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Cambridge University Press eBooks

Nonequilibrium Bose–Einstein condensates

Esteban Calzetta, B. L. Hu

Published Jul 24, 2008• 0 citations
Publisher

Abstract

Bose–Einstein condensation was predicted in 1925 [Ein24, Ein25] but, except for its indirect manifestation in the superfluidity of He 4 [Lon38, Kap38, AllMis38], it remained a purely theoretical construct until 1995, when condensation in alkali gases was achieved in the laboratory [CorWie02, Ket02]. Since then, a great deal of the theoretical work in the previous 70 years has been put to the experimental test, while new avenues have been opening up, such as the superfluid-insulator or Mott transition [FWGF89, GMEHB02, CaHuRe06], the BEC-BCS cross-over [Leg06, Reg04] and the Tonks gas regime [Gir60, Par04]. Because of the great experimental control over the relevant parameters and the deep understanding of the fundamental physics, BECs have become a field of choice to perform experiments of interest not just in atomic and molecular physics, but also in quantum optics, condensed matter physics, quantum critical dynamics, even field theory, gravitation, cosmology and black hole physics [CoEnWi99, PetSmi02, Sou02, BonSen04, ParZha93, ParZha95, UnrSch07, BaLiVi05]. Moreover, cold Bose gases on optical lattices have been proposed as a possible implementation of a quantum information processing (QIP) device [JakZo104], boosting new interest in these systems. As introductions and reviews on this fast evolving subject abound, it is perhaps more fitting for us to focus on certain aspects of the nonequilibrium field theory of BEC, specifically [And04] the application of quantum field theoretic methods described in this book to the description of nonequilibrium evolution of condensed gases in magneto-optical traps [ChCoPh95].

DOI
10.1017/cbo9780511535123.014

Related papers

Foreword

Tony Leggett

At the time of the first workshop in this series in 1993, the only experimentally realized Bose condensate (at least in the simple sense conjectured by Einstein) was liquid 4 He. In the intervening twenty-plus years, much has happened in the world of Bose-Einstein condensation (BEC). Probably the most exciting development has been the attainment of condensation in ultracold bosonic atomic gases such as 87 Rb and 23 Na in 1995, followed a few years later by the achievement of degeneracy and eventually Bardeen-Cooper-Schrieffer (BCS) pairing in their fermionic counterparts, and the experimental realization of the theoretically long-anticipated “BEC-BCS crossover” by using the magnetic field degree of freedom to tune the system through a Feshbach resonance. One particularly fascinating aspect of the latter has been the realization of a “unitary gas” at the resonance itself – a system which prima facie has no characteristic length scale other than the interparticle separation, and is therefore a major challenge to theorists. Other systems in which BEC has been realized, sometimes transiently, include exciton-polariton complexes in semiconducting microcavities and, at least in a formal sense, the magnons in a magnetic insulator, as well as ultracold gases with a nontrivial and sometimes large “spin” degree of freedom. As compared with our “traditional” Bose condensate, liquid 4 He, these new systems typically have many more (and more rapidly adjustable) control parameters, and have therefore permitted qualitatively new types of experiment. One particularly fascinating development has been the use of optical techniques to generate “synthetic gauge fields” and thus mimic some of the topologically nontrivial systems which have recently been of such intense interest in a condensed-matter setting. At the same time, there remain long-standing issues from helium physics, such as the nature and consequences of “spontaneously broken U(1) symmetry,” the “Kibble-Zurek” mechanism, and more generally the relaxation of strongly nonequilibrium states to equilibrium; in some cases, the new systems have been used to address these more quantitatively than was possible with 4 He. The chapters in this volume address all of these questions and more, and should be of intense interest to both the experimental and the theoretical sides of the BEC community.

Cambridge University Press eBooks•Oct 4, 2018•0 citations

Bose–Einstein condensation

P. van der Straten, Harold J. Metcalf

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.

Cambridge University Press eBooks•Dec 31, 2015•8 citations

Bose-Condensed Gases at Finite Temperatures

Allan Griffin, Tetsuro Nikuni, E. Zaremba

The discovery of Bose Einstein condensation (BEC) in trapped ultracold atomic gases in 1995 has led to an explosion of theoretical and experimental research on the properties of Bose-condensed dilute gases. The first treatment of BEC at finite temperatures, this book presents a thorough account of the theory of two-component dynamics and nonequilibrium behaviour in superfluid Bose gases. It uses a simplified microscopic model to give a clear, explicit account of collective modes in both the collisionless and collision-dominated regions. Major topics such as kinetic equations, local equilibrium and two-fluid hydrodynamics are introduced at an elementary level. Explicit predictions are worked out and linked to experiments. Providing a platform for future experimental and theoretical studies on the finite temperature dynamics of trapped Bose gases, this book is ideal for researchers and graduate students in ultracold atom physics, atomic, molecular and optical physics and condensed matter physics.

