Optics, as defined in this book, is concerned with the interaction of electromagnetic radiation with matter. The theoretical description of the phenomena and the analysis of the experimental results are based on Maxwell's equations and on their solution for time-varying electric and magnetic fields. The optical properties of solids have been the subject of extensive treatises [Sok67, Ste63, Woo72]; most of these focus on the parameters which are accessible with conventional optical methods using light in the infrared, visible, and ultraviolet spectral range. The approach taken here is more general and includes the discussion of those aspects of the interaction of electromagnetic waves with matter which are particularly relevant to experiments conducted at lower frequencies, typically in the millimeter wave and microwave spectral range, but also for radio frequencies.

After introducing Maxwell's equations, we present the time dependent solution of the equations leading to wave propagation. In order to describe modifications of the fields in the presence of matter, the material parameters which characterize the medium have to be introduced: the conductivity and the dielectric constant. In the following step, we define the optical constants which characterize the propagation and dissipation of electromagnetic waves in the medium: the refractive index and the impedance. Next, phenomena which occur at the interface of free space and matter (or in general between two media with different optical constants) are described. This discussion eventually leads to the introduction of the optical parameters which are accessible to experiment: the optical reflectivity and transmission.

In the previous chapters the response of the medium to the electromagnetic waves was described in a phenomenological manner in terms of the frequency and wavevector dependent complex dielectric constant and conductivity. Our task at hand now is to relate these parameters to the changes in the electronic states of solids, brought about by the electromagnetic fields or by external potentials. Several routes can be chosen to achieve this goal. First we derive the celebrated Kubo formula: the conductivity given in terms of current–current correlation functions. The expression is general and not limited to electrical transport; it can be used in the context of different correlation functions, and has been useful in a variety of transport problems in condensed matter. We use it in the subsequent chapters to discuss the complex, frequency dependent conductivity. This is followed by the description of the response to a scalar field given in terms of the density–density correlations. Although this formalism has few limitations, in the following discussion we restrict ourselves to electronic states which have well defined momenta. In Section 4.2 formulas for the so-called semiclassical approximation are given; it is utilized in later chapters when the electrodynamics of the various broken symmetry states is discussed. Next, the response to longitudinal and transverse electromagnetic fields is treated in terms of the Bloch wavefunctions, and we derive the well known Lindhard dielectric function: the expression is used for longitudinal excitations of the electron gas; the response to transverse electromagnetic fields is accounted for in terms of the conductivity.

Introductory remarks

The theoretical concepts of metals, semiconductors, and the various broken symmetry states were developed in Part 1. Our objective in this part is to subject these theories to test by looking at some examples, and thus to check the validity of the assumptions which lie behind the theories and to extract some important parameters which can be compared with results obtained by utilizing other methods.

We first focus our attention on simple metals and simple semiconductors, on which experiments have been conducted since the early days of solid state physics. Perhaps not too surprisingly the comparison between theory and experiment is satisfactory, with the differences attributed to complexities associated with the electron states of solids which, although important, will not be treated here. We also discuss topics of current interest, materials where electron–electron, electron–phonon interactions and/or disorder are important. These interactions fundamentally change the character of the electronic states – and consequently the optical properties. These topics also indicate some general trends of condensed matter physics.

The discussion of metals and semiconductors is followed by the review of optical experiments on various broken symmetry ground states. Examples involving the BCS superconducting state are followed by observations on materials where the conditions of the weak coupling BCS approach are not adequate, and we also describe the current experimental state of affairs on materials with charge or spin density wave ground states.

This book has its origins in a set of lecture notes, assembled at UCLA for a graduate course on the optical studies of solids. In preparing the course it soon became apparent that a modern, up to date summary of the field is not available. More than a quarter of a century has elapsed since the book by Wooten: Optical Properties of Solids – and also several monographs – appeared in print. The progress in optical studies of materials, in methodology, experiments and theory has been substantial, and optical studies (often in combination with other methods) have made definite contributions to and their marks in several areas of solid state physics. There appeared to be a clear need for a summary of the state of affairs – even if with a somewhat limited scope.

