Central field approximation, angular momentum and spherical functions

We have already mentioned the central field approximation in Chapter 1 while discussing the self-consistent field method and the zero-order Hamiltonian of many-electron atoms. Let us recall briefly its main idea. The central field approximation means that any given electron in the N-electron atom moves independently in the electrostatic field of the nucleus, which is considered to be stationary, and of the other N – 1 electrons. This field is assumed to be time-averaged over the motion of these N – 1 electrons and, therefore, to be spherically symmetric. Then the wave function of this electron will be described by a formula of the type (1.14). In any such central field, the wave function (1.14) will be an eigenfunction of L2 and Lz, where L is angular momentum of an electron, and Lz its projection. Thus, the angular momentum of the electron is a constant of motion, and the wave function of the type (1.14) is an eigenfunction of the one-electron angular and spin momentum operators L2, Lz and s2, sz with eigenvalues l(l + 1), mL and s(s + 1) = ¾, ms, correspondingly (in units of ħ2).

Thus, in the central field approximation the wave function of the stationary state of an electron in an atom will be the eigenfunction of the operators of total energy, angular and spin momenta squared and one of their projections. These operators will form the full set of commuting operators and the corresponding stationary state of an atomic electron will be characterized by total energy E, quantum numbers of orbital l and spin s momenta as well as by one of their projections.

Atomic spectroscopy continues to be one of the most important branches of contemporary physics, and has a wealth of practical applications. Many domains of modern physics and other fields of science and technology utilize atomic spectroscopy. Modern atomic spectroscopy studies the structure and properties of all atoms of the Periodical Table, and ions of any ionization degree.

Spectra are fundamental characteristics of atoms and ions, the main source of information on their structure and properties. Diagnostics of both laboratory and astrophysical plasma is carried out, as a rule, on the basis of these spectra. Nowadays the possibilities of theoretical spectroscopy are much enlarged thanks to wide usage of powerful computing devices. This enables one to investigate fairly complicated mathematical models of the system under consideration and to obtain in this way results which are in close agreement with experimental measurements. Interest in spectroscopy has increased particularly in connection with the rise of non-atmospheric astrophysics and laser physics. New important applied problems have arisen: diagnostics of thermonuclear plasmas, creation of lasers generating in the X-ray region, identification of solar and stellar spectra, studies of Rydberg, ‘hollow’ atoms, etc. Due to the use of laser-produced plasmas, powerful thermonuclear equipments (Tokamaks), low inductance vacuum sparks, exploding wires, beam-foil spectroscopy, and other advanced ion sources and ion traps (such as EBIS, EBIT, ECR, etc.) there was discovered a new, extremely interesting and original world of very highly ionized atoms, their radiation being, as a rule, in the far ultraviolet and even the X-ray wavelengths region. The spectra due to the transitions between highly excited (Rydberg) levels are studied intensively.

Overview

Resonant vapours as optically nonlinear media

Mutual interactions

So far, we have considered the effect of the laser on the atomic medium separately from the measurement of microscopic dynamics by the polarisation selective detection of transmitted light. In this approximation, the pump laser drives the atoms without suffering significant attenuation or polarisation changes. Conversely, the probe beam, which monitors the optical properties of the atomic medium, changes the microscopic state of the atoms only infinitesimally. This approach guarantees, e.g., that the response of the medium to the probe beam is linear. As stressed before, this assumption of one-sided interactions is always an approximation, since the conservation of energy and angular momentum make it impossible to change either of the two partial systems without compensating changes in the other part.

Atoms

Historical

Early models: atoms as building blocks

The term “atom” was coined by the Greek philosopher Democritus of Abdera (460–370 B.C.), who tried to reconcile change with eternal existence. His solution to this dilemma was that matter was not indefinitely divisible, but consisted of structureless building blocks that he called atoms. According to Democritus and other proponents of this idea, the diverse aspects of matter, as we know it, are a result of different arrangements of the same building blocks in empty space (Melsen 1957; Simonyi 1990). The most important opponent of this theory was Aristotle (384–322 B.C.), and his great influence is probably the main reason that the atomic hypothesis was not widely accepted, but lay dormant for two thousand years. It reappeared only in the eighteenth century, when the emerging experimental science found convincing evidence that matter does indeed consist of elementary building blocks.

Isotropic atoms

The Lorentz–Lorenz model

Outline

The interaction of electromagnetic radiation with matter, in particular light, has inspired philosophers of nature for many centuries and has led to heated debates like the one between Newton and Goethe. The original interest was with the difference in absorption of various materials as a function of the optical wavelength and with the refraction of light, i.e., its dispersion.

The first theoretical analysis that connected these macroscopic effects to microscopic properties of the material was the Lorentz–Lorenz theory of dispersion (Lorentz 1880; Lorenz 1881), which was put forward shortly after the Maxwell equations and published in the year after Maxwell's death. (See Figure 6.1.) This theory models the material as a collection of dipoles, driven by the electromagnetic wave. We give here a brief summary of the theory, since its physical content is still the foundation of today's description although the mathematical formalism has changed significantly.