Cambridge University Press eBooks•Feb 19, 2009•351 citations

Bose gas: Theory and Experiment

Alexander L. Fetter, Christopher J. Foot

For many years, $^4$He typified Bose-Einstein superfluids, but recent advances in dilute ultra-cold alkali-metal gases have provided new neutral superfluids that are particularly tractable because the system is dilute. This chapter starts with a brief review of the physics of superfluid $^4$He, followed by the basic ideas of Bose-Einstein condensation (BEC), first for an ideal Bose gas and then considering the effect of interparticle interactions, including time-dependent phenomena. Extensions to more exotic condensates include magnetic dipolar gases, mixtures of two components, and spinor condensates that require a focused infrared laser for trapping of all the various hyperfine magnetic states in a particular hyperfine $F$ manifold of $m_F$ states. With an applied rotation, the trapped BECs nucleate quantized vortices. Recent theory and experiment have shown that laser coupling fields can mimic the effect of rotation. The resulting synthetic gauge fields have produced vortices in a nonrotating condensate.

arXiv:1203.3183•Mar 14, 2012•2 citations

The first BEC conference in Levico in 1993

Allan Griffin

In opening the Levico BEC 2003 meeting, the directors thought it would be useful to say a little about the first BEC workshop held in the same hotel, in the same lecture room, almost exactly ten years ago. I want to give you a personal view of what led up to the historic BEC 93 workshop at Levico and contrast the situation then with the fantastic developments which have occurred over the last decade. BEC 93 was organized by Griffin, Snoke, Stringari, Laloe and Baym. The purpose of the workshop was to bring together people working in different fields of physics but with a common interest in the phenomenon of Bose–Einstein condensation. It is useful to briefly recall the history of BEC studies before the BEC 93 workshop, to understand why it was organized. In 1938 Fritz London suggested that the mysterious phase transition which occurred in liquid 4 He at T = 2.17 K might be related to the formation of a Bose–Einstein condensate (BEC) of 4 He atoms, a phenomenon first predicted by Einstein in 1925 in an ideal gas. After Tisza suggested that the spectacular superfluid effects discovered in 1938 were related to the motion of the Bose condensate, BEC became a hot topic in the years before World War II. In 1941, Landau developed a very successful two-fluid theory of superfluid 4 He, based on the relatively new concept of elementary excitations of a many-body system to describe the 'normal fluid' component. The Landau picture was spectacularly successful in explaining various thermodynamic and transport properties, albeit in a phenomenological manner. However, Landau's magnificent theory made no use of the concept of BEC, or even the fact that the 4 He atoms obeyed Bose statistics. Interest in BEC faded from low temperature conferences in the period 1945–1955, in spite of Bogoliubov's seminal work in 1947 showing that a dilute weakly interacting Bose gas still exhibited BEC. Quite separate from the study of BEC in superfluid 4 He, interest in finding a BEC in a dilute, weakly interacting Bose gas started to develop in the early 1960s. The first system was optically excited excitons in semiconductors (electron-hole pairs, with zero total spin, played the role of the bosons). However, the short lifetime of the excitons in most semiconductors frustrated the early experiments. A second major effort developed in the 1980s with attempts to produce an atomic BEC in spin-polarized hydrogen gas. The initial experiments were very promising, but various problems ultimately stopped people from ever producing (until 1998) high enough densities needed for BEC. However, this work revived interest in the theory of a dilute, weakly interacting Bose gas, and also led to several workshops on 'spin-polarized' quantum gases in the 1980s in which BEC was a major topic. The preceding summary sets the stage for why BEC 93 was organized, namely, to bring together people (both theorists and experimentalists!) interested in BEC in quite different systems, for encouragement as well as stimulation. The topic was largely ignored in the large international low temperature conferences. The organizers were mainly from the condensed matter and low temperature physics communities. Fortunately Franck Laloe was involved, and he was aware of the attempts to produce a BEC in trapped alkali gases using the novel technique of laser-cooling. The titles of the review talks given at BEC 93 show that the popular Bose systems in 1993 were liquid 4 He, excitons, spin-polarized hydrogen and small Cooper pairs in high-temperature superconductors. Work on laser-cooled alkali gases was reviewed by Yvan Castin, but people seemed pessimistic at the time about producing a stable BEC in these very dilute systems in the near future. We also had several excellent talks about Bose condensates involving mesons, as well as Cooper pairing in nuclei. The main reason for this was that crucial financial support for BEC 93 came from the newly formed ECT* Institute in Trento, thanks to Stringari. As a result, several theorists interested in condensates in nuclear and high energy physics were incorporated into the schedule of BEC 93. To my surprise, their talks, relating to Bose condensates in fairly exotic settings, fitted in very well with the other speakers. It was decided to hold BEC 93 in Levico-Terme