Bose–Einstein condensation (BEC) in cold atomic gases was first achieved experimentally in 1995. Since then there has been a surge of activity in this field, with ingenious experiments putting forth more and more astonishing results about the behaviour of matter at very cold temperatures. The theoretical investigation of BEC goes back much further, and even predates the modern formulation of quantum mechanics. It was investigated in two papers by Einstein in 1924 and 1925, respectively, following up on a work by Bose on the derivation of Planck's radiation law. Einstein's result, in its modern formulation, can be found in any textbook on quantum statistical mechanics, and was concerned with ideal, i.e., non-interacting gases. The understanding of BEC in the presence of interparticle interactions poses a formidable challenge to mathematical physics. Some progress has been made in the last ten years or so, and the purpose of this chapter is to explain part of what was achieved and how it is related to the actual experiments on cold gases.
Bose–Einstein condensation (BEC) in cold atomic gases was first achieved experimentally in 1995. Since then there has been a surge of activity in this field, with ingenious experiments putting forth more and more astonishing results about the behaviour of matter at very cold temperatures. The theoretical investigation of BEC goes back much further, and even predates the modern formulation of quantum mechanics. It was investigated in two papers by Einstein in 1924 and 1925, respectively, following up on a work by Bose on the derivation of Planck's radiation law. Einstein's result, in its modern formulation, can be found in any textbook on quantum statistical mechanics, and was concerned with ideal, i.e., non-interacting gases. The understanding of BEC in the presence of interparticle interactions poses a formidable challenge to mathematical physics. Some progress has been made in the last ten years or so, and the purpose of this chapter is to explain part of what was achieved and how it is related to the actual experiments on cold gases.
Introduction Bose–Einstein condensation (BEC) is one of the most intriguing phenomena in physics. The basic idea goes back to the 1920s, when Einstein made predictions for the behavior of atoms at sufficiently low temperatures and high densities. However, it took seven decades for experimental physicists to realize this phase under the low-density conditions envisioned by Einstein. In the intermediate period there were many experiments in which aspects of BEC played important roles. Therefore, phenomena such as superfluidity and superconductivity were connected to BEC long before its 1995 realization in ultra-cold, low-density atomic vapors. Since BEC can be discussed with very many different backgrounds and applications in mind, the subject can sometimes be overwhelming for interested students. There is a very good reason why a chapter on BEC does not belong in this book. In all the other chapters here, the processes described involve one or at most two atoms, and the interactions that play a role are between the particles that constitute the atom, namely between the nucleus (or nuclei) and their electrons. The interactions are described quantum mechanically by their Hamiltonian that can always be written exactly in non-relativistic terms, and in some cases can even be solved exactly. This makes atoms and molecules a branch of physics where comparisons between theory and experiments become very meaningful since theoretical predictions are often confirmed to very high accuracy. In BEC, it is the interactions among many atoms that are the key for understanding the phenomenon. Although in principle one could write a Hamiltonian for a Bose condensed gas, its large number of particles makes this not very practical. The starting point for the description of a BEC is in the area of many-body physics that belongs to the expertise of condensed-matter physicists. The interactions between ~10 23 electrons in a solid state with a lattice formed by ~10 23 ions resemble the kind of interactions in a BEC much more closely than the simple fewbody interactions in an atom or molecule. Thus all the theoretical models that have been described up to now are not applicable for the contents of this chapter.
