Classical evolution of quantum elliptic states

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 dyna mics of a hydrogenic electron in external fields is based on the elliptic s tates of the hydrogen atom, i.e., the coherent states of SO(4), which is th e dynamical symmetry group of the Kepler problem. Moreover, we also show th at classical perturbation theory yields the exact evolution in time of thes e quantum states, and so we explain the surprising match between purely cla ssical perturbative calculations and experiments. Finally, as a first appli cation, we propose a fast method for the excitation of circular states; the se are ultrastable hydrogenic eigenstates that have maximum total angular m omentum and also maximum projection of the angular momentum along a fixed d irection. [S1050-2947(99)05203-8].