In the 1980s the first successful experiments [312] and theory [98], demonstrating that light could be used to cool and confine atoms to submillikelvin temperatures, opened several exciting new chapters in atomic, molecular, and optical (AMO) physics. Atom interferometry [6, 8], matter–wave holography [294], optical lattices [192], and Bose–Einstein condensation in dilute gases [18, 95] all exemplified significant new physics where collisions between atoms cooled with light play a pivotal role. The nature of these collisions has become the subject of intensive study not only because of their importance to these new areas of AMO physics but also because their investigation has led to new insights into how cold collision spectroscopy can lead to precision measurements of atomic and molecular parameters and how radiation fields can manipulate the outcome of a collision itself. As a general orientation Fig. 1.1 shows how a typical atomic de Broglie wavelength varies with temperature and where various physical phenomena situate along the scale. With de Broglie wavelengths on the order of a few thousandths of a nanometer, conventional gas-phase chemistry can usually be interpreted as the interaction of classical nuclear point particles moving along potential surfaces defined by their associated electronic charge distribution. At one time liquid helium was thought to define a regime of cryogenic physics, but it is clear from Fig. 1.1 that optical and evaporative cooling have created “cryogenic” environments below liquid helium by many orders of magnitude.
In the 1980s the first successful experiments [312] and theory [98], demonstrating that light could be used to cool and confine atoms to submillikelvin temperatures, opened several exciting new chapters in atomic, molecular, and optical (AMO) physics. Atom interferometry [6, 8], matter–wave holography [294], optical lattices [192], and Bose–Einstein condensation in dilute gases [18, 95] all exemplified significant new physics where collisions between atoms cooled with light play a pivotal role. The nature of these collisions has become the subject of intensive study not only because of their importance to these new areas of AMO physics but also because their investigation has led to new insights into how cold collision spectroscopy can lead to precision measurements of atomic and molecular parameters and how radiation fields can manipulate the outcome of a collision itself. As a general orientation Fig. 1.1 shows how a typical atomic de Broglie wavelength varies with temperature and where various physical phenomena situate along the scale. With de Broglie wavelengths on the order of a few thousandths of a nanometer, conventional gas-phase chemistry can usually be interpreted as the interaction of classical nuclear point particles moving along potential surfaces defined by their associated electronic charge distribution. At one time liquid helium was thought to define a regime of cryogenic physics, but it is clear from Fig. 1.1 that optical and evaporative cooling have created “cryogenic” environments below liquid helium by many orders of magnitude.
Introduction This chapter describes remarkable new experiments that have been made possible through recent advances in laser cooling of atoms. Cooling of atoms in a gas by laser light is in itself surprising and this technique is important in a range of applications. In particular the experiments described here take atoms that have been laser cooled and then use other methods to reach even lower temperatures, at which wonderful and fascinating quantum effects occur. As well as revealing new phenomena, these experiments explore the deep analogies between the roles played by light waves in lasers and matter waves. These techniques are widely accessible and enable skilled researchers in laboratories all over the world to explore exciting new physics. The key physics and techniques of laser cooling are described in depth in Chapter 6. What is “temperature”? We use the words “cold” and “hot” every day; for example to refer to the temperature of the air outside. But what do “cold” and “hot” mean for the air’s molecules? Broadly speaking, we can say that atoms in a hot gas move faster than those in a cold gas and that cooling, by taking energy out of the gas, slows the atoms down. In this description of the gas as a collection of atoms behaving like billiard balls, or little hard spheres, the lowest possible temperature occurs when the atoms stop moving but, as we shall see, this picture does not give an accurate description at very low energies, for which quantum effects become important. In this chapter we will look at the fascinating properties of atomic gases cooled to temperatures within one millionth of a degree Kelvin from absolute zero. We shall use the term “ultra-cold” to refer to such extremely low temperatures that have only recently been achieved in experiments on atomic gases. The techniques used to cool liquids and solids have a longer history and have made available a different set of possibilities (see Chapter 8). Note that most of this chapter refers to gases in which the individual particles are atoms, but the general statements about temperature scales apply equally well to gases of molecules, e.g. air, whose major constituents, oxygen and nitrogen, exist as diatomic molecules.
