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.

Introduction. The principal aim of this chapter is to familiarize the reader with the notation adopted in the text, as well as to introduce some concepts, such as the energy-momentum tensor of the electromagnetic field, the partition of its total angular momentum into an orbital and a spin contribution and its expansion in vector spherical harmonics, which are not usually included in an undergraduate course on electrodynamics. The chapter is entirely dedicated to the classical electromagnetic field in the absence of charges and currents. In the first two sections we present Maxwell's equations, the vector potential and different forms of the Lagrangian density of the free field from which Maxwell's equations can be obtained as Euler-Lagrange equations. In Section 1.3 we discuss briefly the properties of the field under pure Lorentz transformation and tensor notation. Then we introduce the concept of local gauge invariance and of gauge transformation, and we define the constraints leading to the Lorentz and to the Coulomb gauge. Using a canonical formalism, in Section 1.5 we obtain the Hamiltonian density of the field in the Coulomb gauge. The energy-momentum tensor of the field, the momentum and the angular momentum, along with their important conservation properties, are discussed in Section 1.6.

Introduction. The purpose of Chapter 6 is to discuss from a general point of view the dressing of a source by the vacuum fluctuations of the field coupled to the source. In Section 6.1 we show that in quantum optics, as well as in different branches of physics, virtual quanta of the field are present in the ground state of the source-field system. Three examples are considered: a two-level atom coupled to the vacuum electromagnetic field, a static model of a nucleon coupled to the vacuum meson field and an electron coupled to the optical phonon modes of a semiconductor (Fröhlich polaron). Section 6.2 is dedicated to a qualitative discussion of the physical nature of these virtual quanta and of their spatial distribution around the source. The dressed source is then defined as the bare source together with the virtual quanta surrounding it. This virtual cloud is shown is Section 6.3 to lead to a change of the energy levels of a nonrelativistic free electron interacting with the vacuum electromagnetic field. This kind of self-energy effect can be represented by a mass renormalization of the free electron. Self-energy effects due to the virtual cloud are discussed in Section 6.4 for each of the three examples of dressed sources considered in Section 6.1.

Introduction. The main theme of this chapter is the explicit calculation of the shape of the virtual cloud surrounding different kinds of ground-state sources. In Section 7.1 two of the three examples considered in Section 6.1 are taken up again, and it is argued that a convenient description of the shape of the virtual cloud is given by the energy density of the field around the source. This energy density is evaluated in detail for the static source of mesons and for the Fröhlich polaron. Section 7.2 is dedicated to an analogous calculation of the electric energy density around a two-level atom within a perturbation scheme. The virtual cloud around a two-level atom is again the subject of Section 7.3, where we evaluate the magnetic energy density as well as the coarse-grained energy density. From the results of the first three sections we conclude that the qualitative picture of the virtual cloud proposed in Section 6.2 is well founded. Moreover, the discussion of the two-level atom leads us to an important conclusion: the space surrounding an atom can be divided into a near zone and a far zone, where the energy density of the virtual cloud behaves differently as a function of the distance from the atom. In Section 7.4 we evaluate the energy density of the virtual cloud surrounding a slow free electron, separating classical and quantum contributions.

Introduction. In this final chapter we review two topics of the literature concerning dressed atoms which are of conceptual relevance and capable of shedding some light on the physical meaning and significance of atomic dressing. The first section is devoted to some recent work in connection with the quantum theory of measurement. We include the measuring apparatus in the Hamiltonian along with an appropriate apparatus-atom coupling. We argue that the theory of measurement of finite duration provides us with a tool for detecting the spectral composition of the virtual cloud surrounding an atom. In fact, we show that in a measurement of duration T on a two-level atom fully dressed by the vacuum fluctuations, as discussed in Chapters 6 and 7, the apparatus perceives the atom as dressed only by photons of frequency larger than T-1. In the case of a two-level atom dressed by a single-mode field populated by real photons, discussed in Chapter 5, we show that if T is smaller than the inverse Rabi frequency ħΔ-1, the atom is perceived by the apparatus as bare; on the contrary, if T is larger than ħΔ-1, the atom is perceived as dressed. The time scales for the two cases of dressing, by vacuum fluctuations or by a real single-mode field, are very different, but the similarity of the effects indicates a common physical aspect of the two kinds of dressing.

The interaction with the transverse modes of the vacuum electromagnetic field has been shown to yield a radiative (or self-energy) shift of the ground state of a two-level atom in Section 6.4. In this appendix we will show that the same effect occurs also in the energy of the states of a multilevel atom of the hydrogenic kind, which can be modelled by an electron bound to a fixed nucleus of charge Ze. This self-energy shift will turn out to depend on the form of the wavefunction of the state of the electron, leading to the possibility of lifting some of the accidental degeneracies which occur in hydrogenic atoms. Indeed the first experimental observation of this effect is related to the lifting of the well-known 2s-2p degeneracy in atomic hydrogen, and it is due to Lamb and Retherford (1947). Its nonrelativistic QED explanation, on the other hand, is due to Bethe (1947), and this appendix is simply a short account of his theory.

Consider a bare one-electron atom, whose energy levels and corresponding wavefunctions are denoted by En and un(x) = 〈x | n〉. Here n indicates the triplet of quantum numbers N, L, M and the electron spin is disregarded.