We showed in Section 5.2 of the present chapter that according to quantum theory the value of the electric field of an electromagnetic wave could not be predicted to arbitrarily high precision, this being a consequence of the uncertainty relation satisfied by the two quadrature components of the field (Equation 5.32), which imposed a finite limit on the product of their variances. This limit is more than an abstract theoretical limit; it is often the overriding factor determining the resolution of high-precision optical measurements. In such situations measurements by photodetectors on the field exhibit uncontrollable fluctuations of quantum origin known as quantum noise. Fortunately, as we shall show in this complement, it is possible to overcome this by using non-classical states of the radiation field, namely the squeezed states, provided measurements are made of a single one of a pair of conjugate field variables. A field mode prepared in such a state exhibits reduced quantum noise in one of the variables at the expense of increased noise in the other, so that the uncertainty relation involving their variances is still satisfied. The potential of such states for increasing the obtainable precision of optical measurements is self-evident. However, the practical realization of the potential benefits requires delicate experimental techniques that we describe only briefly here.

The advent of lasers has revolutionized the methods and possibilities of spectroscopy. Their monochromaticity permits the study of extremely narrow spectral features which would not have been resolved using classical techniques. Moreover, their extreme spectral power density, in giving rise to the field of nonlinear optics, which will be studied in Chapter 7, has resulted in the development of new and powerful spectroscopic techniques. These techniques, of which we describe briefly some examples in this complement, have as their basis the fact that an atom does not respond linearly when subjected to a very intense light field.

First of all we shall describe the broadening mechanisms that explain the fact that the experimentally measured widths of atomic or molecular resonances are larger than those that would be observed if the atoms were isolated and at rest. We shall introduce in Section 3G.1 the concept of homogeneous width, which can be considered as the intrinsic width of a transition, and inhomogeneous width which concerns the variation of the frequency of a given transition that occurs because of the state of motion of the atom, or because of factors in its environment. We shall explain why it is possible, using nonlinear optical techniques to eliminate the effects of inhomogeneous broadening on recorded spectra, resulting in an improvement over the resolution of conventional techniques. We shall then concentrate more specifically on Doppler broadening, which is the most commonly encountered source of inhomogeneous broadening, describing two nonlinear optical techniques, saturated absorption spectroscopy (in Section 3G.2) and two-photon spectroscopy (in Section 3G.3), which enable it to be overcome.

Hamiltonian formalism and canonical quantization

In this book, we use the classical Hamiltonian formalism to write the energy of a system in a form that lends itself to canonical quantization. For this to work, we must be sure to identify pairs of conjugate canonical variables which, after quantization, will give pairs of operators with commutators equal to iħ. To achieve this, we write down Hamilton's equations for the relevant energy expression and check that they do indeed return the previously established dynamical equations. This is what was done in Chapter 4 for the electromagnetic field in vacuum. We expressed the electromagnetic energy in terms of the real and imaginary parts of normal variables, and showed that Hamilton's equations were equivalent to Maxwell's equations. In Chapter 6, we used an analogous approach for an interacting system of charges and fields, but we did not supply all the details of the calculations.

The aim in this complement is to show explicitly that the Hamiltonian for classical electrodynamics, expressed in the form given at the end of Section 6.1, does indeed lead back to the equations of motion of the particles under the effect of the fields, but also to the dynamical equations of the fields themselves in the presence of the charges, i.e. the Maxwell–Lorentz equations as given in Section 6.1.1. This is then used to confirm that we have indeed identified pairs of conjugate canonical variables.

The simplified description of the operation of lasers, presented in Section 3.1 of Chapter 3, rests on the assumption of the infinite transverse extent of the laser cavity, so that the circulating light fields can be represented by plane waves. This is obviously a somewhat unrealistic assumption; the various components of a laser cavity are of limited spatial extent, more usually transverse dimensions are of the order of a centimetre. If the light waves were really plane waves, the diffraction at one of these aperture-limiting components would make impossible reproduction of the form of the wavefront after a complete cavity round trip and would, furthermore, introduce severe losses. In practice diffraction losses are compensated for by the use of focusing elements such as concave mirrors, but a theoretical treatment of the light field based on plane waves is then inappropriate.

A more useful description of the intracavity light field is one in terms of a wave with a non-uniform transverse spatial distribution which also takes into account the question of the stability of the wave propagating in the cavity. This description must also account for diffraction effects and the reflections on the cavity mirrors. Such a stable light field is known as a transverse mode of the cavity.

In general, finding an expression for the transverse modes of an arbitrary cavity is a complicated problem. Fortunately for the cavity geometries most commonly employed in continuous-wave lasers (and most particularly for a linear cavity composed of two concave mirrors) classes of simple solutions exist: the transverse Gaussian modes.