at the Grand Hotel Bellavista. Stringari scouted many hotels in the area around Trento and Sandro clearly chose wisely, since we returned ten years later for Levico BEC 2003. Because of the interdisciplinary quality of the BEC 93 workshop and the slightly controversial nature of BEC in real systems, however, we found it very difficult to obtain funding for BEC 93. Many of the funding agencies turned down requests for funding so the BEC 93 workshop was funded by the ECT*, a small grant from CNRS in France, a payment from Cambridge University Press (in lieu of royalties for the conference proceedings) and registration fees. We were therefore overjoyed when we received some late applications from people who said they would pay the full registration fee. One of these was Eric Cornell, a young AMO physicist from 'someplace' called JILA in Boulder, Colorado. We gladly accepted his money and his participation. The opening session of BEC 93 in the Grand Hotel Bellavista was a very historic occasion, since almost all the people in the world interested in BEC were present. The excellent review talks and energetic discussions quickly made us realize that the workshop was going to be a success. People heard about the search for BEC in systems quite different from what they were familiar with. It probably was the first time that the CMP and AMO communities were in the same room together. Eric Cornell later told me that it was the first meeting where he met a lot of well known CMP theorists, and heard them arguing about the role of interactions, broken symmetry and order parameters, Goldstone modes, phase rigidity, etc. It dawned on Eric that the physics of a Bose condensate had not been completely settled in Einstein's 1925 paper! One of the star invited speakers was Nozières but at the last moment, he could not attend because of family reasons. However, he gave a lecture a month later at the 60th birthday party for Cohen-Tannoudji in Paris, based on his planned talk for BEC 93. I convinced him that his lecture 'belonged' to BEC 93 so he wrote it up for the BEC book we were preparing. His insightful article became one of the most quoted articles in the book, and many of his phrases (fragmentation of the condensate, phase locking, enemies of Bose–Einstein condensation, etc.) have entered our language. At the conference dinner, we gave out small prizes to some of the best speakers and participants who had made a significant contribution. One of the themes of BEC 93 was to pin down what an unambiguous signature of a Bose condensate was, and the term 'smoking gun' became very popular in discussions. Shlyapnikov received the coveted 'Smoking Gun' award, the prize being a package of cigarettes. The 'Prix d'Or' (1000 Lire), for giving the most convincing evidence for a Bose condensate, was awarded to Wolfe for his group's work on exciton gases. Some years later, in 1999, further work by his group at Urbana showed that an unexpected rapid exciton decay process had not been included in the original data analysis. As a result, the exciton densities had been over-estimated and they had not crossed into the BEC region. Shortly after, I received a letter from Wolfe telling me about the new results, and also the original 1000 Lire banknote he had received as a prize at BEC 93. It made me proud having colleagues like this. Brown, who argued that kaon condensates played a crucial role in the collapse of large stars, received the very special 'Smoking Universe' award. He gave the most impressive 'thank you' speech, worthy of the Academy Awards. At the end of BEC 93, we had an open discussion about whether we should organize another BEC workshop two years later, in 1995. Many people felt that, while BEC 93 had been very stimulating, there probably would not be enough progress to have another BEC meeting 'so soon', but a committee was set up to monitor developments. Walraven took the initiative to organize BEC 95, which was held in Mt St Odile near Strasbourg in France, in June 1995. As luck would have it, a few weeks before BEC 95 convened, rumours circulated that the JILA group at Boulder had clear evidence for a Bose condensate in a trapped gas of 87 Rb atoms. When Eric arrived in Mt St Odile from a meeting in Capri and presented the JILA data 'hot off the press', we all gave him a standing ovation. The earth had moved and BEC was about to move to centre stage. Another piece of luck was that our BEC 93 book ( Bose–Einstein Condensation 1995 ed A Griffin, D W Snoke and S Stringari (Cambridge: Cambridge University Press)) finally appeared in March 1995. The discovery of BEC in atomic gases a few months later made the book a modest 'bestseller' and it was quickly re-issued as a paperback (almost unheard of for this type of conference proceedings). Several people later congratulated us on producing such an excellent book on BEC so quickly after the discovery in June 1995! The BEC book (often called the 'Green Book') did fill a vacuum in the early days, providing carefully written reviews about many aspects of BEC. Several of the articles were well ahead of their time, such as the discussion of the BCS–BEC crossover in Fermi gases by Randeria. We are only now seeing a lot of references to his review article, as the search for BCS pairing in trapped Fermi gases has developed over the last year. Probably over half of all publications in the area of BEC and ultracold quantum gases since 1995 have made one or more references to the Green Book. Beginning with BEC 95, these biennial meetings have