Light-induced forces

That light exerts mechanical forces on massive particles like atoms may appear surprising. To motivate the existence of such an effect two different approaches are possible. The first approach considers the light as a collection of photons that carry, apart from energy and angular momentum, linear momentum as well. Photons interacting with atoms can therefore change the momentum of the atoms. The second approach considers light as a wave, i.e., an inhomogeneous electromagnetic field interacting with the atomic dipole moment. Both approaches provide a possible description for the numerous phenomena that can occur in this context, but in many situations, one of them turns out to be more intuitive or more useful for calculations than the other.

Overview

Raman processes

Introduction

In the preceding chapters, we mentioned several types of Raman processes. Their common feature is a resonant change of the energy of the photons that interact with the material system. The energy of the scattered photons may be lower (Stokes process) or higher (anti-Stokes) than that of the incident photons. The energy difference is transferred to the material system, where it must match an energy level separation. The photon energy itself, however, does not have to match exactly a transition frequency of the medium. This is commonly expressed by the statement that the Raman scattering proceeds through a virtual state, represented by the dashed line in Figure 7.1. The presence of a real state of the atom, indicated by the full line, nevertheless increases the coupling efficiency, as discussed in Chapter 3.

The earlier sections on three-level effects and optical anisotropy dealt with the mathematical formalism of Raman processes, using generic level systems to describe them.

Experimental arrangement

General considerations

Laser-induced dynamics

This chapter surveys the coherent evolution of coherences between angular momentum sublevels. Optical pumping excites this microscopic order, and it evolves under the influence of external magnetic fields and the laser radiation. The primary goal of this section is to show how the mathematical models developed in the preceding sections apply to real physical systems. We discuss how the observed signals arise and by which parameters the experimenter can control the dynamics of these systems.

Principle and overview

Phenomenology

Optical pumping (Happer 1972) is one of the earliest examples wherein optical radiation qualitatively modifies the properties of a material system. In its original implementation, it corresponds to a selective population of specific angular momentum states, starting from thermal equilibrium.

In the idealised process depicted in Figure 5.1, the light brings the atomic system from the initially disordered state, in which the populations of degenerate levels are equal, into an ordered state where the internal state of all atoms is the same. If we consider only the material system, it appears as if the evolution from the initially disordered state into an ordered state, where the population of one level is higher than that of another level, violated the second law of thermodynamics. This process does not proceed spontaneously, however. It is the interaction with polarised light that drives the system and increased disorder in the radiation field compensates for the increase in the population difference in the material system (Enk and Nienhuis 1992).

Rotational symmetry

Motivation

The number of energy levels that contribute to the dynamics of a quantum mechanical system is a direct measure of the number of degrees of freedom required for a full description of the system. The systems in which we are interested always include electric dipole moments – the degrees of freedom that couple to the radiation field. The second most important contribution is the magnetic dipole moment associated with the atomic angular momentum. Electric and magnetic dipole moments are those degrees of freedom that couple to external fields. Other degrees of freedom do not couple strongly to external fields but they may still modify the behaviour of the system and its optical properties.

Fundamentals

Motivation and principle

Motivation

The preceding chapter showed how light drives the internal dynamics of resonant atomic media and how the measurement of optical anisotropies allows us to monitor these dynamics. The experiments discussed in the preceding chapter, however, can provide only limited information about the system. Most physical systems have more degrees of freedom than we can observe by measurements on transmitted light. As another limitation, we have primarily considered atoms that evolve under their internal Hamiltonian, only weakly perturbed by the probe laser beam. The example of light-induced spin nutation showed that the dynamics of optically pumped atoms differ significantly from those of a free atom. Although it is possible to observe spin nutation for systems with more than two ground-state sublevels, such an experiment suffers from the damping that accompanies optical pumping. The damping drives the system rapidly to an equilibrium, too fast for detailed dynamical observations.

Light interacting with material substances is one of the prerequisites for life on our planet. More recently, it has become important for many technological applications, from CD players and optical communication to gravitational-wave astronomy. Physicists have therefore always tried to improve their understanding of the observed effects. The ultimate goal of such a development is always a microscopic description of the relevant processes. For a long time, this description was identical with a perturbation analysis of the material system in the external fields. More than a hundred years ago, such a microscopic theory was developed in terms of oscillating dipoles. After the development of quantum mechanics, these dipoles were replaced by quantum mechanical two-level systems, and this is still the most frequently used description.

However, the physical situation has changed qualitatively in the last decades. The development of intense, narrowband or pulsed lasers as tunable light sources has provided not only a new tool that allows much more detailed investigation, but also the observation of qualitatively new phenomena. These effects can no longer be analysed in the form of a perturbation expansion. One consequence is that the actual number of quantum mechanical states involved in the interaction becomes relevant. It is therefore not surprising that many newly discovered effects are associated with the details of the level structure of the medium used in the experiment. Two popular examples are the discovery of sub-Doppler laser cooling and the development of magnetooptical traps, which rely on the presence of angular momentum substates.