Analytic structure of the ac quasienergy in the complex-field plane

Motivated by the possibility of developing new computational techniques for studying multiphoton ionization of atoms by monochromatic radiation, we have analyzed, in a nonrigorous fashion, the behavior of the ac quasienergy as a function of the complex field strength F. We elaborate upon earlier work of Manakov and Fainshtein [Theor. Math. Phys. 48, 815 (1981)] and conjecture that the ac quasienergy is a multivalued analytic function ${\mathit{E}}_{\mathrm{ac}}$(F) whose branches originate from different unperturbed (real or ‘‘shadow’’) atomic levels for real values of F. We further conjecture that the branch points of ${\mathit{E}}_{\mathrm{ac}}$(F) are of the square-root type. These branch points occur at complex values of F where two branches coalesce, and, as F sweeps along the real axis, the passage past a branch point coincides with the passage past either an intermediate multiphoton resonance (R) or a multiphoton ionization threshold (T). Branch points of type-R group into quadruplets, while branch points of type-T group into pairs. The two branches that intersect at a type-R branch point are both physically accessible, and originate from real levels, while only one of the two branches that intersect at a type-T branch point is physically accessible—the unphysical type-T branch is a ‘‘shadow’’ eigenvalue, which corresponds to a state with unphysical boundary conditions. We discuss the probability for the atom to make a transition from one branch to another when F is a slowly varying function of time. Normally, a type-R branch point enhances the ionization signal, while a type-T branch point diminishes the signal. In partial support of some of our conjectures, we present results of an accurate numerical study of the quasienergy (and its perturbation expansion) for the ground state of both the hydrogen atom and a model atom (an electron bound to a zero-range potential).