This first complement is devoted to a completely classical approach of lightmatter interaction which was proposed by Lorentz at the end of the nineteenth century, before the advent of quantum mechanics, but after the discovery of the electron. Lorentz' phenomenological model is based on the experimental fact that atoms have well-defined and sharp absorption lines: he assumed that atoms behaved like harmonic oscillators, in which the electrons are bound to the atomic nucleus by a restoring force which varies linearly with its displacement (from its equilibrium point close to a nucleus), and makes them oscillate at a given frequency ω0 equal to the experimentally determined absorption frequency.

Within the frame of this model, we first calculate the electromagnetic field radiated by an oscillating electron. We show that in the absence of an externally applied force the free oscillations of the electron are damped, because electromagnetic energy is radiated at the expense of mechanical energy. We then study the characteristics of the radiation that is emitted when the oscillations are forced by the application of an external oscillatory electromagnetic field of angular frequency ω. We characterize the different regimes of this scattering of the incident electromagnetic wave and finally determine the polarization induced in the atomic medium by the incident electromagnetic wave.

The Lorentz model can be considered as a lowest order approximation to a description of the light–matter interaction, a better approximation being the semi-classical treatment presented in Chapter 2, and the rigorous treatment being the completely quantum mechanical model presented in Chapter 6.

The difference between laser light and the light emitted by an incoherent source can only be fully appreciated with reference to certain notions of energetic photometry, which are spelt out in the first part of this complement. It will be shown in Section 3C.2 how the laws of photometry drastically reduce the energy density that can be obtained from a conventional incoherent source (such as a heated filament, or a discharge lamp) in comparison with a laser source (Section 3C.3). Far from being merely circumstantial, these laws for classical sources are of a fundamental kind that can be deduced from the basic principles of thermodynamics. Another way to relate these properties of light to the fundamental principles of physics is to examine them in the context provided by the statistical physics of photons, as will be discussed in Sections 3C.4 and 3C.5.

Conservation of radiance for an incoherent source

Étendue and radiance

An incoherent source comprises a large number of independent, elementary emitters, emitting electromagnetic waves with a random distribution of uncorrelated phases. It emits light in every direction. A light beam produced by this source can be decomposed into elementary pencils of light. Since the light is incoherent, the total power carried by the beam is the sum of the powers carried by the elementary pencils.

An elementary pencil is defined by the element dS of the source from which it originates, and a second surface element dS′, as shown in Figure 3C.1.

Introduction

In 1961, just a few months after Maiman invented the ruby laser, Franken focused the pulses emitted from such a laser, of wavelength 694 nm, on a quartz plate, and examined the spectrum of the light transmitted using a simple prism (see Figure 7.1). He thus discovered that ultra-violet light of wavelength 347 nm was emerging from the quartz plate. Clearly, as it propagated through the quartz, the light of frequency ω had generated the second harmonic, of frequency 2ω.

It thus transpires that in optics, as in any other part of physics, a system subjected to a strong enough sinusoidal excitation will leave the linear response regime. Nonlinearities cause harmonics of the excitation frequency to appear.

But what intensity is needed before nonlinear effects will appear? One might think that a natural scale would be the electric field of the nucleus at the location of an atomic electron. In the case of the hydrogen atom in its ground state, this field is about, or 3 × 1011 V.m−1. (Here e is the charge of the electron, and a0 the Bohr radius, of the order of 5 × 10−11 m). In fact, experiment shows that, in the transparency zone of a dielectric material like quartz, a field of just 107 V.m−1 (corresponding to a light intensity of 2.5 kW.cm−2) is sufficient for nonlinear effects to appear perturbatively.

One-photon sources are important elements in quantum optics. The archetypal example is an atom raised to an excited state at time t = 0, then de-exciting with emission of a single photon. The development of this kind of source depends on progress with experimental techniques, e.g. the possibility of isolating a single atom, molecule or quantum well. In this complement, we present the formalism for describing the corresponding radiation, and use it to discuss some spectacular experiments which bring out properties quite incompatible with a classical description of the electromagnetic field. We begin in Section 5B.2 by describing the anti-correlation between detections on either side of a semi-reflecting mirror, establishing the quantitative difference with a classical field. Section 5B.3 discusses a quantum optical effect that was only demonstrated at the beginning of the twenty-first century, namely the quantum coalescence of two one-photon wave packets on a semi-reflecting mirror, which occurs even when the two photons were emitted by independent atoms. An analogous effect, the Hong–Hou–Mandel effect, is discussed in Chapter 7. These effects exemplify quantum interference involving two photons. Finally, Section 5B.4 is concerned with quantum calculations involving quasi-classical states. As we now know, this leads to results that are identical to the predictions of semi-classical theory.

