approaches zero and many Floquet channels are nested together and are coupled by the laser field. This is the main motivation for the development of perturbation theory for QE (Floquet) solutions when the laser frequency is small, to avoid the need to represent the Floquet operator by a matrix when the Fourier functions are used as a basis set. A way to calculate the radius of convergence of the perturbational expansion of the Floquet solutions in $ω$ is given.