Quantum Anomaly Dissociation of Quasibound States Near the Saddle-Point Ionization Limit of a Rydberg Electron in Crossed Electric and Magnetic Fields

In the combination of crossed electric and magnetic fields and the Coulomb field of the atomic nucleus the spectrum of the Rydberg electron in the vicinity of the Stark saddle-point are investigated at a quantum mechanical level. The results expose a quantum anomaly dissociation: quasibound states near and above the saddle-point ionization limit predicted at the semi-classical level disappear at a quantum mechanical level. PACS numbers: 32.60.+i ∗ Corresponding author Twenty years ago Clark et al. [1] claimed that the combination of crossed electric and magnetic fields and the Coulomb field of the atomic nucleus can lead to the localization of the Rydberg electron in the vicinity of the Stark saddle-point. When the characteristic parameter ω t > 0 in the classical equation of motion the electron motion is periodic in an elliptical orbit. such orbits give rise to electron states which are localized above the saddlepoint and whose spectrum is that of a harmonic oscillator. Divergent hyperbolic trajectories are obtained in the case when ω t < 0. The classical equation of motion includes both type of solutions. The periodic orbits are unstable with respect to small perturbations, thus assume the character of quasibound states. Ref. [1] focused attention on the energy region near the ionization threshold for the first time in literature. The distinctive character of this portion of the spectrum makes it an attractive target of experimental investigation. Their treatment is semi-classical. The determination of the lifetimes of these states and their associated transition moments awaits a full quantum mechanical treatment. In this Letter we investigate the above system at a quantum mechanical level. The results expose a quantum anomaly dissociation: bound states which exist at a semi-classical level may disappear at a quantum mechanical level. For the present example, we clarify that quasibound states near and above the saddle-point ionization limit predicted in Ref. [1] do not exist at a quantum mechanical level. This explains the reason that non of the suggested experiments yet has been realized. A particle trap using static fields must confine the electron about the Stark saddle-point where the net electric force vanishes. We therefore consider the Schrödinger equation in coordinates centered about the saddle point rather than about the atomic nucleus. In this coordinate system the coordinates of the electron are (x1, x2, x3). The electric field E = −Ex̂1. The coordinate of the saddle point is x10 = √ e/E. The constant magnetic field ~ B aligns the x3 axis. We can choose a gauge so that the corresponding vector potential Ai reads (henceforth summation convention is used) Ai = 1 2 ǫijBixj , where ǫij is a 2dimensional antisymmetric unit tensor, ǫ12 = −ǫ21 = 1, ǫ11 = ǫ22 = 0. The electrostatic potential is given by Φ = e/[(x1+x10) +x2+x 2 3] +E(x1+x10). A harmonic approximation of the potential in the region about the saddle point is enough. For small x1, x2 and x3 the electrostatic potential is approximated by Φ = −Vc/e−(e/2x10)(−2x1+x2+x3), where Vc =