Chapter 6 transforms the optical wave into a quantized form. The electric field is pictured as an infinite ladder of discrete states: this structure allows optical interactions to be described photon by photon. The origin of spontaneous emission emerges as a transition process that is stimulated by electric field fluctuations of the vacuum; Weisskopf–Wigner theory explains why it is irreversible. Squeezed states are introduced and qualitatively unique properties are identified. Statistical analysis of light fields serves 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. 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 a subset of the components or dynamics of a complex system, ignoring aspects that are not of primary interest. 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.
Chapter 6 transforms the optical wave into a quantized form. The electric field is pictured as an infinite ladder of discrete states: this structure allows optical interactions to be described photon by photon. The origin of spontaneous emission emerges as a transition process that is stimulated by electric field fluctuations of the vacuum; Weisskopf–Wigner theory explains why it is irreversible. Squeezed states are introduced and qualitatively unique properties are identified. Statistical analysis of light fields serves 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. 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 a subset of the components or dynamics of a complex system, ignoring aspects that are not of primary interest. 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.
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.
Introduction . The second part of the book is dedicated to the dressed atom, and it begins with this chapter, which deals mainly with the quantum-dynamical description of an atom dressed by a real electro-magnetic field. Here the emphasis is on the adjective ‘real’, by which we mean that the field is in an excited state populated by real photons and not just by the zero-point photon background. Due to the coupling discussed in the first part of the book, atom-photon correlations are established which admix, shift and split the levels of the system atom plus radiation field. The admixed and correlated states are called dressed-atom states. In Section 5.2 we obtain the Hamiltonian for an atom in a cavity with perfectly reflecting walls. The cavity selects a discrete set of field modes, and this leads us naturally to consider the simplest possible nontrivial atom-field system: the Jaynes-Cummings model describing a two-level atom coupled to a single-mode system. In Section 5.3 we develop the theory, based on a unitary transformation, to dress a two-level atom by a mode of the cavity populated by real photons. A necessary preliminary for dealing with more complicated atom-field models is the theory of spontaneous emission in free space, which is discussed in Section 5.4 in a Wigner-Weisskopf framework.
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.
In Chapter 3 the interaction Hamiltonian which determines the way light interacts with matter is derived. Simple perturbative analysis is applied to see if basic dynamics of atoms can be explained. The partial successes of perturbation theory are compared with predictions of an “exact” method of calculating the occupation probabilities of various atomic states. Even the “exact” method is shown to fail, establishing the need for improved approaches that yield correct results in later chapters. Some elementary results for the transition rates, including Fermi's golden rule are covered. Then the density matrix is introduced as a tool for describing not only the populations of atomic energy levels but coherence that can be created and lost during the dynamic evolution of atoms in time. A vector model based on the Bloch vector is presented as a useful way of picturing coherent atom–field interactions in an optical “spin” space, particularly multiple‐pulse interactions. Mechanisms are described that cause line‐broadening in optical spectroscopy, such as the Doppler effect. The chapter closes by showing that it is usually possible to reduce the complexity of real atoms on an experimental basis to facilitate direct comparisons with theoretical calculations based on simple models with only two or three energy levels.
Radiation can change the energy state of matter via three optical processes: spontaneous emission, absorption and stimulated emission. In spontaneous emission, an excited atomic or molecular state decays to a lower energy state with the emission of a photon of energy equal to the difference in the quantum state energies. Photon absorption is the reverse process, with a photon causing excitation from a lower to an excited quantum state. Stimulated emission was first proposed by Einstein in a paper published in 1917. He attempted to balance spontaneous emission and photon absorption, but could not get the correct balance without the postulated process of stimulated emission (see Section 4.2). With stimulated emission, an excited quantum state is stimulated to decay to a lower quantum state by an impinging photon. The existing photon survives with another identical photon being created (so that energy is conserved). The rates at which radiative transitions occur can be calculated using quantum mechanics. The time-dependent Schrodinger equation is used with the system wavefunction comprising time-varying linear combinations of the two wavefunctions for the lower and upper quantum states. In semi-classical treatments, an atom is treated quantum mechanically with the radiation field treated classically. There is a perturbation in energy associated with, say, the electric field of the radiation interacting with the electric dipole of the atom, though other interactions can be significant if the electric dipole interaction energy is small. In absorption, for example, a differential equation representing the change in ‘weighting’ of the wavefunction from the lower to upper state is obtained. A ‘time constant’ for the speed of change represents the rate of the radiative process. We examine the quantum mechanical nature of radiation absorption in Section 10.1 before discussing (Section 10.3) another important parameter arising with radiative transitions between discrete bound quantum states: the lineshape function. Quantum Theory of the Atom–Radiation Interaction An expression for the Einstein B-coefficient for photo-absorption is derived using quantum mechanics in this section. Einstein determined the detailed balance relationship between the rates of spontaneous emission, photo-absorption and stimulated emission when light interacts with an atom by considering an atomic system of two energy levels which are in equilibrium with a black-body radiation field. We use a semi-classical treatment where the atom is treated quantum mechanically and the radiation field is treated as a classical oscillating electric field.
Introduction . The purpose of this chapter is to present the quantum theory of the electromagnetic field in the absence of charges and currents. Thus the classical field discussed in the previous chapter is subjected to canonical quantization in Section 2.1, where creation and annihilation operators for plane-wave and spherical wave modes, as well as their commutation relations, are derived along with various field-field commutators related to field propagators. In the next section we introduce the concept of the photon as an elementary excitation of the electromagnetic field. The attention is focused on the ground state of the quantized electromagnetic field in the absence of sources, which is the photon vacuum. The amplitude fluctuations, or zero-point fluctuations of this vacuum, are evaluated. Excited states of the field are examined in the next sections. In particular, Section 2.3 is concerned with number states and coherent states, the latter being obtained by a Glauber transformation of the vacuum, and with their statistical properties. Squeezed states of the field are introduced in Section 2.4 by a unitary transformation leading from the normal to the squeezed vacuum, whose statistical properties are compared with those of a coherent state. Section 2.5 is devoted to a brief discussion of thermal states. The final section of this chapter is dedicated to a discussion of the nonlocalizability of the photon.