Collision physics plays a key role in understanding the properties of ultracold gases. First, collisions ensure thermalization in trapped atomic samples, which is essential for the success of evaporative cooling. Second, interactions between atoms determine the equilibrium shape and the dynamics of the condensate once it is formed. Finally collisions limit the achievable spatial densities via inelastic processes, which ultimately determine whether a given atomic species may lead to Bose-Einstein condensation. The purpose of the following lectures is to give a basic introduction to collision physics with a special emphasis on the low temperature aspect of these interactions. We will consider here only binary interactions between atoms since one of the remarkable prop-erties of the physics of ultra-cold dilute gases is that all the macroscopic properties of the system can be directly deduced from the two-body interaction potential. In particular the essential concept of scattering length will be introduced and developped for some simple potentials, using as practical examples experimental results obtained on atomic cesium and rubidium. The present lectures are complementary to the ones given by D. Heinzen where a detailed comparison between the collisional properties of the various alkali atoms will be found, together with explanations of the spectroscopic approaches used in the precise determination of interatomic potentials. 1. – Scattering theory: a brief reminder 1.1. Collision between two particles. – We consider here an elementary collision process between two particles 1 and 2 with the same mass M, interacting through the potential
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.
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 10 13 –10 15 cm −3 . By contrast, the density of molecules in air at room temperature and atmospheric pressure is about 10 19 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 10 38 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 10 4 –10 5 K, and for phonons below the De bye temperature, which is typically of order 10 2 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 10 11 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.
The experimental realization in 1995 of Bose–Einstein condensation in dilute atomic gases marked the beginning of a very rapid development in the study of quantum gases. The initial experiments were performed on vapours of rubidium, sodium, and lithium. So far, the atoms 1 H, 7 Li, 23 Na, 39 K, 41 K, 52 Cr, 85 Rb, 87 Rb, 133 Cs, 170 Yb, 174 Yb and 4 He * (the helium atom in an excited state) have been demonstrated to undergo Bose–Einstein condensation. In related developments, atomic Fermi gases have been cooled to well below the degeneracy temperature, and a superfluid state with correlated pairs of fermions has been observed. Also molecules consisting of pairs of fermionic atoms such as 6 Li or 40 K have been observed to undergo Bose–Einstein condensation. Atoms have been put into optical lattices, thereby allowing the study of many-body systems that are realizations of models used in condensed matter physics. Although the gases are very dilute, the atoms can be made to interact strongly, thus providing new challenges for the description of strongly correlated many-body systems. In a period of less than ten years the study of dilute quantum gases has changed from an esoteric topic to an integral part of contemporary physics, with strong ties to molecular, atomic, subatomic and condensed matter physics. The dilute quantum gases differ from ordinary gases, liquids and solids in a number of ways, as we shall now illustrate by giving values of physical quantities.
Abstract The invention and development of laser methods for cooling and trapping atoms have led to the advent of the physics of ultracold matter. Low-temperature quantum physics has so far dealt with condensed media, wherein quantum phenomena (superfluidity and superconductivity) usually manifest themselves at temperatures of a few kelvin. Laser cooling and trapping of atoms have made it possible to successively attain millikelvin, microkelvin, and nanokelvin temperatures, at which the de Broglie wavelength of the atoms becomes commensurable with the interatomic separation, even in dilute gases. The interaction of atoms in ultracold dilute gases becomes essentially quantum mechanical. The physics of such quantum gases is more akin to the well-developed physics of condensed matter at low temperatures. It is in essence a boundary field between atomic, laser, and condensed-matter physics. Following the first, pioneering experiments on the Bose–Einstein condensation of alkali-metal atoms, this field has continued to develop at a very fast pace, which can be explained by the vast variety of particles (atoms, isotopes, and molecules) that can be laser cooled and trapped under degeneracy conditions. This is much more than physicists had at their disposal previously (3,4He). Moreover, laser radiation and magnetic fields additionally make it possible to control not only the intrinsic quantum state of the interacting ultracold particle species, but also the interaction of the particles. All this makes the research in the field of quantum gases a great breakthrough, both in atomic–molecular physics and in many-body-interaction physics, which heretofore was the prerogative of low-temperature condensed-matter physics.