The work of John Bell in the mid 1960s and experiments carried out to test his famous inequalities in the following decades have led to a detailed re-examination of the concepts of quantum mechanics, and revealed the full importance of the notion of entanglement. This reconsideration helped to generate the new and extremely rich field of research known as quantum information in the 1980s. The guiding idea behind this field of activity is that, by exploiting the specific rules of quantum physics, one can conceive of new ways of calculating and communicating, in which the rules of play are no longer the well-known classical rules.

One can thus develop new methods of cryptography in which the message is protected by the basic principles of quantum mechanics, and new computation methods that can be exponentially more efficient than classical algorithms. Quantum information is not therefore a mere sideline for physics, but concerns information theory, algorithmics and the mathematics of complexity theory. This research has already led to proposals for new algorithms and new computation architectures based on quantum logic gates with no classical equivalent. Still on the fundamental level, the meeting of information theory and quantum mechanics which lies at the heart of quantum information has led to a very stimulating regeneration of the theoretical tools used on both sides.

In this chapter we shall be using the formalism of Chapter 4 to describe some properties of the quantized free electromagnetic field, emphasizing the analogies and differences with classical electromagnetism. The analogies explain why classical optics was so successful, to the extent that, even though light was not quantized, it was able to describe almost all known optical phenomena up until the 1970s. The differences highlight typical quantum behaviour, with no classical equivalent, which lies at the heart of modern quantum optics.

The chapter begins by presenting formal ways of describing the main components of quantum optics experiments, namely, photodetectors and semi-reflecting mirrors. One can then understand homodyne detection, used to measure field quadrature components, even at optical frequencies for which no detector is fast enough to follow the oscillations of the field itself.

Section 5.2 deals with the quantum vacuum where, in contrast to the classical vacuum, radiation has properties, in particular, fluctuations, with which one can associate physical effects.

We then discuss in Section 5.3 some quantum radiation states in the simple case where a single mode l of the field is excited (non-vacuum). We introduce the idea of a quasiclassical state, or coherent state, used to make the link with classical optics. We shall also study other states, those with a definite number of photons, or squeezed states, which exhibit typically quantum properties, quite inconceivable in classical electromagnetism, and we shall show how these properties can be understood by reference to the Heisenberg relations, with an extremely useful graphical representation.

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’).

Since its invention in 1960, the laser has revolutionized both the study of optics and our understanding of the nature of light, prompting the emergence of a new field, quantum optics. Actually, it took decades until the words quantum optics took their current precise meaning, referring to phenomena which can be understood only by quantizing the electromagnetic field describing light. Surprisingly enough, such quantum optics phenomena hardly existed at the time that the laser was invented, and almost all optics effects could be fully understood by describing light as a classical electromagnetic field; the laser was no exception. As a matter of fact, to understand how a laser works, it suffices to use the semi-classical description of matter–light interaction, where the laser amplifying medium, made of atoms, molecules, ions or semi-conductors, is given a quantum mechanical treatment, but light itself is described by classical electromagnetic waves.

The first part of our book is devoted to presentation of the semi-classical approach and its use in describing various optical phenomena. It includes an elementary exposition of the physics of lasers, and some applications of this ubiquitous device. After recalling in Chapter 1 some basic results of the quantum mechanical description of interaction induced transitions between the atomic energy levels, we use these results in Chapter 2 to show how the interaction of a quantized atom with a classical electromagnetic wave leads to absorption or stimulated emission, and to derive the process of laser amplification that happens when a wave propagates in an inverted medium.

The arguments of Chapter 2, as well as of those of subsequent chapters, have as their foundation the formalism based on the state vector of a system of which the evolution is described by the Schrödinger equation. In fact, such an approach is badly suited to the case in which the coupling between an atom and its environment (for example through collisions with other atoms or spontaneous emission into formerly empty modes of the electromagnetic field) cannot be neglected. If the correlations induced by these interactions between the atom and its environment do not concern us and we are only interested in the evolution of the atom, the formalism of the density matrix must be employed. This provides a description at all times of the state of the atom, although a state vector for the atom alone cannot be defined. In this formalism the effect of the environment on the atom is accounted for by the introduction of suitable relaxation terms (Section 2C.1) in the equation of evolution of the density matrix. An important application of the density matrix is to the case of a two-level atomic system for which the relaxation terms lead to its deexcitation to a level of lower energy. We shall show that in this case the density matrix can be represented by a vector, known as the Bloch vector, which will allow us to give simple geometrical pictures of the evolution of the system.