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American Journal of Physics

Simultaneous Stark and Zeeman effects in atoms with hyperfine structure

Wilton L. Virgo

Published Nov 21, 2013• 8 citations
Publisher

Abstract

A quantum model for calculating the combined Stark and Zeeman effects of simultaneously applied electric and magnetic fields is presented. Our focus here is on atoms with hyperfine structure, such as Cesium. Matrix representations of the Stark, Zeeman, and hyperfine interaction operators are constructed using angular momentum theory and spherical tensor algebra. Matrix elements are evaluated in order to determine the energy-level dependence on the applied fields and reveal intriguing state dynamics in both parallel and orthogonal electric and magnetic fields. The fundamental physics is relevant for an advanced undergraduate or graduate quantum mechanics course.

DOI
10.1119/1.4823999

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A quantum model for calculating the combined Stark and Zeeman effects of simultaneously applied electric and magnetic fields is presented. Our focus here is on atoms with hyperfine structure, such as Cesium. Matrix representations of the Stark, Zeeman, and hyperfine interaction operators are constructed using angular momentum theory and spherical tensor algebra. Matrix elements are evaluated in order to determine the energy-level dependence on the applied fields and reveal intriguing state dynamics in both parallel and orthogonal electric and magnetic fields. The fundamental physics is relevant for an advanced undergraduate or graduate quantum mechanics course.
The well-known Zeeman and Stark effects can lead to unusual atomic state dynamics when both magnetic and electric fields are present. In this paper we analyze quantum mechanical and classical models of the time evolution of the angular momentum of an atom in the presence of weak, orthogonal electric and magnetic fields. The anisotropic electric polarizability of the atom plays a crucial role in the dynamics. The classical model leads to nonlinear evolution equations, and we investigate how quantum mechanics ‘‘avoids’’ the nonlinearity. Finally, we note that the classical equations of motion are identical to those used to describe the time evolution of Stokes vectors for polarized light propagating in an optically nonlinear medium. The treatment is appropriate for an undergraduate course in quantum mechanics.
The problem of Rydberg atomic spectra in crossed electric F and magnetic B fields (combined Stark–Zeeman effects), as well as an oscillating electric field are under consideration. The main problem is the great array of radiative transitions between the Rydberg atomic states. The different versions of semiclassical approaches are applied for the solution of the problem. New approximate selection rules are established to simplify the matrix elements of radiative transitions, making it possible to reduce the complex expressions of Gordon’s hypergeometric series to the trivial trigonometric functions. Simple analytical formulas for the dipole matrix elements of a Rydberg hydrogen-like atom in external crossed electric and magnetic fields for Hnα are obtained. The specific calculations are presented for n = 10 to 9 transition both for parallel and perpendicular orientations of F–B fields. The result of Blokhintsev for a single Stark component in oscillating field is generalized for the array of radiative transitions between Rydberg atomic states. Such spectra with large values of modulation index are compared with static ones.
A Rydberg atom has a highly excited valence electron which is weakly bound and far from the nucleus. These atoms have exaggerated properties that make them attractive candidates for quantum computation and studies of fundamental quantum mechanics. The discrete energy levels of Rydberg atoms are shifted in the presence of an electric field by the Stark effect and are similarly shifted due to a magnetic field by the Zeeman effect. These effects couple the energy levels together, creating avoiding crossings. At these avoided crossings, an electron in one energy level can jump to the other. Our goal is to be able to use these avoided crossings to put the electron in a superposition state of both energy levels. In order to achieve this we created new software that enables us to calculate the energy levels of an electron in both a magnetic and an electric field. We present energy level maps visualizing the results of the Stark and Zeeman effects.
We investigate single-particle energy spectra of the hydroxyl free radical (OH) in the lowest electronic and rovibrational level under combined static electric and magnetic fields, as an example of heteronuclear polar diatomic molecules. In addition to the fine-structure interactions, the hyperfine interactions and centrifugal distortion effects are taken into account to yield the zero-field spectrum of the lowest 2 Π 3 / 2 manifold to an accuracy of less than 2 kHz. We also examine level crossings and repulsions in the hyperfine structure induced by applied electric and magnetic fields. Compared to previous work, we found more than 10% reduction of the magnetic fields at level repulsions in the Zeeman spectrum subjected to a perpendicular electric field. In addition, we find new level repulsions, which we call Stark-induced hyperfine level repulsions, that require both an electric field and hyperfine structure. It is important to take into account hyperfine structure when we investigate physics of OH molecules at micro-Kelvin temperatures and below.
A comprehensive set of lecture and study notes covering fundamental atomic and molecular physics. The text systematically works through the breakdown of the classical Rutherford atom and the Bohr-Sommerfeld model, followed by a full 3D Schrödinger treatment of the hydrogen atom using spherical harmonics and Laguerre polynomials. It details relativistic corrections, spin-orbit coupling, and fine structure via Clebsch-Gordan addition and Kramers' recursion relations. The notes further explore magnetic field interactions (normal and anomalous Zeeman effect, Paschen-Back effect), time-dependent perturbation theory for light-matter transitions, electric dipole selection rules, the 21-cm hyperfine splitting of hydrogen, and two-electron systems (Helium, exchange integrals, and Slater determinants). Includes extensive reference tables and angular momentum formulas.