Floquet perturbation theory: Applicability of the finite level approximation in different gauges

The Floquet perturbation theory represents an important technique for studying the response of atoms or molecules exposed to weak monochromatic fields. In this paper, we discuss an explicit implementation of the Floquet perturbation theory starting from different gauge representations of the matter-light Hamiltonian, namely, from the velocity gauge, the length gauge, and the acceleration gauge. Interestingly, the development of the second-order Floquet perturbation theory in different gauges gives rise to formally different quasienergy corrections, whose mutual equivalence is not self-evident. Nevertheless, our derivation shows how the perturbation formulas associated with different gauges can be converted to one another, provided that a complete basis set of the atomic-molecular electronic states has been used. On the other hand, it turns out that an inappropriate basis set truncation employed for a particular gauge (such as, e.g., the two-level approximation applied within the velocity gauge) may sometimes lead toward completely wrong results. This fact should serve as a warning against an uncautious use of various gauge transformations in practical calculations with a finite basis set