In this complement we discuss several examples of optical phenomena in media where the refractive index depends nonlinearly on the intensity, known as optical Kerr media. This nonlinear effect exists in all materials, even isotropic ones, like glass or fused silica, but it is particularly marked in certain physical systems to be exemplified in Section 7B.1. After investigating the propagation of light through such media in Section 7B.2, we shall discuss three applications of the optical Kerr effect (which can be studied in any order). We begin by describing a bistable optical system, when this nonlinear medium is inserted in a Fabry–Perot cavity (Section 7B.3). We then study phase conjugate mirrors and examine their potential applications in adaptive optics (Section 7B.4). Finally, we discuss certain effects occurring during the propagation of an isolated wave, bounded either transversely or temporally, in a Kerr medium, and describe self-focusing effects (Section 7B.5) and self-phase-modulation effects (Section 7B.6). In particular, we shall show that nonlinear effects and dispersion effects can compensate to produce stable structures known as solitons, which maintain their shape during propagation.

Examples of third-order nonlinearities

Nonlinear response of two-level atoms

We begin by studying a simple case of a nonlinear interaction, namely a two-level quantum system under the effects of a plane wave.

The spectral width of the output of most single-mode lasers is determined by technical limitations associated with the stability of the optical length of the laser cavity (see Section 3.3.3). However, in the absence of these, there is a more fundamental limit to the degree of monochromaticity that can be achieved. This limit, known as the Schawlow–Townes limit is, in fact, rather narrower than the passive bandwidth of the laser cavity or the width of the gain curve of the active medium it contains. We calculate in a heuristic fashion in this complement the Schawlow–Townes limit for a laser operating far above threshold.

The fundamental mechanism for the spectral broadening of a laser output beam is the spontaneous emission by the gain medium of photons into the laser mode. Spontaneous emission adds to the complex field of the laser mode εL a fluctuating field, εsp corresponding to the addition of a single photon with a random phase. The total field therefore undergoes amplitude and phase fluctuations. The fluctuations of the amplitude are damped by the gain saturation of the amplifying medium and only the phase fluctuations persist, because the mechanism responsible for laser oscillation does not impose any phase to the generated field. Thus, in the course of successive spontaneous emission events, the phase of the laser field undergoes a random walk. After a time τc (the field correlation time) the phase of the laser field can no longer be predicted; it has lost all memory of its initial value.

In this chapter we shall describe the principle of the operation of lasers, their common features and the properties of the light they emit. Our aim is not to provide an exhaustive catalogue of the types of laser available at the time of writing. Such an account would, in any case, soon be obsolete. Rather, we shall use concrete examples of existing systems to illustrate important features or general principles. We do not want either to give an extensive theoretical description of a laser's properties and of its dynamics. We restrict ourselves here to a rather simplified approach to its main features and refer the reader to more specialized handbooks for further information (see the further reading section at the end of the main chapter).

The physical principles accounting for laser operation can appear quite straightforward. This impression stems from the fact that the essential concepts are now well understood, whilst the detail and some incorrect notions are passed over in silence. It is interesting to note, however, how painstaking our progress in understanding lasers has been. It is usually considered that the prehistory of the laser commenced in 1917 when Einstein introduced the notion of stimulated emission. In fact, Einstein was led to the conclusion that such a phenomenon must occur from considerations of the thermodynamic equilibrium of the radiation field and a sample of atoms at a finite temperature T.

We have seen in the present chapter that the light emitted by a laser has properties that are radically different from those of the light emitted by classical sources. These properties have been the basis for the myriad applications found for lasers since their advent in the 1960s; they have escaped the confines of the research laboratory to become ubiquitous in industrial production and modern consumer society. Lasers now have innumerable applications in such disparate areas as medicine, metallurgy and telecommunications and are at the heart of new developments in commercial and consumer electronics (CD and DVD players, bar-code readers and printers, to name but a few examples).

