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.

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.