Our intention was to summarize those aspects of the optical studies which have by now earned their well deserved place in various fields of condensed matter physics, and, at the same time, to bring forth those areas of research which are at the focus of current attention, where unresolved issues abound. Prepared by experimentalists, the rigors of formalism are avoided. Instead, the aim was to reflect upon the fact that the subject matter is much like other fields of solid state physics where progress is made by consulting both theory and experiment, and invariably by choosing the technique which is most appropriate.

The focus of this chapter is on the optical properties of band semiconductors and insulators. The central feature of these materials is the appearance of a single-particle gap, separating the valence band from the conduction band. The former is full and the latter is empty at zero temperature. The Fermi energy lies between these bands, leading to zero dc conduction at T=0, and to a finite static dielectric constant. In contrast to metals, interband transitions from the valence band to the conduction band are of superior importance, and these excitations are responsible for the main features of the electrodynamic properties. Many of the phenomena discussed in this chapter also become relevant for higher energy excitations in metals when the transition between bands becomes appreciable for the optical absorption.

Following the outline of Chapter 5, we first introduce the Lorentz model, a phenomenological description which, while obviously not the appropriate description of the state of affairs, reproduces many of the optical characteristics of semiconductors. The transverse conductivity of a semiconductor is then described, utilizing the formalisms which we have developed in Chapter 4, and the absorption near the bandgap is discussed in detail, followed by a summary of band structure effects. After discussing longitudinal excitations and the q dependent optical response, we briefly mention indirect transitions and finite temperature and impurity effects; some of the discussion of these phenomena, however, is relegated to Chapter 13.

The configurations which guide the electromagnetic radiation and allow the interaction of light with matter vary considerably, with a wide variation of techniques developed over the years and employed today. We can distinguish between single-path and interferometric arrangements, and – as far as the interaction of light with the material under study is concerned – single-bounce from multiple-bounce, so-called resonant techniques. In this chapter we present a short summary.

Single-path arrangements sample the change of the electromagnetic wave if only one interaction with matter takes place; for instance if the light is reflected off the sample surface or is transmitted through the specimen in a single path. In general, part of the radiation is absorbed, and from this the optical properties of the material can be evaluated. However, only at low frequencies (i.e. at long wavelengths) does this simple configuration allow the determination of the phase change of the radiation due to the interaction; for higher frequencies only the attenuation in power is observed.

Interferometric techniques compare one part of the radiation, which undergoes the interaction with the material (i.e. reflection from or transmission through the material), with a second part of the signal, which serves as a reference. In this comparative approach – the so-called bridge configuration – the mutual coherence of the two beams is crucial. The interference of the two beams is sensitive to both the change in amplitude and in phase upon interaction, and thus allows calculation of the complex response of the specimen.

The role of electron–electron and electron–phonon interactions in renormalizing the Fermi-liquid state has been mentioned earlier. These interactions may also lead to a variety of so-called broken symmetry ground states, of which the superconducting ground state is the best known and most studied. The ground states are superpositions of electron–electron or electron–hole pairs all in the same quantum state with total momenta of zero or 2kF; these are the Cooper pairs for the superconducting case. There is an energy gap ∧, the well known BCS gap, introduced by Bardeen, Cooper, and Schrieffer [Bar57], which separates the ground state from the single-particle excitations. The states develop with decreasing temperature as the consequence of a second order phase transition.

After a short review of the various ground states, the collective modes and their response will be discussed. The order parameter is complex and can be written as ∧ exp{iφ}; the phase plays an important role in the electrodynamics of the ground state. Many aspects of the various broken symmetry states are common, but the distinct symmetries also lead to important differences in the optical properties. The absorption induced by an external probe will then be considered; it is usually discussed in terms of the so-called coherence effects, which played an important role in the early confirmation of the BCS theory. Although these effects are in general discussed in relation to the nuclear magnetic relaxation rate and ultrasonic attenuation, the electromagnetic absorption also reflects these coherence features, which are different for the various broken symmetry ground states.

With a few exceptions we have considered mainly bulk properties in the book. The physics of reduced dimensions is not only of theoretical interest, for many models can be solved analytically in one dimension only. A variety of interesting phenomena are bounded to restricted dimensions. On the other hand, fundamental models such as the theory of Fermi liquids developed for three dimensions break down in one or two dimensions. In recent years a number of possibilities have surfaced to explain how reduced dimensions can be achieved in real systems. One avenue is the study of real crystals with an extremely large anisotropy. The second approach considers artificial structures such as interfaces which might be confined further to reach the one-dimensional limit.

