Introduction

In our discussion of the classical electromagnetic field we found it convenient to describe the field by a complex amplitude, both in the frequency and in the time domains. The complex representation is convenient partly because it contains information about the magnitude and about the phase of the electromagnetic disturbance, and partly because of its analytic properties. These features turn out to be particularly useful for the description of the optical coherence properties of the field.

We shall see that an analogous quantum state of the field exists, leading to an interesting representation that is also particularly useful for the treatment of optical coherence. This coherent-state representation leads to a close correspondence between the quantum and classical correlation functions. The coherent states of the field come as close as possible to being classical states of definite complex amplitude. We shall find that coherent states turn out to be particularly appropriate for the description of the electromagnetic fields generated by coherent sources, like lasers and parametric oscillators; indeed it turns out that the field produced by any deterministic current source is in a coherent state. In the following sections we shall examine some of the properties of coherent states, and then go on to discuss representations based on these states.

Coherent states were first discovered in connection with the quantum harmonic oscillator by Schrödinger (1926), who referred to them as states of minimum uncertainty product.

In the preceding chapters we have solved a number of specific problems in which electromagnetic fields interact with charges, atoms, or molecules, and these have been approached in several different ways. For example, for the problem of photoelectric detection, which involves short interaction times, we found it convenient to use a perturbative method, whereas the resonance fluorescence problem was treated by solving the Heisenberg equations of motion. In the following sections we shall encounter a number of general methods for tackling interaction problems that can often simplify the problems substantially when they are applicable. We shall illustrate the utility of these methods by recalculating a number of results that were obtained in a different manner before.

The quantum regression theorem

It was shown by Lax (1963; see also Louisell, 1973, Sec. 6.6) that, with the help of a certain factorization assumption, it is often possible to express multi-time correlation functions of certain quantum mechanical operators in terms of single-time expectations. The result is now known as the regression theorem. As multi-time correlation functions play a rather important role in quantum optics, the theorem is often of great utility, and it can drastically simplify certain calculations. In the following we largely adopt the procedure given by Lax.

We consider two coupled quantum systems, to which, for the sake of convenience, we shall refer as the system (S) and the reservoir (R).

Introduction

In Chapter 5 we discussed some applications of second-order coherence theory to problems involving radiation from localized sources of any state of coherence. In the present chapter we will describe some other applications of the second-order theory. The first two concern classic interferometric techniques for determining the angular diameters of stars and the energy distribution in spectral lines. Both techniques were introduced by Albert Michelson many years ago and the underlying principles were explained by him without the use of any concept of coherence theory (which was formulated later). However, second-order coherence theory provided a deeper understanding of the physical principles involved and also suggested useful modifications of these techniques, some of which will be discussed in Section 9.10.

Another application which will be described in this chapter concerns the determination of the angular and the spectral distribution of energy in optical fields scattered from fluctuating linear media. The analysis will be based on the second-order coherence theory of the full electromagnetic field, which we developed in Chapter 6.

Stellar interferometry

As is well known, the angular diameters that stars subtend at the surface of the earth are so small that no available telescopes can resolve them. In the focal plane of a telescope, the star light gives rise to a diffraction pattern which is indistinguishable from that which would be produced by light from a point source, diffracted at the aperture of the telescope and degraded by the passage of the light through the earth's atmosphere.

Introduction

Up to now the electromagnetic field has been treated as a classical field, describable by c-number functions. The great success of classical electromagnetic theory in accounting for a variety of optical phenomena, particularly those connected with wave propagation, interference and diffraction, amply justifies the classical approach. Moreover, as we have seen in the preceding chapters, in some cases the classical wave theory also gives a good account of itself in the treatment of the interaction of electromagnetic fields. For example, it is able to describe such seemingly non-classical effects as photoelectric bunching and the photo-electric counting statistics. It might almost seem that there is little justification for going beyond the domain of classical wave theory in optics.

On the other hand, it can be argued that optics lies well and truly in the quantum domain, in the sense that we often encounter situations in which very few quanta or photons are present. In the microwave region of the electromagnetic spectrum, and at still longer wavelengths, the number of photons in each mode of the field is usually very large, and we are justified in treating the system classically. However, in the optical region the situation is usually just the opposite. As we show in Section 13.1, for light produced by practically all sources other than lasers, the average number of photons per mode is typically much less than unity.

Definitions

The concept of probability is of considerable importance in optics, as in any situation in which the outcome of a given trial or measurement is uncertain. Under these conditions it is desirable to be able to associate a measure with the likelihood of the outcome or the event in question; such a measure is called the probability of the event.

Several different definitions of probability have been adopted at various times in the past. The classical definition is based on an exhaustive enumeration of the possible outcomes of an experiment or trial. If the trial has N distinguishable, mutually exclusive outcomes, which are equally likely to occur, and if n out of these N possible outcomes have an attribute or characteristic that we call ‘success’, then the probability of success in any one trial is given by the ratio n/N. For example, if we roll a die, and if each of the six digits is equally likely to be on top when the die comes to rest, there are N = 6 distinguishable outcomes. If we identify success with an even number, for example, then since there are three different ways in which success can be achieved, it follows that the probability of success when the die is rolled is given by 3/6 = 1/2. Unfortunately, an exhaustive enumeration of all possibilities is not always feasible.