now become a regular affair, generously financed and sponsored by the European Science Foundation (ESF). BEC 99, 2001 and 2003 have all been held in San Feliu de Guixols, in Spain. Since BEC 95, these workshops have been dominated by people working on trapped atomic gases, for obvious reasons. However, it is nice to see that work has picked up recently in the search for BEC in exciton gases (especially in quantum wells), and the study of trapped Fermi gases has stimulated new connections with people working in nuclear physics as well as highly correlated electron systems. So one can see that the exciting interdisciplinary atmosphere of BEC 93 may return in the near future. The papers given at Levico BEC 2003 already show strong evidence of this trend. In their 2001 Nobel Prize lectures, Cornell, Wieman and Ketterle all emphasized how 'surprised' they were on how rich BEC physics turned out to be. Initially, they expected that these atomic condensates would be small and somewhat delicate, certainly only of minor interest to the larger physics community. In fact, as we all know, interest in ultracold gases has exploded since 1995 like a 'Bosenova' and, like the universe, there is no sign of this expansion stopping. It is rapidly producing a whole new field of research which is an amalgamation of AMO and CMP physics. This in turn is producing a major re-organization of all major physics departments, in terms of the hiring of new faculty and resource allocations. International conferences in more traditional research areas in CMP and AMO have been trying to incorporate this growing subject, but it is probably time to start a new 'International Conference on Ultracold Quantum Gases' which is open to all. Of course, the young physicists who have organized and are the main speakers at Levico BEC 2003 illustrate this on-going revolution. I predict that Bose matter waves, atomic optics and, more generally, phenomena in ultracold quantum gases, will be the basis for all really new technology in the 21st century. I say this with confidence, since I don't see any real competition from developing technologies based on electrons (discovered about 1900) and laser beams (discovered about 1960). As with these older technologies, it will take decades to see how cold quantum gases can be used, in ways that we cannot even imagine today. A crucial reason for the exciting future of ultracold gases is based on the fact that it involves a creative interaction between two major areas of physics, CMP and AMO. Each has their own well developed paradigms for thinking about physics, and this difference surfaces in dealing with cold quantum gases. This creates a certain tension but ultimately it is the basis of a new creativity. Many CMP theorists who had previously worked in the area of quantum fluids and superconductors have entered this new field. This is shown by the recent publication of two excellent books on cold quantum gases (Pethick C and Smith H 2002 Bose–Einstein Condensation in Dilute Gases (Cambridge: Cambridge University Press); Pitaevskii L and Stringari S 2003 Bose–Einstein Condensation (Oxford: Oxford University Press)), both by CMP theorists. This is easy to understand since trapped atomic gases (Bose or Fermi) are beautiful examples of interacting, highly coherent, quantum many-body systems, precisely the subject that CMP theorists have been struggling to understand since the 1950s. The new concepts and techniques needed to understand such many-body systems were developed and fought over in the period 1955–1965, when it was realized that there was more to the quantum theory of matter than solving Schrödinger's wave equation. A major contribution was made by the Landau school in Moscow. This battle of ideas was focused on the attempts to understand both superconductivity and superfluid 4 He. While the BCS theory of superconductivity in metals which appeared in 1957 clearly had the right physics to explain the key properties of superconductors, it still took some time for most theorists to really understand the BCS theory at a deeper level, in terms of broken symmetry, order parameters and Goldstone modes. The BCS theory gave CMP physics a simple microscopic model for superfluidity in a real system, in terms of the coherent motion of Cooper pairs. Very quickly, the close analogy was made between Bogoliubov's theory of a Bose-condensed gas and the BCS theory of superconductors. Strangely enough, however, it was only in the 1980s that pioneering work by Leggett, Nozières and others re-emphasized that one should think of the BCS theory as a Bose condensate of Cooper pairs. This led to an understanding of the BCS–BEC crossover, and also that the Bogoliubov theory of a weakly interacting Bose-condensed gas was 'buried' in the full BCS theory. It is fitting that, in 2003, this BCS–BEC crossover is becoming a central topic in the field of ultracold atoms. I say this because it represents a beautiful synthesis of both AMO physics (the formation of diatomic molecules using Feshbach resonances) and CMP physics (many-body effects giving rise to Cooper pairs) in a novel setting. As Levico BEC 2003 and the articles published in this special section show, we can expect more exciting advances of this kind in the near future. Acknowledgments I would like to thank the organizers of the Levico BEC 2003 workshop for inviting me to participate as a senior lecturer and also to commemorate the tenth anniversary of the first BEC workshop held in Levico-Terme. My research is funded by NSERC of Canada.

