Residual Symmetries in the Spectrum of Periodically Driven Alkali Rydberg States

– We identify a fundamental structure in the spectrum of microwave driven alkali Rydberg states, which highlights the remnants of the Coulomb symmetry in the presence of a non-hydrogenic core. Core-induced corrections with respect to the hydrogen spectrum can be accounted for by a perturbative approach. Introduction. – The excitation and subsequent ionization of Rydberg states of atomic hydrogen by microwave fields is one of the most prominent examples of the manifestation of classically nonlinear dynamics in a realistic physical system [1]. Given a driving field frequency comparable to the classical Kepler frequency of the unperturbed Rydberg electron, the electron’s classical trajectory goes chaotic for sufficiently large driving field amplitudes, finally leading to its ionization on a finite time scale [2]. Correspondingly, large ionization rates are observed in experiments on real (i.e., quantum) Rydberg states of atomic hydrogen, in the appropriate parameter range [1, 3]. As a matter of fact, already before the onset of classically chaotic motion, i.e. at not too large driving field amplitudes, individual quantum eigenstates of the atom in the field exhibit energies and ionization rates which are determined only by the orbital parameters of the classical trajectory they are associated with [4]. Those orbits which are the least stable under the external perturbation (i.e., which turn chaotic for the lowest values of the driving field amplitude, such as straight line orbits parallel to the field polarization axis for a linearly polarized drive) induce the largest ionization rates for their associated eigenstates. Consequently, in this near-integrable regime of classical dynamics, it is possible to classify the eigenstates of the atom in the field through quantum numbers associated with the orbital parameters of unperturbed Kepler ellipses, i.e. with the angular momentum and the RungeTypeset using EURO-TEX 2 EUROPHYSICS LETTERS Lenz vector. An adiabatic invariant governs the slow evolution of these parameters under external driving [4]. It should be noted, however, that a considerable part of experimental data has been accumulated in experiments on Rydberg states of alkali atoms rather than of atomic hydrogen [5, 6, 7, 8, 9, 10]. A priori, a classical-quantum correspondence as briefly sketched above for atomic hydrogen cannot be established here, due to the absence of a well and uniquely defined classical Hamiltonian. In particular, the atomic core destroys the symmetry characteristic for the hydrogen atom and the Runge-Lenz vector is no more a constant of motion. Indeed, experimental data systematically suggest strongly enhanced ionization rates of nonhydrogenic (i.e., low angular momentum) alkali Rydberg states as compared to atomic hydrogen [5, 6, 7, 9, 10], though they also exhibit qualitatively similar features, e.g. of the dependence of the ionization yield on the principal quantum number of the atomic state the atoms are initially prepared in [9, 10]. On the other hand, a direct comparison of available hydrogen and alkali data is somewhat questionable, since relevant experimental parameters such as the interaction time of the atom with the field are typically different for different experiments. Furthermore, a rigourous theoretical treatment of alkali atoms exposed to microwave fields was not accomplished until now. It is the purpose of the present letter to outline such a rigourous treatment which allows for the first time for a direct comparison of hydrogen and alkali ionization dynamics under precisely the same conditions, without adjustable parameters. First results of our numerical experiments directly address the above question of quantum-classical correspondence for periodically driven alkali atoms. Theory. – Let us start with the nonrelativistic Hamiltonian of a one-electron atom exposed to a linearly polarized microwave field of (constant) amplitude F and frequency ω, in length gauge, employing the dipole approximation and atomic units: H(t) = p 2 + Vatom(r) + Fz cosωt, r > 0. (1) As this Hamiltonian is periodic in time, we can use the Floquet theorem [11] to find the eigenstates (“dressed states”) of the atom in the field. After integration over the solid angle we have to solve the time-independent, radial eigenvalue equation