Abstract In the chapter « Quantum electrodynamics », both the particles and the electromagnetic field are quantized. An elementary quantization procedure is first proposed in which the classical field normal modes are naïvely replaced by photon operators. The full quantization procedure is then presented that makes use of the Lagrangian formalism and the resulting Hamilton equations. This chapter ends with a discussion on photon number states and coherent Glauber states.
Abstract In the chapter « Quantum electrodynamics », both the particles and the electromagnetic field are quantized. An elementary quantization procedure is first proposed in which the classical field normal modes are naïvely replaced by photon operators. The full quantization procedure is then presented that makes use of the Lagrangian formalism and the resulting Hamilton equations. This chapter ends with a discussion on photon number states and coherent Glauber states.
At the conceptual level, the quantization procedure replaces the pure particle description of classical mechanics by a particle-wave treatment satisfying the requirements of the fundamental wave–particle dualism. Technically, one can always start from a generalized wave equation and use it as a basis to discuss wave-front propagation in the form of rays following classical particle trajectories, see Sections 2.2.1 and 3.1. Consequently, any wave theory inherently contains that aspect of the wave–particle dualism. Especially, Maxwell's equations of electrodynamic theory include both wave and particle aspects of electromagnetic fields. Hence, it could be that there are no additional “quantum,” i.e., wave–particle dualistic aspects. However, as we discuss in this chapter, Maxwell's equations can be expressed in the framework of Hamilton's formulation of particle mechanics. This then suggests that in addition to the approximate ray-like behavior, electromagnetic fields do have a supplementary level of particle-like properties. Due to the fundamental axiom of wave–particle dualism, the additional particle-like aspects of electromagnetic fields must then also have a wave counterpart that can be axiomatically introduced via the canonical quantization scheme described in Section 3.2.3. Since the spatial dependency in Maxwell's equation contains wave–particle dualism already at the classical level, it is clear that the new level of quantization cannot be a simple real-space feature but has to involve some other space describing the structure of the electromagnetic fields. In fact, this quantization yields several very concrete implications such as quantization of the light energy, elementary fluctuations in the light-field amplitude and intensity, and more.
In a concise, but accessible for the first acquaintance form the procedure for the quantization of linear oscillator is set out. By analogy with this procedure the procedure of quantization (second quantization) of classical Maxwell’s electrodynamics is set up. The physical sense of the wave functions arguments of transverse electromagnetic field and its Fourier transformation is set up. One pay attention as for quantum coherent (almost a classic) states of the electromagnetic field and for photonics Fock states. Attention is drawn to the fact of absence the power of the universal content of such concepts as field amplitude, phase and number of particles (photons), which are used by experimenter’s to describe the states of a quantized field. The semi quantitative description the interaction processes of a quantum electromagnetic field with substance is set up. Specified situations are shown in which the discrepancy between the predictions of classical and quantum electrodynamics is noticeable at the macroscopic level.
In Chapter 1, we treated the electromagnetic field as classical. Henceforth, we will want to treat it as a quantum field. In order to do so, we will first present some of the formalism of quantum field theory. This formalism is very useful in describing many-particle systems and processes in which the number of particles changes. Why this is important to a quantum description of nonlinear optics can be seen by considering a parametric amplifier of the type discussed in the last chapter. A pump field, consisting of many photons, amplifies idler and signal fields by means of a process in which a pump photon splits into two lower-energy photons, one at the idler frequency and one at the signal frequency. Therefore, what we would like to do in this chapter is to provide a discussion of some of the basics of quantum field theory that will be useful in the treatment of the quantization of the electromagnetic field. In particular, we will begin with a summary of quantum theory notation, and a discussion of many-particle Hilbert spaces. These provide the arena in which all of the action takes place. We will then move on to a treatment of the canonical quantization procedure for fields. This will allow us to develop a scattering theory for fields, which is ideally what we need. This relates the properties of a field entering a medium to those of the field leaving it, and this corresponds to what is done in an experiment.
According to modern quantum field theory all elementary particles are quantum excitations of an underlying quantum field. In the case of the photon the underlying field is the electromagnetic field. In photon wave mechanics the photon concept was reached by a reinterpretation of the physical meaning of the free classical electromagnetic field. Such a reinterpretation does not give us a quantum theory of light quanta because the elementary quantum of action is foreign to classical physics. Although Planck’s constant does occur in the various Schro¨dinger-like wave equations we have presented in Part III it just plays the role of a multiplicative factor, see e.g., Eq. (10.36). The “justification” for bringing ~ into the formalism is the wish to make contact to the Einstein-de Broglie relations [Eqs. (10.40) and (10.41)]. In a sense one may call the light particle appearing from classical electrodynamics the classical photon.
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.
In this chapter we explain the procedure for quantizing classical fields, based on the preparatory work we did in the previous chapter. We have seen that classical fields can generally be treated as an (infinite) set of oscillator modes. In fact, for all fields we considered in the previous chapter, we were able to rewrite the Hamiltonian in a way such that it looked identical to the Hamiltonian for a generic (mechanical) oscillator, or a set of such oscillators. We therefore understand that, if we can find an unambiguous way to quantize the mechanical oscillator, we get the quantization of all classical fields for free! We thus start considering the mechanical harmonic oscillator and refresh the quantum mechanics of this system. The new idea is to associate the oscillator with a level for bosonic particles representing the quantized excitations of the oscillator. We then turn to a concrete example and show how the idea can be used to quantize the deformation field of the elastic string discussed in Section 7.2. In this case, the bosonic particles describing the excitations of the field are called phonons: propagating wave-like deformations of the string. As a next example, we revisit the low-lying spinfull excitations in a ferromagnet, already described in Section 4.5.3. Here, we start from a classical field theory for the magnetization of the magnet, and show how the quantization procedure leads to the same bosonic magnons we found in Chapter 4. Finally, we implement the quantization technique for the electromagnetic field, leading us to the concept of photons – the quanta of the electromagnetic field.