Journal of Physics B Atomic Molecular and Optical Physics•Mar 24, 2004•2 citations

Introduction

C. J. Pethick, H. Smith

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.

Cambridge University Press eBooks•Sep 11, 2008•0 citations
Bose–Einstein condensation was predicted in 1925 [Ein24, Ein25] but, except for its indirect manifestation in the superfluidity of He 4 [Lon38, Kap38, AllMis38], it remained a purely theoretical construct until 1995, when condensation in alkali gases was achieved in the laboratory [CorWie02, Ket02]. Since then, a great deal of the theoretical work in the previous 70 years has been put to the experimental test, while new avenues have been opening up, such as the superfluid-insulator or Mott transition [FWGF89, GMEHB02, CaHuRe06], the BEC-BCS cross-over [Leg06, Reg04] and the Tonks gas regime [Gir60, Par04]. Because of the great experimental control over the relevant parameters and the deep understanding of the fundamental physics, BECs have become a field of choice to perform experiments of interest not just in atomic and molecular physics, but also in quantum optics, condensed matter physics, quantum critical dynamics, even field theory, gravitation, cosmology and black hole physics [CoEnWi99, PetSmi02, Sou02, BonSen04, ParZha93, ParZha95, UnrSch07, BaLiVi05]. Moreover, cold Bose gases on optical lattices have been proposed as a possible implementation of a quantum information processing (QIP) device [JakZo104], boosting new interest in these systems. As introductions and reviews on this fast evolving subject abound, it is perhaps more fitting for us to focus on certain aspects of the nonequilibrium field theory of BEC, specifically [And04] the application of quantum field theoretic methods described in this book to the description of nonequilibrium evolution of condensed gases in magneto-optical traps [ChCoPh95].
At the time of the first workshop in this series in 1993, the only experimentally realized Bose condensate (at least in the simple sense conjectured by Einstein) was liquid 4 He. In the intervening twenty-plus years, much has happened in the world of Bose-Einstein condensation (BEC). Probably the most exciting development has been the attainment of condensation in ultracold bosonic atomic gases such as 87 Rb and 23 Na in 1995, followed a few years later by the achievement of degeneracy and eventually Bardeen-Cooper-Schrieffer (BCS) pairing in their fermionic counterparts, and the experimental realization of the theoretically long-anticipated “BEC-BCS crossover” by using the magnetic field degree of freedom to tune the system through a Feshbach resonance. One particularly fascinating aspect of the latter has been the realization of a “unitary gas” at the resonance itself – a system which prima facie has no characteristic length scale other than the interparticle separation, and is therefore a major challenge to theorists. Other systems in which BEC has been realized, sometimes transiently, include exciton-polariton complexes in semiconducting microcavities and, at least in a formal sense, the magnons in a magnetic insulator, as well as ultracold gases with a nontrivial and sometimes large “spin” degree of freedom. As compared with our “traditional” Bose condensate, liquid 4 He, these new systems typically have many more (and more rapidly adjustable) control parameters, and have therefore permitted qualitatively new types of experiment. One particularly fascinating development has been the use of optical techniques to generate “synthetic gauge fields” and thus mimic some of the topologically nontrivial systems which have recently been of such intense interest in a condensed-matter setting. At the same time, there remain long-standing issues from helium physics, such as the nature and consequences of “spontaneously broken U(1) symmetry,” the “Kibble-Zurek” mechanism, and more generally the relaxation of strongly nonequilibrium states to equilibrium; in some cases, the new systems have been used to address these more quantitatively than was possible with 4 He. The chapters in this volume address all of these questions and more, and should be of intense interest to both the experimental and the theoretical sides of the BEC community.
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.
The discovery of Bose Einstein condensation (BEC) in trapped ultracold atomic gases in 1995 has led to an explosion of theoretical and experimental research on the properties of Bose-condensed dilute gases. The first treatment of BEC at finite temperatures, this book presents a thorough account of the theory of two-component dynamics and nonequilibrium behaviour in superfluid Bose gases. It uses a simplified microscopic model to give a clear, explicit account of collective modes in both the collisionless and collision-dominated regions. Major topics such as kinetic equations, local equilibrium and two-fluid hydrodynamics are introduced at an elementary level. Explicit predictions are worked out and linked to experiments. Providing a platform for future experimental and theoretical studies on the finite temperature dynamics of trapped Bose gases, this book is ideal for researchers and graduate students in ultracold atom physics, atomic, molecular and optical physics and condensed matter physics.