Dielectric response function in two dimensions

Reducing the dimension from three to two significantly changes many properties of the electron gas. If the thickness of the layer is smaller than the extension of the electronic wavefunction, the energy of the system is quantized (size quantization). We consider only the ground state to be occupied. For any practical case, just the electrons are confined to a thin sheet, while the field lines pass through the surrounding material which usually is a dielectric. A good approximation of a two-dimensional electron gas can be obtained in surfaces, semiconductor interfaces, and inversion layers; a detailed discussion which also takes the dielectric properties of the surrounding media into account can be found in [And82, Hau94].

Ever since Euclid, the interaction of light with matter has aroused interest – at least among poets, painters, and physicists. This interest stems not so much from our curiosity about materials themselves, but rather to applications, should it be the exploration of distant stars, the burning of ships of ill intent, or the discovery of new paint pigments.

It was only with the advent of solid state physics about a century ago that this interaction was used to explore the properties of materials in depth. As in the field of atomic physics, in a short period of time optics has advanced to become a major tool of condensed matter physics in achieving this goal, with distinct advantages – and some disadvantages as well – when compared with other experimental tools.

The focus of this book is on optical spectroscopy, defined here as the information gained from the absorption, reflection, or transmission of electromagnetic radiation, including models which account for, or interpret, the experimental results. Together with other spectroscopic tools, notably photoelectron and electron energy loss spectroscopy, and Raman together with Brillouin scattering, optics primarily measures charge excitations, and, because of the speed of light exceeding substantially the velocities of various excitations in solids, explores in most cases the Δq = 0 limit. While this is a disadvantage, it is amply compensated for by the enormous spectral range which can be explored; this range extends from well below to well above the energies of various single-particle and collective excitations.

The experimental discovery of Bose–Einstein condensation in trapped atomic clouds opened up the exploration of quantum phenomena in a qualitatively new regime. Our aim in the present work is to provide an introduction to this rapidly developing field.

The study of Bose–Einstein condensation in dilute gases draws on many different subfields of physics. Atomic physics provides the basic methods for creating and manipulating these systems, and the physical data required to characterize them. Because interactions between atoms play a key role in the behaviour of ultra cold atomic clouds, concepts and methods from condensed matter physics are used extensively. Investigations of spatial and temporal correlations of particles provide links to quantum optics, where related studies have been made for photons. Trapped atomic clouds have some similarities to atomic nuclei, and insights from nuclear physics have been helpful in understanding their properties.

In presenting this diverse range of topics we have attempted to explain physical phenomena in terms of basic principles. In order to make the presentation self-contained, while keeping the length of the book within reasonable bounds, we have been forced to select some subjects and omit others. For similar reasons and because there now exist review articles with extensive bibliographies, the lists of references following each chapter are far from exhaustive. A valuable source for publications in the field is the archive at Georgia Southern University: http://amo.phy.gasou.edu/bec.html

This book originated in a set of lecture notes written for a graduate level one-semester course on Bose-Einstein condensation at the University of Cop enhagen. We have received much inspiration from contacts with our colleagues in both experiment and theory. In particular we thank Gordon Baym and George Kavoulakis for many stimulating and helpful discussions over the past few years.

The phenomena of superfluidity and superconductivity are intimately connected with the existence of a condensate, a macroscopically occupied quantum state. Such condensates occur in a variety of different physical systems, as described in Chapter 1. The foundation for the description of superfluidity is a picture of the system as being comprised of a condensate and elementary excitations. In Chapter 8 we have seen how physical properties such as the energy and the density of a Bose–Einstein condensed system may be expressed in terms of a contribution from the condensate, plus one from the elementary excitations, and in this chapter we shall consider further developments of this basic idea to other situations. As a first application, we determine the critical velocity for creation of an excitation in a homogeneous system (Sec. 10.1). Following that, we show how to express the momentum density in terms of the velocity of the condensate and the distribution function for excitations. This provides the basis for a two-component description, the two components being the condensate and the thermal excitations (Sec. 10.2). In the past, this framework has proved to be very effective in describing the properties of superfluids and superconductors, and in Sec. 10.3 we apply it to dynamical processes.