Another common definition of probability is based on the notion of relative frequency of success.

Interactions of a quantized electromagnetic field

In the preceding chapters we have studied the quantum properties of the electromagnetic field, but we have treated it as a free or non-interacting quantum system until now. However, both the emission and the absorption of light imply interactions with other quantum systems, and we now turn to the treatment of such interaction problems.

It is true that, to a limited extent, we have already succeeded in treating some interactions of light in previous chapters, without invoking the full apparatus of the quantum theory of interacting systems. For example, in Chapter 9 the electromagnetic field was treated as a classical potential acting on an atomic quantum system, and in Sections 12.2 and 12.9 we invoked simple heuristic arguments to describe the operation of certain optical detectors. But these are limited applications, and, in any case, the validity of results obtained in this way needs to be confirmed.

To describe the state of a quantized electromagnetic field (F) in interaction with some other quantum system (A), we evidently require an enlarged, or product Hilbert space, which encompasses both F and A, of which the Hilbert spaces of F and A are subspaces. We shall find it convenient to refer to the other system A as an ‘atomic system’, simply to give a name, without restricting the nature of A. The dynamical variables of the field ÔF(t) and of the atomic system ÔA(t) commute at the same time t, and at the beginning, when the interaction between them is assumed to be turned on, each operator acts only on the state vectors within its respective Hilbert space.

Introduction

In the preceding chapters we have been largely concerned with the simplest coherence effects of optical fields, namely those which depend on the correlation of the field variable at two space-time points (r1, t1) and (r2, t2). As we have seen, these effects include the most elementary coherence phenomena involving interference, diffraction and radiation from fluctuating sources.

In this chapter we present an extension of the theory to cover more complicated situations, which have to be described by correlations of higher order, i.e. by correlations of the field variables at more than two space-time points or the expectation values involving various powers and products of the field variables. Situations of this kind have become of considerable importance since the development of the laser and of nonlinear optics. The basic difference between the statistical properties of thermal light and laser light can, in fact, only be understood by going beyond the elementary second-order correlation theory.

Many of the higher-order coherence phenomena are most clearly manifest in the photoelectric detection process, which can only be adequately described by the quantum theory of detection or by taking into account the quantum features of the field, both of which will be studied in the succeeding chapters. However, because the field is still described classically in the semi-classical theory of light detection, and also because the classical field description provides a natural stepping stone to the quantum description of field correlations, we will now discuss the general description of field correlations of all orders on the basis of the classical theory of the fluctuating wavefield.

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.

Quantum Optics is a branch of physics which has developed recently in different directions relevant to fundamental physics as well as to highly sophisticated technological applications. The scientific roots of quantum optics, however, originate from the broader subject of Quantum Electrodynamics and, more generally, from quantum field theory. Thus the boundary between quantum optics and quantum field theory is a particularly delicate conceptual ground which should be properly mastered by any prospective quantum optician, theorist or experimentalist alike. This book is intended to foster understanding and knowledge of this boundary region by presenting in a pedagogical fashion the basic theory of dressed atoms, which has been established as a concept of central importance in quantum optics, since it is capable of shedding light on such diverse physical phenomena as resonance fluorescence, the Lamb shift and van der Waals forces.

Coherently with the aims outlined above, the first part of this book, consisting of the first four chapters, is dedicated to the foundations of atom-field interactions. Both radiation and matter are treated from the quantum field theory point of view, and the coupling between matter and the electromagnetic field is derived using the principle of gauge invariance. The atom-photon Hamiltonian is obtained by specializing this general treatment to a nonrelativistic electron field describing the electrons around an atomic nucleus.

Introduction. In the previous two chapters we have discussed electrodynamics in the absence of charges and currents. We are now ready to investigate the nature of these charges and currents. Thus in this chapter we introduce the concept of matter field, both classical and quantized, which as we will see acts as a source of the electromagnetic field. The difficulties encountered in the definition of convenient wave equations (Klein-Gordon and Dirac) for a relativistic particle are examined in Section 3.1, and they lead naturally to consider these equations as equations of motion of a field, obtainable from an appropriate field Lagrangian. Thus the probabilistic single-particle interpretation of the wave equations is abandoned, and Section 3.2 is dedicated to the Klein-Gordon field, which is introduced by an appropriate Klein-Gordon Lagrangian, yielding the Klein-Gordon equation. The Klein-Gordon field is then second-quantized, both in its real and complex versions. The eigenstates of the Hamiltonian of this second-quantized field are shown to correspond to many-particle states satisfying Bose-Einstein statistics. An analogous procedure is followed in Section 3.3 for the Dirac equation, leading to the definition of a Dirac field which upon second quantization yields a field Hamiltonian whose eigenstates correspond to many-particle states satisfying Fermi-Dirac statistics. For both fields the energy-momentum tensors are defined and various conservation properties are obtained.