For many years, 4^44He typified Bose-Einstein superfluids, but recent advances in dilute ultra-cold alkali-metal gases have provided new neutral superfluids that are particularly tractable because the system is dilute. This chapter starts with a brief review of the physics of superfluid 4^44He, followed by the basic ideas of Bose-Einstein condensation (BEC), first for an ideal Bose gas and then considering the effect of interparticle interactions, including time-dependent phenomena. Extensions to more exotic condensates include magnetic dipolar gases, mixtures of two components, and spinor condensates that require a focused infrared laser for trapping of all the various hyperfine magnetic states in a particular hyperfine FFF manifold of mFm_FmF​ states. With an applied rotation, the trapped BECs nucleate quantized vortices. Recent theory and experiment have shown that laser coupling fields can mimic the effect of rotation. The resulting synthetic gauge fields have produced vortices in a nonrotating condensate.
In opening the Levico BEC 2003 meeting, the directors thought it would be useful to say a little about the first BEC workshop held in the same hotel, in the same lecture room, almost exactly ten years ago. I want to give you a personal view of what led up to the historic BEC 93 workshop at Levico and contrast the situation then with the fantastic developments which have occurred over the last decade. BEC 93 was organized by Griffin, Snoke, Stringari, Laloe and Baym. The purpose of the workshop was to bring together people working in different fields of physics but with a common interest in the phenomenon of Bose–Einstein condensation. It is useful to briefly recall the history of BEC studies before the BEC 93 workshop, to understand why it was organized. In 1938 Fritz London suggested that the mysterious phase transition which occurred in liquid 4 He at T = 2.17 K might be related to the formation of a Bose–Einstein condensate (BEC) of 4 He atoms, a phenomenon first predicted by Einstein in 1925 in an ideal gas. After Tisza suggested that the spectacular superfluid effects discovered in 1938 were related to the motion of the Bose condensate, BEC became a hot topic in the years before World War II. In 1941, Landau developed a very successful two-fluid theory of superfluid 4 He, based on the relatively new concept of elementary excitations of a many-body system to describe the 'normal fluid' component. The Landau picture was spectacularly successful in explaining various thermodynamic and transport properties, albeit in a phenomenological manner. However, Landau's magnificent theory made no use of the concept of BEC, or even the fact that the 4 He atoms obeyed Bose statistics. Interest in BEC faded from low temperature conferences in the period 1945–1955, in spite of Bogoliubov's seminal work in 1947 showing that a dilute weakly interacting Bose gas still exhibited BEC. Quite separate from the study of BEC in superfluid 4 He, interest in finding a BEC in a dilute, weakly interacting Bose gas started to develop in the early 1960s. The first system was optically excited excitons in semiconductors (electron-hole pairs, with zero total spin, played the role of the bosons). However, the short lifetime of the excitons in most semiconductors frustrated the early experiments. A second major effort developed in the 1980s with attempts to produce an atomic BEC in spin-polarized hydrogen gas. The initial experiments were very promising, but various problems ultimately stopped people from ever producing (until 1998) high enough densities needed for BEC. However, this work revived interest in the theory of a dilute, weakly interacting Bose gas, and also led to several workshops on 'spin-polarized' quantum gases in the 1980s in which BEC was a major topic. The preceding summary sets the stage for why BEC 93 was organized, namely, to bring together people (both theorists and experimentalists!) interested in BEC in quite different systems, for encouragement as well as stimulation. The topic was largely ignored in the large international low temperature conferences. The organizers were mainly from the condensed matter and low temperature physics communities. Fortunately Franck Laloe was involved, and he was aware of the attempts to produce a BEC in trapped alkali gases using the novel technique of laser-cooling. The titles of the review talks given at BEC 93 show that the popular Bose systems in 1993 were liquid 4 He, excitons, spin-polarized hydrogen and small Cooper pairs in high-temperature superconductors. Work on laser-cooled alkali gases was reviewed by Yvan Castin, but people seemed pessimistic at the time about producing a stable BEC in these very dilute systems in the near future. We also had several excellent talks about Bose condensates involving mesons, as well as Cooper pairing in nuclei. The main reason for this was that crucial financial support for BEC 93 came from the newly formed ECT* Institute in Trento, thanks to Stringari. As a result, several theorists interested in condensates in nuclear and high energy physics were incorporated into the schedule of BEC 93. To my surprise, their talks, relating to Bose condensates in fairly exotic settings, fitted in very well with the other speakers. It was decided to hold BEC 93 in Levico-Terme at the Grand Hotel Bellavista. Stringari scouted many hotels in the area around Trento and Sandro clearly chose wisely, since we returned ten years later for Levico BEC 2003. Because of the interdisciplinary quality of the BEC 93 workshop and the slightly controversial