To describe the state of a superfluid, one must specify the condensate velocity, in addition to the variables needed to characterize the state of an ordinary fluid. As a consequence, the collective behaviour of a superfluid is richer than that of an ordinary one. Collective modes are most simply examined when excitations collide frequently enough that they are in local thermodynamic equilibrium.

Bose–Einstein condensates of particles behave in many ways like coherent radiation fields, and the realization of Bose–Einstein condensation in dilute gases has opened up the experimental study of many aspects of interactions between coherent matter waves. In addition, the existence of these dilute trapped quantum gases has prompted a re-examination of a number of theoretical issues. This field is a vast one, and in this chapter we shall touch briefly on selected topics.

In Sec. 13.1 we describe the classic interference experiment, in which two clouds of atoms are allowed to expand and overlap. Rather surprisingly, an interference pattern is produced even though initially the two clouds are completely isolated. We shall analyse the reasons for this effect. The marked decrease in density fluctuations in a Bose gas when it undergoes Bose–Einstein condensation is demonstrated in Sec. 13.2. Gaseous Bose–Einstein condensates can be manipulated by lasers, and this has made possible the study of coherent matter wave optics. We describe applications of these techniques to observe solitons, Bragg scattering, and non-linear mixing of matter waves in Sec. 13.3. The atom laser and amplification of matter waves is taken up in Sec. 13.4. How to characterize Bose–Einstein condensation microscopically is the subject of Sec. 13.5, where we also consider fragmented condensates.

Interference of two condensates

One of the striking manifestations of the wave nature of Bose–Einstein condensates is the observation of an interference pattern when two condensed and initially separated clouds are allowed to overlap. An example is shown in Fig. 13.1.

Atomic properties of the alkali atoms play a key role in experiments on cold atomic gases, and we shall discuss them briefly in the present chapter. Basic atomic structure is the subject of Sec. 3.1. Two effects exploited to trap and cool atoms are the influence of a magnetic field on atomic energy levels, and the response of an atom to radiation. In Sec. 3.2 we describe the combined influence of the hyperfine interaction and the Zeeman effect on the energy levels of an atom, and in Sec. 3.3 we review the calculation of the atomic polarizability. In Sec. 3.4 we summarize and compare some energy scales.

Atomic structure

The total spin of a Bose particle must be an integer, and therefore a boson made up of fermions must contain an even number of them. Neutral atoms contain equal numbers of electrons and protons, and therefore the statistics that an atom obeys is determined solely by the number of neutrons N: if N is even, the atom is a boson, and if it is odd, a fermion. Since the alkalis have odd atomic number Z, boson alkali atoms have odd mass numbers A. In Table 3.1 we list N, Z, and the nuclear spin quantum number I for some alkali atoms and hydrogen.

Bose–Einstein condensates in dilute atomic gases, which were first realized experimentally in 1995 for rubidium, sodium, and lithium, provide unique opportunities for exploring quantum phenomena on a macroscopic scale. These systems differ from ordinary gases, liquids, and solids in a number of respects, as we shall now illustrate by giving typical values of some physical quantities.

The particle density at the centre of a Bose-Einstein condensed atomic cloud is typically 1013–1015 cm−3. By contrast, the density of molecules in air at room temperature and atmospheric pressure is about 1019 cm−3. In liquids and solids the density of atoms is of order 1022 cm−3, while the density of nucleons in atomic nuclei is about 1038 cm−3.

To observe quantum phenomena in such low-density systems, the temperature must be of order 10−5 K or less. This may be contrasted with the temperatures at which quantum phenomena occur in solids and liquids. In solids, quantum effects become strong for electrons in metals below the Fermi temperature, which is typically 104–105 K, and for phonons below the De bye temperature, which is typically of order 102 K. For the helium liquids, the temperatures required for observing quantum phenomena are of order 1 K. Due to the much higher particle density in atomic nuclei, the corresponding degeneracy temperature is about 1011 K.

The path that led in 1995 to the first realization of Bose–Einstein condensation in dilute gases exploited the powerful methods developed over the past quarter of a century for cooling alkali metal atoms by using lasers.