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CRC Press eBooks

Interaction of Electrons and Photons

Toshiaki Suhara

Published Mar 16, 2004• 0 citations
Publisher

Abstract

This chapter provides the basis for the discussion in the following chapters by summarizing the fundamental concepts and the quantum theory concerning the interaction between electrons and photons in a form that is convenient for theoretical analysis of semiconductor lasers [1-9]. First, quantization of electromagnetic fields of optical waves is outlined, and the concept of a photon is clarified. Quantum theory expressions for coherent states are also given. Then the quantum theory of electronphoton interactions and the general characteristics of optical transitions are explained. Fundamental mathematical expressions for absorption, spontaneous emission, and stimulated emission of photons are deduced, and the possibility of optical wave amplification in population-inverted states is shown.

DOI
10.1201/9780203020470-6

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This chapter provides the basis for the discussion in the following chapters by summarizing the fundamental concepts and the quantum theory concerning the interaction between electrons and photons in a form that is convenient for theoretical analysis of semiconductor lasers [1-9]. First, quantization of electromagnetic fields of optical waves is outlined, and the concept of a photon is clarified. Quantum theory expressions for coherent states are also given. Then the quantum theory of electronphoton interactions and the general characteristics of optical transitions are explained. Fundamental mathematical expressions for absorption, spontaneous emission, and stimulated emission of photons are deduced, and the possibility of optical wave amplification in population-inverted states is shown.
This chapter considers electron–photon interaction, i.e., light–matter coupling. It introduces the theoretical treatment of electron–photon interaction for different and mostly strongly simplified cases. In resonator structures, calculation of the appropriate photon modes is required to allow access to the strength of electron–photon coupling. The general guideline is that light may be treated as a wave when propagating in vacuum, yet it has to be quantized in the case of the interaction with matter. In any ensemble of atoms or molecules in interaction with an electromagnetic field, there are three processes occurring: spontaneous emission, absorption, and stimulated emission. In order to get a flavor of a many-body treatment of electron–photon interaction, the derivation of the semiconductor Bloch equations will be discussed briefly. The strength of the light field at the location of the electron determines the interaction strength between light and matter.
Abstract The chapter first develops the wave equation in a rectangular cavity and gives the definition and number of density-of-modes within it. Prescription for converting the classical expressions of the EM field to quantized forms in terms of creation and annihilation operators is then outlined. After introducing the concept of photons, the interactions of an atomic system with photons are described in terms of the interaction Hamiltonian. The time-dependent perturbation theory, using the Fermi Golden Rule and the dipole approximation is employed to obtain expressions for the absorption and emission rates. The theory is then applied to the case of a two-level atomic system to obtain the expression for Einstein’s A-coefficient. Broadening of emission line is pointed out and the oscillator strength is defined.
Abstract The quantum mechanical description of the interaction of light with atoms treats absorption, gain, and recombination as upward and downward induced transitions of electrons between two states by perturbation due to the optical electric field. This provides the basis for a quantitative theory of diode lasers. The chapter begins with Einstein’s description of absorption, stimulated emission, and spontaneous emission, setting the scene for calculation of the transition probability by solving the time-dependent Schrödinger equation for a coherent superposition of the wavefunctions of two states. The resulting absorption spectrum has a Lorentzian lineshape due to dephasing of these wavefunctions. Expressions for the optical absorption cross section and Einstein coefficients are derived in terms of the dipole matrix element, and the dipole is illustrated using a hypothetical one-dimensional atom. Fermi’s Golden Rule for transitions to a continuum of states (as in quantum wells) is introduced in terms of the momentum matrix element.
Chapter 6 begins by transforming the optical wave itself into a quantized form, and the full implications of the wave‐particle duality of Chapter 2 become clear. The electric field may be pictured as an infinite ladder of discrete states and this structure allows optical interactions to be described photon by photon. The origin of spontaneous emission emerges from this treatment as a transition process that is stimulated by electric field fluctuations of the vacuum, and Weisskopf–Wigner theory explains why it is irreversible. The subject of coherent states is introduced and intriguing possibilities outlined regarding the noise properties of components of these states that form different quadratures in terms of the optical phase. Next, statistical analysis of light fields is considered to serve as a basis for more quantitative comparisons of fields and their noise properties. Methods of calculating the first and second degrees of coherence are presented, and the ranges of values corresponding to “classical” versus “quantum” light sources are discussed. The reduced density matrix is developed to justify analysis that focuses on only certain aspects of optical dynamics while ignoring others, as an important tool for simplifying complex problems. Finally, in view of its historical importance, the problem of calculating the fluorescence spectrum of resonantly excited two‐level atoms is considered using both a method of moments and dressed atom theory.
All optical oscillators and amplifiers can be analyzed as an electromagnetic structure, such as a cavity or a transmission line, which contains an amplifying medium. The operating characteristics of lasers are governed by both the electromagnetic properties of the structures and the properties of the amplifying medium. Stimulated emission and absorption of radiation from energy states are the physical basis of amplification in all laser materials. In order to study lasers, it is necessary to understand the energy levels of the amplifying medium and the stimulated transitions involving them. Quantum mechanical analysis of the energy states and the stimulated emission and absorption is presented in most books on lasers. However, quantum mechanical analyses cover only the analysis of individual atoms. We need to relate the effects of the quantum mechanical interactions to the macroscopic properties of the materials so that properties of the lasers may be analyzed. Therefore, the macroscopic susceptibility of materials related to stimulated emission and absorption is the focus of discussion in this chapter. It is not the purpose of this book to teach quantum mechanics. There are already many excellent books on this topic. The readers are assumed to have a fundamental knowledge of quantum mechanics. However, it is necessary first to review some of the major steps in order to understand precisely the notation used and the meaning of the results. This is presented in Section 5.1.
In this chapter, we discuss a purely quantum approach to the interaction between an atom and the electromagnetic field. In this treatment, the atom and electromagnetic field form a single quantum system, whose evolution is handled globally within a unified formalism. This will thus have the merit of being fully consistent from the theoretical standpoint. But the main advantage in an entirely quantum approach is that it can treat the full range of matter–radiation interaction phenomena. In particular, it provides a rigorous description of the spontaneous emission of light by an excited atom , something that falls outside the scope of the semi-classical framework applied in Chapters 2 and 3, where the lifetime of an excited atomic state had to be fed in phenomenologically. It can describe other phenomena of the same type, such as parametric fluorescence by a nonlinear crystal subjected to pumping radiation (see Chapter 7), which underlies many recent developments in quantum optics. The fully quantum approach also has the merit of allowing a simple interpretation in terms of photons for the various matter–radiation interaction processes, such as absorption, stimulated emission, scattering, and also the basic processes of nonlinear optics. Indeed, it provides a unified framework for both stimulated and spontaneous processes. Finally, it can be used to tackle completely new situations where matter and radiation interact, which lie outside the scope of any semi-classical description, such as cavity quantum electrodynamics or the production of single-photon or entangled-photons states .