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.

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 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.

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.

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.

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.

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 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.

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.

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.