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Cambridge University Press eBooks

The laser–atom interaction

Bruce W. Shore

Published Sep 29, 2011• 0 citations
Publisher

Abstract

As light passes an atom, it exerts forces on the charges, electrons, and nuclei that alter the atomic structure. These may be slight distortions (perturbations) of the electron cloud or they may be more severe, as described in subsequent sections of this monograph. We wish to determine those changes, given the radiation field, or to devise a radiation field that will produce specified changes. The changes to the atomic structure also affect the radiation that subsequently passes the atom. To describe those effects we must consider wave equations for radiation in the presence of altered atomic structure. More generally we must find self-consistent equations for the atoms and the field together, as discussed in Chaps. 21 and 22. Here we consider mathematical descriptions of the influence of coherent radiation on individual atoms, molecules, or other single quantum systems. Individual atoms Traditional sources of emission and absorption spectra, though revealing energy states of the constituent atoms and molecules, are macroscopic samples. One observes averaged characteristics of many individual particles, see Chap. 16. Quantum theory offers the basic formalism for dealing with individual atoms exposed to controlled radiation fields. Several experimental techniques provide acceptable approximations to this ideal. The following paragraphs note some of these examples. Vapors Neutral atoms or molecules in a vapor move freely along straight-line paths, interrupted by brief collisions that redirect the two collision partners. When the kinetic energies of the two partners are small, there can be no transfer of kinetic energy into internal energy of either particle – the collision is elastic .

