Classical evolution of quantum elliptic states

Abstract

The hydrogen atom in weak external fields is a very accurate model for the multiphoton excitation of ultrastable high angular momentum Rydberg states, a process which classical mechanics describes with astonishing precision. In this paper we show that the simplest treatment of the intramanifold dynamics of a hydrogenic electron in external fields is based on the elliptic states of the hydrogen atom, i.e., the coherent states of $\mathrm{SO}(4),$ which is the dynamical symmetry group of the Kepler problem. Moreover, we also show that classical perturbation theory yields the exact evolution in time of these quantum states, and so we explain the surprising match between purely classical perturbative calculations and experiments. Finally, as a first application, we propose a fast method for the excitation of circular states; these are ultrastable hydrogenic eigenstates that have maximum total angular momentum and also maximum projection of the angular momentum along a fixed direction.