Bose-Einstein condensation is a unique phase transition in that it is not driven by interparticle interactions, but can theoretically occur in an ideal gas, purely as a consequence of quantum statistics. This chapter addresses the question, ‘ How is this ideal Bose gas condensation modified in the presence of interactions between the particles?’ This seemingly simple question turns out to be surprisingly difficult to answer. Here we outline the theoretical background to this question and discuss some recent measurements on ultracold atomic Bose gases that have sought to provide some answers. Introduction Unlike the vast majority of phase transitions, Bose-Einstein condensation (BEC) is not driven by interparticle interactions but can theoretically occur in an ideal (noninteracting) gas, purely as a consequence of quantum statistics. However, in reality, interactions are needed for a Bose gas to remain close to thermal equilibrium. It is thus interesting to discuss if something close to ideal gas BEC can be observed in a real system and what happens in the vicinity of the BEC transition in the presence of interparticle interactions. These simple questions have not been easy to answer, either theoretically or experimentally. The theoretical foundations for studying the effect of interactions on Bose condensed systems were laid over half a century ago by Bogoliubov [1], Penrose and Onsager [2], and Belieav [3], among others. These works initially focused on zerotemperature properties and were extended to nonzero temperature in the pioneering papers of Lee, Huang, and Yang [4, 5, 6, 7]. At that time, the main experimental system was liquid He-4 in which the inter-particle interactions are strong, making connections with theory difficult. The realisation in 1995 of BEC in weakly interacting ultracold atomic gases [8, 9] thus opened up the possibility to experimentally revisit some of these long-discussed questions. This was further aided by, among other advances, the use of Feshbach resonances to tune the interaction strength in atomic gases [10, 11]. Thus, in the last twenty years the study of ultracold Bose gases has been very successful (see Chapter 3 for an insightful account of selected topics and surprising developments over this exciting period). However, the fact that, until recently, ultracold atoms were confined using harmonic potentials has hindered the study of the BEC transition itself.
The field of atomic quantum gases has seen rapid and sometimes surprising developments since its beginnings in 1995. In this chapter, I summarize, highlight, and comment on selected topics. Introduction Bose-Einstein condensation (BEC) in atomic gases was first observed in 1995, and has changed the face of atomic physics. It is for atoms or matter waves what the laser is for photons: a macroscopically occupied quantum state. It was regarded as an elusive goal until it was discovered in 1995. Although BEC was immediately viewed as a major accomplishment, its impact has far exceeded expectations. Now, twenty years on, there is no question that the field remains exciting. It is impossible to give a review over the developments during those twenty years. Instead, in this chapter, I want to illustrate how often predictions or expectations changed in the pursuit of Bose-Einstein condensation. There were many surprises, and some advances and breakthroughs happened although they were predicted to be impossible. A lesson we can learn from this is that we should always carefully read the fine print when something is proven or assumed impossible, and try to figure out if the assumptions can be circumvented! Since the early history of Bose-Einstein condensation provides several examples for such “impossibility theorems,” I digress into those earlier developments in the first part of this chapter. Early Theoretical Questions Validity of the Prediction of Bose-Einstein Condensation Einstein predicted Bose-Einstein condensation in 1924 in the second of two papers where he generalized Bose's treatment from photons to massive particles [1]. Using statistical arguments introduced by Bose, he found that below a critical temperature, bosonic particles condense in the lowest energy state of the system. In contrast, at very low temperature photons can simply disappear. However, for about a decade, this prediction was not taken at face value. Einstein himself wrote to Ehrenfest, “From a certain temperature on, the molecules ‘condense’ without attractive forces; that is, they accumulate at zero velocity. The theory is pretty, but is there some truth in it?” [2]. In 1927, George E. Uhlenbeck concluded that the predicted BEC phase transition was an artifact of the replacement of the summation over states by an integral and wrote that no “splitting into two phases” would occur [3].
Since the first experimental realization of Bose-Einstein condensation in cold atomic gases in 1995 there has been a surge of activity in this field. Ingenious experiments have allowed us to probe matter close to zero temperature and reveal some of the fascinating effects quantum mechanics has bestowed on nature. It is a challenge for mathematical physicists to understand these various phenomena from first principles, that is, starting from the underlying many-body Schrödinger equation. Recent progress in this direction concerns mainly equilibrium properties of dilute, cold quantum gases. We shall explain some of the results in this article, and describe the mathematics involved in understanding these phenomena. Topics include the ground state energy and the free energy at positive temperature, the effect of interparticle interaction on the critical temperature for Bose-Einstein condensation, as well as the occurrence of superfluidity and quantized vortices in rapidly rotating gases.