nature of BEC in real systems, however, we found it very difficult to obtain funding for BEC 93. Many of the funding agencies turned down requests for funding so the BEC 93 workshop was funded by the ECT*, a small grant from CNRS in France, a payment from Cambridge University Press (in lieu of royalties for the conference proceedings) and registration fees. We were therefore overjoyed when we received some late applications from people who said they would pay the full registration fee. One of these was Eric Cornell, a young AMO physicist from 'someplace' called JILA in Boulder, Colorado. We gladly accepted his money and his participation. The opening session of BEC 93 in the Grand Hotel Bellavista was a very historic occasion, since almost all the people in the world interested in BEC were present. The excellent review talks and energetic discussions quickly made us realize that the workshop was going to be a success. People heard about the search for BEC in systems quite different from what they were familiar with. It probably was the first time that the CMP and AMO communities were in the same room together. Eric Cornell later told me that it was the first meeting where he met a lot of well known CMP theorists, and heard them arguing about the role of interactions, broken symmetry and order parameters, Goldstone modes, phase rigidity, etc. It dawned on Eric that the physics of a Bose condensate had not been completely settled in Einstein's 1925 paper! One of the star invited speakers was Nozières but at the last moment, he could not attend because of family reasons. However, he gave a lecture a month later at the 60th birthday party for Cohen-Tannoudji in Paris, based on his planned talk for BEC 93. I convinced him that his lecture 'belonged' to BEC 93 so he wrote it up for the BEC book we were preparing. His insightful article became one of the most quoted articles in the book, and many of his phrases (fragmentation of the condensate, phase locking, enemies of Bose–Einstein condensation, etc.) have entered our language. At the conference dinner, we gave out small prizes to some of the best speakers and participants who had made a significant contribution. One of the themes of BEC 93 was to pin down what an unambiguous signature of a Bose condensate was, and the term 'smoking gun' became very popular in discussions. Shlyapnikov received the coveted 'Smoking Gun' award, the prize being a package of cigarettes. The 'Prix d'Or' (1000 Lire), for giving the most convincing evidence for a Bose condensate, was awarded to Wolfe for his group's work on exciton gases. Some years later, in 1999, further work by his group at Urbana showed that an unexpected rapid exciton decay process had not been included in the original data analysis. As a result, the exciton densities had been over-estimated and they had not crossed into the BEC region. Shortly after, I received a letter from Wolfe telling me about the new results, and also the original 1000 Lire banknote he had received as a prize at BEC 93. It made me proud having colleagues like this. Brown, who argued that kaon condensates played a crucial role in the collapse of large stars, received the very special 'Smoking Universe' award. He gave the most impressive 'thank you' speech, worthy of the Academy Awards. At the end of BEC 93, we had an open discussion about whether we should organize another BEC workshop two years later, in 1995. Many people felt that, while BEC 93 had been very stimulating, there probably would not be enough progress to have another BEC meeting 'so soon', but a committee was set up to monitor developments. Walraven took the initiative to organize BEC 95, which was held in Mt St Odile near Strasbourg in France, in June 1995. As luck would have it, a few weeks before BEC 95 convened, rumours circulated that the JILA group at Boulder had clear evidence for a Bose condensate in a trapped gas of 87 Rb atoms. When Eric arrived in Mt St Odile from a meeting in Capri and presented the JILA data 'hot off the press', we all gave him a standing ovation. The earth had moved and BEC was about to move to centre stage. Another piece of luck was that our BEC 93 book ( Bose–Einstein Condensation 1995 ed A Griffin, D W Snoke and S Stringari (Cambridge: Cambridge University Press)) finally appeared in March 1995. The discovery of BEC in atomic gases a few months later made the book a modest 'bestseller' and it was quickly re-issued as a paperback (almost unheard of for this type of conference proceedings). Several people later congratulated us on producing such an excellent book on BEC so quickly after the discovery in June 1995! The BEC book (often called the 'Green Book') did fill a vacuum in the early days, providing carefully written reviews about many aspects of BEC. Several of the articles were well ahead of their time, such as the discussion of the BCS–BEC crossover in Fermi gases by Randeria. We are only now seeing a lot of references to his review article, as the search for BCS pairing in trapped Fermi gases has developed over the last year. Probably over half of all publications in the area of BEC and ultracold quantum gases since 1995 have made one or more references to the Green Book. Beginning with BEC 95, these biennial meetings have now become a regular affair, generously financed and sponsored by the European Science Foundation (ESF). BEC 99, 2001 and 2003 have all been held in San Feliu de