The interaction of nonrelativistic matter with the quantized electromagnetic field is investigated in the classical limit of large photon numbers. Quantization of both matter, say an atom, and the field results in a time-independent Schr\"odinger equation (TISE). However, for very strong fields (quantum mechanically, large photon numbers) this is impractical to solve. The standard approach then is simply to replace the quantized field by a classical field to give a time-dependent Schr\"odinger equation (TDSE) for the atom alone. Here we show how this TDSE can be derived from the TISE for atom plus field, illustrating at each stage the approximations that are necessary to treat the field classically. An important difficulty at the semiclassical stage is a breakdown of the approximation at classical turning points. We show how the use of coherent field states can circumvent this problem. In the limit that the field can be treated classically, time emerges from the Maxwell equations and a TDSE for the atom alone results.
The interaction of nonrelativistic matter with the quantized electromagnetic field is investigated in the classical limit of large photon numbers. Quantization of both matter, say an atom, and the field results in a time-independent Schr"odinger equation (TISE). However, for very strong fields (quantum mechanically, large photon numbers) this is impractical to solve. The standard approach then is simply to replace the quantized field by a classical field to give a time-dependent Schr"odinger equation (TDSE) for the atom alone. Here we show how this TDSE can be derived from the TISE for atom plus field, illustrating at each stage the approximations that are necessary to treat the field classically. An important difficulty at the semiclassical stage is a breakdown of the approximation at classical turning points. We show how the use of coherent field states can circumvent this problem. In the limit that the field can be treated classically, time emerges from the Maxwell equations and a TDSE for the atom alone results.
We study the general problem of matter-laser interaction within the framework of nonrelativistic quantum electrodynamics, with particular emphasis on strong laser field effects. Consequently, we formulate a well-defined approximation leading in a straightforward manner toward the conventional semiclassical mode of description. Namely, we arrive naturally to two coupled equations of motion: (i) the Schr\"odinger equation which governs the quantum dynamics of an atomic system driven by classically described radiation field (composed of an incoming laser pulse plus radiation emitted from the atom), and (ii) the classical Maxwell wave equation which describes the emission of radiation from the mentioned atomic source. Employing the formalism of adiabatic Floquet theory, we derive a simple criterion of validity of the just described semiclassical approach. It shows that the semiclassical treatment is justified in most situations. On the other hand, it turns out that the semiclassical approximation breaks down completely in certain special but realistic cases, regardless of the fact that the incoming laser pulse contains a huge number of photons. Under such special circumstances, we anticipate new effects arising due to the quantized nature of the radiation field, to be observable, for example, in harmonic generation spectra. Our considerations are illustrated more explicitly using a simple model of a two-level atom strongly driven by a laser. The quantum dynamics of this model problem is resolved within the framework of quantum electrodynamics while adopting well-defined and physically justifiable approximations. As an outcome, analytic formulas are found serving as a quantitative criterion of (non)applicability of the semiclassical approach and demonstrating the breakdown of semiclassical theory under well-defined conditions. An illustrative numerical calculation is provided.
We investigate in this chapter the general problem of the interaction of an atom (or a molecule) and a classical electromagnetic field. Its importance derives firstly from the fact that a large part of our knowledge of atoms is obtained from a study of the radiation they absorb or emit (we shall consider equally visible radiation and radio-frequency and X-rays) and, secondly, that the interaction with matter modifies the propagation of the electromagnetic field itself, notably through absorption, refraction or scattering. The interactions of atoms with light therefore encompass a vast range of physical effects that we could not hope to cover adequately in a single chapter. In this chapter we shall therefore present the fundamental features of the interaction of an atom, which will be treated quantum mechanically, with a classical electromagnetic field, that is an electromagnetic field described by real electric and magnetic vectors obeying Maxwell's equations. A rigorous description of the atom–light interaction would have to take into account the quantum nature of light. This we shall leave to Chapter 6. It turns out, nevertheless, that many important results can be obtained from the semi-classical viewpoint adopted here (although it is perhaps more appropriate to refer to this treatment as ‘semi-quantized’).
A rigorous description of the light–matter interaction requires the use of quantized Maxwell equations to describe the light field and the Schrödinger equation to describe the material medium. This book does not attempt to deal with quantum optical problems that necessitate a field quantization. Instead we focus on properties that are generally grouped under the umbrella of nonlinear optics . Although many definitions of this expression exist, we use it in the sense that the properties we deal with do not depend in a crucial way on the field quantization. For practical purposes, this means in general that the average photon number is large and that we can neglect spontaneous emission as a dynamical process. Rather, we include the consequence of spontaneous emission, that is, the instability of the atomic levels, in a purely phenomenological way. When this is done, we call the material equations the Bloch equations instead of the Schrödinger equation. References [1] through [4] are the classical textbooks that contain a tentative justification for the transition from the Schrödinger to the Bloch equations. None of these attempts is satisfactory from a fundamental viewpoint because the real difficulty is to incorporate the finite lifetime (or natural linewidth) of the atomic energy levels in the Schrödinger equation. This remains an open problem as of now. The Maxwell–Schrödinger equations We consider atoms interacting with an intense electromagnetic field. By intense we mean a field for which the quantization is not necessary.
Beginning with an entangled state of a time-independent (TI) quantum system coupled to its TI quantum environment, we show that a time-dependent Schrödinger equation (TDSE) for the quantum system alone can be derived in the limit that one of the environment variables becomes a classical variable. In the same limit the TI amplitude of the environment wavefunction becomes the TD amplitude of an eigenfunction expansion of the system TD wavefunction. Similarly, the phase of the TI environment wavefunction goes over into the TD phase of the system amplitude. By considering that more and more environment variables become successively classical, each provides a classical clock to give a multiple-time TDSE for the quantum system. Two examples of two-time TDSE given in the literature are derived. When all clocks are synchronized the multiple-time TDSE reduces to the usual one-time TDSE.
The classical or semiclassical theory presented in Chapter 2 gives a satisfactory account of light-matter interactions in some cases. However, a detailed understanding of different processes requires a quantum mechanical theory of radiation. The quantum theory of light, like classical theory, is also based on Maxwell's equations. However, in quantum theory the electric field F and the magnetic field H are treated as operators. This chapter develops the correspondence between the two and outlines the method for converting field vectors F and H into quantum mechanical operators. Topics discussed include wave equation in a rectangular cavity, quantization of the electric field, time-dependent perturbation theory, interaction of an electron with the magnetic field, and second-order perturbation theory. Exercises are provided at the end of the chapter.
The interaction of electromagnetic fields with atomic/molecular systems is the genesis of most optical phenomena observed to date. To explain these effects, we need two broad classifications based on both classical and quantum theories. This strategy involves treating (1) fields as classical systems and atoms as quantum systems having discrete energy, (2) fields as quantum systems (particles rather than waves) and atomic systems as classical, or (3) both as quantum systems. Of course, schemes 1 and 2 are easy to solve and visualize, while scheme 3 is the most accurate description but complex in nature. In general, quantum approaches do not lead to exact solutions and require assumptions and simplifications to achieve a reasonably approximate solution. Thus, it is important to choose an appropriate strategy for solving a particular problem. The problem associated with atom–field interactions will also be subjected to different strategies to understand some of the experimentally observed phenomena such as Rabi oscillations. In this chapter, we will derive a master equation that serves as the starting point for solving a large number of problems associated with atom–field interactions.