After the introduction of the two-level model, we add a third level to the quantum mechanical system to discuss some of the effects that cannot be described with only two states. The main feature is the exchange of order between states and transitions, which may proceed through populations or coherences. We concentrate our attention on the latter possibility. Phenomenological introduction Model atoms In the preceding section, we discussed the interaction between light and two-level atoms – probably one of the most popular physical models. The basis of this popularity is its intuitively simple interpretation combined with the posibility of explaining a wide range of physical phenomena. An interesting aspect of the two-level system is that, although its dynamics are formally equivalent to those of a classical angular momentum, it can explain many aspects of quantum mechanics. Once these aspects are understood, it is tempting to look further into the behaviour of real systems, trying to find patterns inconsistent with the predictions of the two-level model. This section discusses aspects of the interaction between matter and radiation that are incompatible with the two-level model. We do not consider specific atomic systems or attempt a complete analysis of the dynamics that a three-level system can exhibit. Instead we select a number of phenomena that play an important role in the discussion of those physical systems that form the subject of the subsequent sections.
After the introduction of the two-level model, we add a third level to the quantum mechanical system to discuss some of the effects that cannot be described with only two states. The main feature is the exchange of order between states and transitions, which may proceed through populations or coherences. We concentrate our attention on the latter possibility. Phenomenological introduction Model atoms In the preceding section, we discussed the interaction between light and two-level atoms – probably one of the most popular physical models. The basis of this popularity is its intuitively simple interpretation combined with the posibility of explaining a wide range of physical phenomena. An interesting aspect of the two-level system is that, although its dynamics are formally equivalent to those of a classical angular momentum, it can explain many aspects of quantum mechanics. Once these aspects are understood, it is tempting to look further into the behaviour of real systems, trying to find patterns inconsistent with the predictions of the two-level model. This section discusses aspects of the interaction between matter and radiation that are incompatible with the two-level model. We do not consider specific atomic systems or attempt a complete analysis of the dynamics that a three-level system can exhibit. Instead we select a number of phenomena that play an important role in the discussion of those physical systems that form the subject of the subsequent sections.
The two- and three-level model systems considered in the preceding sections can describe a wide range of phenomena. On the other hand, these models are purely mathematical constructs that lack physical content. The connection to physical systems is arbitrary and must be made separately in each case. In this chapter, we take the first step in this direction: We discuss the internal degrees of freedom that an atom has available. By far the most important ones are the electric dipole moment, which is responsible for the coupling to external fields, and the angular momentum, which determines the selection rules for the optical transitions and couples to magnetic fields. 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.
In the preceding chapters, we discussed the manipulation and observation of atomic variables associated with the internal degrees of freedom. A complete separation between internal and external degrees of freedom, like position and momentum, is not always possible, however. It therefore appears appropriate to include a discussion of the mechanical effects of light, which have received increasing attention in recent years. The transfer of momentum between photons and atoms, first observed more than sixty years ago, had to wait for the wide availability of lasers to become a useful experimental tool. In particular the cooling of atoms and atomic ions to very low temperatures has found numerous applications. 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.
In the preceding chapters concerning the interaction of a radiation field with matter, we assumed the field to be classical. In many situations this assumption is valid. There are, however, many instances where a classical field fails to explain experimentally observed results and a quantized description of the field is required. This is, for example, true of spontaneous emission in an atomic system which was described phenomenologically in Chapter 5. For a rigorous treatment of the atomic level decay in free space, we need to consider the interaction of the atom with the vacuum modes of the universe. Even in the simplest system involving the interaction of a single-mode radiation field with a single two-level atom, the predictions for the dynamics of the atom are quite different in the semiclassical theory and the fully quantum theory. In the absence of the decay process, the semiclassical theory predicts Rabi oscillations for the atomic inversion whereas the quantum theory predicts certain collapse and revival phenomena due to the quantum aspects of the field. These interesting quantum field theoretical predictions have been experimentally verified. In this chapter we discuss the interaction of the quantized radiation field with the two-level atomic system described by a Hamiltonian in the dipole and the rotating-wave approximations. For a single-mode field it reduces to a particularly simple form. This is a very interesting Hamiltonian in quantum optics for several reasons. First, it can be solved exactly for arbitrary coupling constants and exhibits some true quantum dynamical effects such as collapse followed by periodic revivals of the atomic inversion.
Humans perceive the world at a macroscopic level where the laws of nature can be explained using classical physics. However, at the sub-atomic level, systems exhibit phenomena that are not explainable using classical physics alone. To model the behaviour of these tiny systems, the pioneers developed a framework known as quantum mechanics. Since the main focus of this book is photons, we must understand basics of quantum mechanics before studying the behaviour of photonic systems. This mechanics has a special mathematical formalism which is the subject of this chapter.
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.
Light is both radiated and absorbed by atoms, and the interaction between the quantized electromagnetic field and an atom represents one of the most fundamental problems in quantum optics. However, real atoms are complicated systems, and even the simplest real atom, the hydrogen atom, has a non-trivial energy level structure. It is therefore often necessary or desirable to approximate the behavior of a real atom by that of a much simpler quantum system. For many purposes only two atomic energy levels play a significant role in the interaction with the electromagnetic field, so that it has become customary in many theoretical treatments to represent the atom by a quantum system with only two energy eigenstates. This is the most basic of all quantum systems, and it generally simplifies the treatment substantially. In a real atom the selection rules limit the allowed transitions between states, so that, in some cases, a certain state may couple to only one other. Moreover, optical pumping techniques have been developed that allow such preferred states to be prepared in the laboratory, and they have been successfully used in experiments (Abate, 1974). The two-level atom approximation is therefore close to the truth and not merely a mathematical convenience in some experimental situations. In the following we begin by developing the algebra for such a two-level atom. Dynamical variables for a two-level atom We consider an atomic quantum system with the two energy levels shown in Fig. 15.1.