The problem of understanding how a coherent, macroscopic Bose- Einstein condensate (BEC) emerges from the cooling of a thermal Bose gas has attracted significant theoretical and experimental interest over several decades. The pioneering achievement of BEC in weakly interacting dilute atomic gases in 1995 was followed by a number of experimental studies examining the growth of the BEC number, as well as the development of its coherence. More recently, there has been interest in connecting such experiments to universal aspects of nonequilibrium phase transitions, in terms of both static and dynamical critical exponents. Here, the spontaneous formation of topological structures such as vortices and solitons in quenched cold-atom experiments has enabled the verification of the Kibble-Zurek mechanism predicting the density of topological defects in continuous phase transitions, first proposed in the context of the evolution of the early universe. This chapter reviews progress in the understanding of BEC formation and discusses open questions and future research directions in the dynamics of phase transitions in quantum gases. Introduction The equilibrium phase diagram of the dilute Bose gas exhibits a continuous phase transition between condensed and noncondensed phases. The order parameter characteristic of the condensed phase vanishes above some critical temperature T c and grows continuously with decreasing temperature below this critical point. However, the dynamical process of condensate formation has proved to be a challenging phenomenon to address both theoretically and experimentally. This formation process is a crucial aspect of Bose systems and of direct relevance to all condensates discussed in this book, despite their evident system-specific properties. Important questions leading to intense discussions in the early literature include the time scale for condensate formation and the role of inhomogeneities and finite-size effects in “closed” systems. These issues are related to the concept of spontaneous symmetry breaking, its causes, and implications for physical systems (see, for example, Chapter 5 by Snoke and Daley). In this chapter, we give an overview of the dynamics of condensate formation and describe the present understanding provided by increasingly well-controlled cold-atom experiments and corresponding theoretical advances over the past twenty years. We focus on the growth of BECs in cooled Bose gases, which, from a theoretical standpoint, requires a suitable nonequilibrium formalism.
The study of Bose-Einstein condensation has undergone a remarkable expansion during the last twenty years. Observations of this phenomenon have been reported in a number of diverse atomic, optical, and condensed matter systems, facilitated by remarkable experimental advances. The synergy of experimental and theoretical work in this broad research area is unique, leading to the establishment of Bose-Einstein condensation as a universal interdisciplinary area of modern physics. This chapter reviews the broad expansion of Bose-Einstein condensation physics in the past two decades. Introduction The field of Bose-Einstein condensation (BEC) has undergone an explosive expansion during the past twenty years. Newcomers to this field are now often introduced to this as a universal phenomenon, which nonetheless exhibits diverse (and sometimes strikingly different) manifestations. Despite such differences, the common underlying theme creates a unique identity across many different energy and length scales. The study of BEC as a universal phenomenon was highlighted in a focused conference in 1993, leading to the publication of the well-known “green book” [1], which surveyed the breadth of the field of condensate physics at that time. The success of the conference led to a second meeting in 1995, at which Eric Cornell and Carl Wieman announced the achievement of Bose-Einstein condensation of ultracold 87 Rb atoms in a harmonic trap. That work began an explosion of new research in the field of cold atoms, which has continued to this day, and this very success inevitably led many of those studying cold atoms to pay less attention to other types of condensates. Wolfgang Ketterle gives a historical overview of this exciting period of time in Chapter 3. Recently, various scientific meetings have worked to re-establish the physical connections across different BEC systems, and in 2013 a workshop was held with the focused goal of improving communications across disciplines. This present book is an outgrowth of that meeting. The Situation Before the Revolution Because of the great success of the cold atom BEC and other BEC systems in the past twenty years, it may be hard for young scientists to understand the climate of BEC research in the early 1990s. At that time, there was only one known example of BEC, namely liquid helium-4, and there was a small but vocal minority of scientists who questioned whether BEC was established even in that system.