The total market in the mid 2000s was estimated to be almost 6 billion dollars. It was dominated by the domains of optical storage (30% of the total amount) and communication (20%), which are mass production markets. In contrast, material processing (25%) and medical applications (8%) involve a smaller number of very expensive lasers. Research and instrumentation amount to 6% of the total sales. The significant fraction of laser sales related to research and development is a testament to the relative youth of the technology. New applications are still coming to light, some of which may have profound economic consequences for the future.

It will not be possible to provide an exhaustive account of these applications here. We shall, therefore, concentrate on a few significant examples selected from the broad categories introduced above.

Many processes, including absorption and stimulated emission occurring in lasers, can be handled using a semi-classical model for the atom–radiation interaction, in which the matter is given a quantum description, but the radiation is represented as a classical electromagnetic field (see Chapter 2). There are other phenomena that cannot be adequately described without quantizing the radiation. For example, it has been known since the 1930s that spontaneous emission can only be treated correctly using a fully quantum framework for the interaction, in which both the matter and the radiation are quantized, as we shall see in Chapter 6.

However, it was not until the 1970s that situations were found in which a free electromagnetic field, far from sources, exhibited properties and behaviour that could not be described by a classical field, but which could be perfectly well interpreted in terms of a quantized field. This chapter is devoted to the quantization of the free electromagnetic field, far from the charges and currents sourcing it. This free electromagnetic field will be called radiation, and in Chapter 6 we shall specify exactly what is meant by radiation when sources are present.

The canonical quantization procedure used here starts from a description of the classical dynamics of the field in the framework of the Hamiltonian formalism, the basic features of which are discussed in Section 4.1.

Entanglement is one of the most surprising features of quantum mechanics. However, it was not until the last decades of the twentieth century that its full importance was understood and it was realized that it could lead to revolutionary applications in the area of quantum information. It was A. Einstein who discovered the extraordinary properties of non-factorizable two-particle states, when seeking to demonstrate that the formalism of quantum mechanics is incomplete. He presented his findings in 1935 in his famous article published jointly with B. Podolsky and N. Rosen, now referred to as the ‘EPR’ paper. Soon afterwards, Schrödinger coined the term ‘entangled states’ to emphasize the fact that the properties of the two particles are inextricably bound together.

In the EPR article, Einstein and his colleagues used quantum predictions to conclude that the formalism of quantum mechanics was incomplete, in the sense that it did not account for the whole of physical reality, and that the task of physics was therefore to find a more complete theory. They did not contest the validity of the quantum formalism, but suggested that a further, more detailed level of description would have to be introduced, in which each particle of the EPR pair would have well-defined properties that were not taken into account in the quantum formalism.

We have seen in Section 1.2 that a transition between two discrete levels |i〉 and |k〉 is possible when the system is submitted to a perturbation which has a non-zero Fourier component close to the Bohr frequency ω0 = (Ek − Ei)/ħ of the transition. This is the case when the perturbation Ĥ1(t) is sinusoidal, of the form Ĥ1(t) = Ŵ cos(ωt + φ), and oscillates at a frequency ω close to ω0. But in many real physical situations, the perturbation is not perfectly mastered: it is a pure sine function only for times shorter than a limit, called the coherence time. At longer times, the oscillating perturbation has a phase and an amplitude that vary randomly. This is, for example, the case when the perturbation is due to an incident electromagnetic wave produced by a thermal lamp, in which the thermal fluctuations induce random variations of the amplitude and phase of the wave. As the perturbation is no longer a pure sinusoidal wave, its Fourier spectrum is no longer a Dirac delta function. It has some finite width around a mean frequency. For this reason a random perturbation is also called a ‘broadband’ perturbation.

We shall determine in this complement the temporal evolution of the quantum system submitted to such a random perturbation. We shall see that the transition probability between two discrete levels is proportional to T for small interaction time T, and has an exponential behaviour at long times, a result which is very similar to that obtained for a transition between a discrete level and a continuum.