DOI
10.1017/cbo9780511675713.005

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Cambridge University Press eBooks•Sep 29, 2011•0 citations
As light passes an atom, it exerts forces on the charges, electrons, and nuclei that alter the atomic structure. These may be slight distortions (perturbations) of the electron cloud or they may be more severe, as described in subsequent sections of this monograph. We wish to determine those changes, given the radiation field, or to devise a radiation field that will produce specified changes. The changes to the atomic structure also affect the radiation that subsequently passes the atom. To describe those effects we must consider wave equations for radiation in the presence of altered atomic structure. More generally we must find self-consistent equations for the atoms and the field together, as discussed in Chaps. 21 and 22. Here we consider mathematical descriptions of the influence of coherent radiation on individual atoms, molecules, or other single quantum systems. Individual atoms Traditional sources of emission and absorption spectra, though revealing energy states of the constituent atoms and molecules, are macroscopic samples. One observes averaged characteristics of many individual particles, see Chap. 16. Quantum theory offers the basic formalism for dealing with individual atoms exposed to controlled radiation fields. Several experimental techniques provide acceptable approximations to this ideal. The following paragraphs note some of these examples. Vapors Neutral atoms or molecules in a vapor move freely along straight-line paths, interrupted by brief collisions that redirect the two collision partners. When the kinetic energies of the two partners are small, there can be no transfer of kinetic energy into internal energy of either particle – the collision is elastic .
The basic phenomena of interaction between atoms and e.m. fields can be discussed in some detail without the need of a detailed model of the atom. Absorption and stimulated emission of radiation can be explained satisfactorily by the theory, where a classical electromagnetic wave provides a time-dependent perturbation to the energy of the atomic electrons. Laser radiation appears to a very good approximation to be a classical travelling wave of well-defined amplitude and phase. The radiation produced by a continuous laser is a good approximation to a coherent state. For conventional sources of radiation, the interaction energy is small compared with the energy an electron possesses by virtue of the attraction to the atomic nucleus, so perturbation theory produces quite acceptable results. The quantum mechanical analogy to the distortion of the classical charge cloud is given by the writing down of a general state function for the system, Ψ.
The quantum world within an atom or molecule that once attracted explorations only by academic physicists now provides fertile sustenance for chemists seeking control of chemical reactions and for engineers developing ever smaller electronic devices or tools for processing information with greater security. Whereas the first pioneers could only discover the most elementary properties – the discrete energy levels that characterize the internal structures of atoms and the radiative transitions that link these structures to our external world – it is now possible to alter that structure at will, albeit briefly. Objective This monograph presents the physical principles that describe such deliberately crafted changes, namely how single atoms or molecules (or other simple quantum systems) are affected by coherent interactions, primarily laser light – a subject that has been regarded first as a part of quantum electronics and then quantum optics but is most generally described as coherent atomic excitation [Sho90]. This physics has relevance to such basic concerns as the detection and quantitative analysis of trace amounts of chemicals, the catalysis or control of chemical reactions, the alignment of molecules, and the processing of quantum information. The physics necessarily involves elementary quantum mechanics, but it has many associations to the classical dynamics that governs macroscopic objects – waves and particles. The mathematics that quantifies the changes is that of differential equations, specifically coupled ordinary differential equations (ODEs), whose parameters incorporate the controls of experimenters and whose solutions, appropriately interpreted, quantify the resulting changes.
From prehistoric times has come recognition that sunlight and firelight provide warmth, and that such illumination casts shadows. Expressed in more contemporary terms one would say that light travels in rays, and that this radiation has the potential to provide heat energy to absorbing material. From the time of Newton it has been known that light sources emit radiation comprising a distribution of colors. During the nineteenth century it was recognized that radiation had characteristics of transverse waves (with wavelength associated to color) but until the late twentieth century, when lasers became laboratory tools, it was hardly necessary to delve into the equations of electromagnetic theory to treat such experiments as were then possible; interest lay primarily with thermodynamic considerations of energy flow or with measurements of the dark or bright lines seen in the distribution of light that, after passing through a slit, was dispersed by a prism or grating into constituent colors. Although laser light sources are essential for the types of atomic excitation considered in this monograph, the legacy from thermal radiation still influences many interpretations of the interaction between radiation and matter, and it is therefore useful to summarize some of those concepts. The mathematics needed for describing laser radiation, or polarized light in general, has much in common with the mathematics of quantum theory discussed starting in Sec. 3.5 and specialized to two-level atoms in Sec. 5.6. In both cases one deals with two complex-valued functions – independent electric field amplitudes or probability amplitudes – whose absolute squares are measurable.
Coherent manipulations of atoms using laser light The internal structure of a particle - an atom or other quantum system in which the excitation energies are discrete - undergoes change when exposed to pulses of near-resonant laser light. This tutorial review presents basic concepts of quantum states, of laser radiation and of the Hilbert-space statevector that provides the theoretical portrait of probability amplitudes - the tools for quantifying quantum properties not only of individual atoms and molecules but also of artificial atoms and other quantum systems. It discusses the equations of motion that describe the laser-induced changes (coherent excitation), and gives examples of laser-pulse effects, with particular emphasis on two-state and three-state adiabatic time evolution within the rotating-wave approximation. It provides pictorial descriptions of excitation based on the Bloch equations that allow visualization of two-state excitation as motion of a three-dimensional vector (the Bloch vector). Other visualization techniques allow portrayal of more elaborate systems, particularly the Hilbert-space motion of adiabatic states subject to various pulse sequences. Various more general multilevel systems receive treatment that includes degeneracies, chains and loop linkages. The concluding sections discuss techniques for creating arbitrary pre-assigned quantum states, for manipulating them into alternative coherent superpositions and for analyzing an unknown superposition. Appendices review some basic mathematical concepts and provide further details of the theoretical formalism, including photons, pulse propagation, statistical averages, analytic solutions to the equations of motion, exact solutions of periodic Hamiltonians, and population-trapping "dark" states.
The processes of absorption and emission of radiation by a system of two interacting atoms is investigated. Separation between the atoms is assumed to be much larger than the atomic diameter, and interactions considered are those between the individual atoms and the radiation field and the electrostatic interaction between the atoms. Calculations include those for the time-dependent probability amplitude of the intermediate and final states as well as that of the emission line shape. The line shape of the emitted radiation consists of two peaks whose amplitude, separation, and half-width are functions of both the internuclear separation R of the atoms and the energy difference of the excited states of the two atoms. For a large energy difference, the peaks are distinct; for small energy differences, the peaks are distinct at smaller separations, but merge toward the line shape characteristic of a single, isolated atom as R increases.
The major portion of this monograph discusses how given pulses of laser radiation affect individual atoms. This chapter inverts that relationship, describing how the matter alters the fields. Those field changes provide quantitative measures of the quantum-state changes produced in the atoms. Their description therefore is an adjunct to Chap. 19. Basically the incident radiation produces excitation which, in turn, alters the various multipole moments of the atoms. When viewed as a macroscopic sample of matter, such changes alter the electric polarization field P and the magnetization field M of the matter through which the radiation must pass. The Maxwell equations, see App. C.1, provide the needed description of how the P and M fields alter the electric and magnetic fields E and B . The combination of the Maxwell equations for the fields and the Schrödinger, Bloch, or Liouville equations for the atoms provide the tools needed to construct a self-consistent description of radiation passing through matter – atoms responding to a pulsed field and traveling waves being modified by the resulting atomic changes [All87; Ale92; Muk99; Die06; Vit01a]. The present chapter, drawing on [Sho90, Chap. 12] and App. C, discusses this theory. Incoherent radiation passing through matter typically undergoes exponential attenuation in accord with eqn. (6.9). A measure of the incremental change of intensity in distance L by absorption coefficient κ is the optical depth κ L . This parameter appears in rateequation treatments of incoherent light attenuation.