Review of Classical Analytical Results for the Motion of a Rydberg Electron around a Polar Molecule under Magnetic or Electric Fields of Arbitrary Strengths in Axially Symmetric Configurations

We review classical studies of the oscillatory-precessional motion of an electron in the field of an electric dipole (the latter representing the polar molecule) with or without external magnetic or electric fields. The focus is on the most recent studies. In one study (at zero external field), it was shown that, generally, the oscillations being in the meridional direction and the precession being along parallels of latitude can take place on the same time scale—contrary to the statement from the previous literature. In another study, it was shown that a magnetic field enables new ranges of the bound oscillatory-precessional motion of the Rydberg electron and that in one of the new ranges, the period of the θ-oscillations has the non-monotonic dependence on primary parameter of the system. This is a counterintuitive result. In yet another study, it was shown that under the electric field there are two equilibrium circular states of a positive energy and one equilibrium state of a negative energy. The existence of the equilibrium state of the negative energy is a counterintuitive result since at the absence of the field, the bound state was possible only for the zero energy. Thus, it is a counterintuitive result that in this case the electric field can play the role of a stabilizing factor.

Circular Rydberg states of helium atoms or helium-like ions in a high-frequency laser field

In the literature, there were studies of Rydberg states of hydrogenic atoms/ions in a high-frequency laser field. It was shown that the motion of the Rydberg electron is analogous to the motion of a satellite around an oblate planet (for a linearly polarized laser field) or around a (fictitious) prolate planet (for a circularly polarized laser field): it exhibits two kinds of precession – one of them is the precession within the orbital plane and another one is the precession of the orbital plane. In this study, we study a helium atom or a helium-like ion with one of the two electrons in a Rydberg state, the system being under a high-frequency laser field. For obtaining analytical results, we use the generalized method of the effective potentials. We find two primary effects of the high-frequency laser field on circular Rydberg states. The first effect is the precession of the orbital plane of the Rydberg electron. We calculate analytically the precession frequency and show that it differs from the case of a hydrogenic atom/ion. In the radiation spectrum, this precession would manifest as satellites separated from the spectral line at the Kepler frequency by multiples of the precession frequency. The second effect is a shift of the energy of the Rydberg electron, also calculated analytically. We find that the absolute value of the shift increases monotonically as the unperturbed binding energy of the Rydberg electron increases. We also find that the shift has a nonmonotonic dependence on the nuclear charge Z: as Z increases, the absolute value of the shift first increases, then reaches a maximum, and then decreases. The nonmonotonic dependence of the laser field-caused energy shift on the nuclear charge is a counterintuitive result.

Classical analytical solution for Rydberg states of muonic–electronic helium and helium-like ions

In our previous papers (Can. J. Phys. 91 (2013) 715; 92 (2014) 1405), we studied Rydberg states of systems consisting of a nucleus of charge Z, a muon, and an electron, both the muon and electron being in circular states. The studies of such quasimolecules μZe were motivated by numerous applications of muonic atoms and molecules, where one of the electrons is substituted by the heavier lepton μ–. We demonstrated that the muonic motion can represent a rapid subsystem, while the electronic motion can represent a slow subsystem. We showed that the spectral lines emitted by the muon in such systems experience a red shift compared to the corresponding spectral lines that would have been emitted by the muon in a muonic hydrogenic atom/ion. In the present paper, we also consider Rydberg states of quasimolecules μZe with Z > 1 (i.e., Rydberg states of muonic–electronic helium and helium-like ions). However, our current approach has important distinctions from our previous papers. The systems considered here are truly stable and the electron orbit is generally elliptical (although the relatively small influence of the electron on the muon is neglected). In our previous papers, the influence of the electron on the muon was taken into account; however, in the rotating frame used in our previous papers, the motion of the muon was only metastable (not truly stable), and furthermore, only circular orbits of the electron were considered in our previous paper. In the present paper, we show that the effective potential energy of the Rydberg electron is mathematically equivalent to the potential energy of a satellite moving around an oblate planet. Based on this, we demonstrate that the unperturbed orbital plane of the Rydberg electron undergoes simultaneously two different precessions: precession within the orbital plane and precession of the orbital plane around the axis of the muonic circular orbit. We provide analytical expressions for the frequencies of both precessions. The shape of the elliptical orbit of the Rydberg electron is not affected by the perturbation, which is the manifestation of the (approximate) conservation of the square of the angular momentum of the Rydberg electron. This means that the above physical systems have a higher than geometrical symmetry (also known as a hidden symmetry) which is a counterintuitive result of general physical interest. We note that the above problem of the motion of the Rydberg electron in muonic–electronic helium atoms or helium-like ions is mathematically equivalent to another problem from atomic physics: a hydrogen Rydberg atom in a linearly-polarized electric field of a high-frequency laser radiation.

Oscillatory-Precessional Motion of a Rydberg Electron Around a Polar Molecule

We provide a detailed classical description of the oscillatory-precessional motion of an electron in the field of an electric dipole. Specifically, we demonstrate that in the general case of the oscillatory-precessional motion of the electron (the oscillations being in the meridional direction (θ-direction) and the precession being along parallels of latitude (φ-direction)), both the θ-oscillations and the φ-precessions can actually occur on the same time scale—contrary to the statement from the work by another author. We obtain the dependence of φ on θ, the time evolution of the dynamical variable θ, the period Tθ of the θ-oscillations, and the change of the angular variable φ during one half-period of the θ-motion—all in the forms of one-fold integrals in the general case and illustrated it pictorially. We also produce the corresponding explicit analytical expressions for relatively small values of the projection pφ of the angular momentum on the axis of the electric dipole. We also derive a general condition for this conditionally-periodic motion to become periodic (the trajectory of the electron would become a closed curve) and then provide examples of the values of pφ for this to happen. Besides, for the particular case of pφ = 0 we produce an explicit analytical result for the dependence of the time t on θ. For the opposite particular case, where pφ is equal to its maximum possible value (consistent with the bound motion), we derive an explicit analytical result for the period of the revolution of the electron along the parallel of latitude.