Guixols, in Spain. Since BEC 95, these workshops have been dominated by people working on trapped atomic gases, for obvious reasons. However, it is nice to see that work has picked up recently in the search for BEC in exciton gases (especially in quantum wells), and the study of trapped Fermi gases has stimulated new connections with people working in nuclear physics as well as highly correlated electron systems. So one can see that the exciting interdisciplinary atmosphere of BEC 93 may return in the near future. The papers given at Levico BEC 2003 already show strong evidence of this trend. In their 2001 Nobel Prize lectures, Cornell, Wieman and Ketterle all emphasized how 'surprised' they were on how rich BEC physics turned out to be. Initially, they expected that these atomic condensates would be small and somewhat delicate, certainly only of minor interest to the larger physics community. In fact, as we all know, interest in ultracold gases has exploded since 1995 like a 'Bosenova' and, like the universe, there is no sign of this expansion stopping. It is rapidly producing a whole new field of research which is an amalgamation of AMO and CMP physics. This in turn is producing a major re-organization of all major physics departments, in terms of the hiring of new faculty and resource allocations. International conferences in more traditional research areas in CMP and AMO have been trying to incorporate this growing subject, but it is probably time to start a new 'International Conference on Ultracold Quantum Gases' which is open to all. Of course, the young physicists who have organized and are the main speakers at Levico BEC 2003 illustrate this on-going revolution. I predict that Bose matter waves, atomic optics and, more generally, phenomena in ultracold quantum gases, will be the basis for all really new technology in the 21st century. I say this with confidence, since I don't see any real competition from developing technologies based on electrons (discovered about 1900) and laser beams (discovered about 1960). As with these older technologies, it will take decades to see how cold quantum gases can be used, in ways that we cannot even imagine today. A crucial reason for the exciting future of ultracold gases is based on the fact that it involves a creative interaction between two major areas of physics, CMP and AMO. Each has their own well developed paradigms for thinking about physics, and this difference surfaces in dealing with cold quantum gases. This creates a certain tension but ultimately it is the basis of a new creativity. Many CMP theorists who had previously worked in the area of quantum fluids and superconductors have entered this new field. This is shown by the recent publication of two excellent books on cold quantum gases (Pethick C and Smith H 2002 Bose–Einstein Condensation in Dilute Gases (Cambridge: Cambridge University Press); Pitaevskii L and Stringari S 2003 Bose–Einstein Condensation (Oxford: Oxford University Press)), both by CMP theorists. This is easy to understand since trapped atomic gases (Bose or Fermi) are beautiful examples of interacting, highly coherent, quantum many-body systems, precisely the subject that CMP theorists have been struggling to understand since the 1950s. The new concepts and techniques needed to understand such many-body systems were developed and fought over in the period 1955–1965, when it was realized that there was more to the quantum theory of matter than solving Schrödinger's wave equation. A major contribution was made by the Landau school in Moscow. This battle of ideas was focused on the attempts to understand both superconductivity and superfluid 4 He. While the BCS theory of superconductivity in metals which appeared in 1957 clearly had the right physics to explain the key properties of superconductors, it still took some time for most theorists to really understand the BCS theory at a deeper level, in terms of broken symmetry, order parameters and Goldstone modes. The BCS theory gave CMP physics a simple microscopic model for superfluidity in a real system, in terms of the coherent motion of Cooper pairs. Very quickly, the close analogy was made between Bogoliubov's theory of a Bose-condensed gas and the BCS theory of superconductors. Strangely enough, however, it was only in the 1980s that pioneering work by Leggett, Nozières and others re-emphasized that one should think of the BCS theory as a Bose condensate of Cooper pairs. This led to an understanding of the BCS–BEC crossover, and also that the Bogoliubov theory of a weakly interacting Bose-condensed gas was 'buried' in the full BCS theory. It is fitting that, in 2003, this BCS–BEC crossover is becoming a central topic in the field of ultracold atoms. I say this because it represents a beautiful synthesis of both AMO physics (the formation of diatomic molecules using Feshbach resonances) and CMP physics (many-body effects giving rise to Cooper pairs) in a novel setting. As Levico BEC 2003 and the articles published in this special section show, we can expect more exciting advances of this kind in the near future. Acknowledgments I would like to thank the organizers of the Levico BEC 2003 workshop for inviting me to participate as a senior lecturer and also to commemorate the tenth anniversary of the first BEC workshop held in Levico-Terme. My research is funded by NSERC of Canada.
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.