Bichromatic dressing of Rydberg atoms and on the correctness of many-mode Floquet theory

Many-mode Floquet theory [T.-S. Ho, S.-I. Chu, and J. V. Tietz, Chem. Phys. Lett. 96, 464 (1983)] was designed as an extension of Floquet theory suitable for solving the time-dependent Schrodinger equation with multiple periodicities, however its limitations are not well understood. I show that for two commensurate frequencies (integer multiples of a common frequency), many-mode Floquet theory always produces an exact expression for the time evolution of a system, despite only part of the eigenvalue spectrum being directly relevant. I show that the rest of the spectrum corresponds to eigenvalues of the same system but at other values of the relative phase between the bichromatic field components. I show by using a Floquet perturbative analysis that dressing a Rydberg atom with a bichromatic field with frequency components ω2 and ω1, such that ω2 = 2ω1, can induce a permanent dipole moment (first order energy shift with dc electric field) without a dc bias field. With frequency ω1 = 2π5.997GHz, ω2 = 2ω1 and field strengths of Eac1 = 0.1 V/cm and Eac2 = 0.05 V/cm, a permanent dipole moment of magnitude 44.06 MHz/(V/cm) is induced in the dressed 65s1/2 state of ⁸⁵Rb. The permanent dipole moment depends on the relative phase between the fields and can be made to be zero at certain values of phase