When the Nobel Prize for Physics was awarded to Claude Cohen-Tannoudji, William D. Philipps and Steven Chu in 1997 for the development of methods for cooling and trapping atoms using lasers, this was the reward for two decades of investigations not only touching upon the fundamental aspects of the light–matter interaction, but also leading to a range of applications. On the theoretical level, the advent of this field of research in the late 1970s stimulated the development of various theoretical methods for describing the radiative forces by which light affects the motion of atoms. On the experimental level, judicious use of the radiative forces exerted by lasers made it possible to obtain a drastic reduction of atomic velocities – in other words, to cool an atomic vapour. These advances were characterized by a remarkable cross-fertilization of theoretical and experimental innovations. It was not long before applications began to appear in the field of high-resolution spectroscopy, since ultracold, i.e. slow, atoms can be observed for longer periods, and this allows greater accuracy when measuring atomic resonance frequencies than could be obtained for atoms at room temperature. Based on such resonances, the most accurate clocks in the world use laser-cooled atoms or ions. Atom interferometry is another application of laser manipulation of atoms, which can be applied to measure inertial effects, due for example to the rotation of the Earth or the motion of a vehicle, with an accuracy that already exceeds traditional methods.

In this chapter, we discuss a purely quantum approach to the interaction between an atom and the electromagnetic field. In this treatment, the atom and electromagnetic field form a single quantum system, whose evolution is handled globally within a unified formalism. This will thus have the merit of being fully consistent from the theoretical standpoint. But the main advantage in an entirely quantum approach is that it can treat the full range of matter–radiation interaction phenomena. In particular, it provides a rigorous description of the spontaneous emission of light by an excited atom, something that falls outside the scope of the semi-classical framework applied in Chapters 2 and 3, where the lifetime of an excited atomic state had to be fed in phenomenologically. It can describe other phenomena of the same type, such as parametric fluorescence by a nonlinear crystal subjected to pumping radiation (see Chapter 7), which underlies many recent developments in quantum optics. The fully quantum approach also has the merit of allowing a simple interpretation in terms of photons for the various matter–radiation interaction processes, such as absorption, stimulated emission, scattering, and also the basic processes of nonlinear optics. Indeed, it provides a unified framework for both stimulated and spontaneous processes. Finally, it can be used to tackle completely new situations where matter and radiation interact, which lie outside the scope of any semi-classical description, such as cavity quantum electrodynamics or the production of single-photon or entangled-photons states.

Atomic, molecular and optical physics is a field which, during the last few decades, has known spectacular developments in various directions, like nonlinear optics, laser cooling and trapping, quantum degenerate gases, quantum information. Atom–photon interactions play an essential role in these developments. This book presents an introduction to quantum optics which, I am sure, will provide an invaluable help to the students, researchers and engineers who are beginning to work in these fields and who want to become familiar with the basic concepts underlying electromagnetic interactions.

Most books dealing with these subjects follow either a semi-classical approach, where the field is treated as a classical field interacting with quantum particles, or a full quantum approach where both systems are quantized. The first approach is often oversimplified and fails to describe correctly new situations that can now be investigated with the development of sophisticated experimental techniques. The second approach is often too difficult for beginners and lacks simple physical pictures, very useful for an initial understanding of a physical phenomenon. The advantage of this book is that it gives both approaches, starting with the first, illustrated by several simple examples, and introducing progressively the second, clearly showing why it is essential for the understanding of certain phenomena. The authors also clearly demonstrate, in the case of non-linear optics and laser cooling, how advantageous it may be to combine both approaches in the analysis of an experimental situation and how one can get from each point of view useful, complementary physical insights.

Up to now in this work we have not been concerned with the environment in which an atom and the radiation field interact. We have implicitly assumed that the radiation propagates in free space and that there are no boundaries to reflect radiation emitted by the atom. We shall show in this complement that when such boundaries do exist and are sufficiently reflecting, or more especially when the atom is enclosed in a resonant cavity, its radiative properties, such as its absorption spectrum and the rate of spontaneous emission, are drastically altered. This can be the case even if the cavity boundaries themselves are very far from the atom, on the atomic scale of distances.

The conditions under which these cavity quantum electrodynamic effects can be observed are actually quite difficult to reach and this is why, usually, it is possible to assume that the radiative properties of a system are independent of the enclosure surrounding it. Nevertheless, thanks to some outstanding technical achievements, these effects can be observed in remarkable experiments. Atoms coupled to cavities then appear as a promising system in quantum information either for quantum processing or for single photon sources (see Complement 5E).

Presentation of the problem

Consider the system sketched in Figure 6B.1, in which an atom at rest at the origin of coordinates is enclosed in a cavity of volume V, with perfectly reflecting walls.