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PART I
Introductory Material
Editors: Simon A. Gardiner and Nick P. Proukakis
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Editorial Notes
Simon A. Gardiner and Nick P. Proukakis
With Introductory Material, we provide important general background material, particularly in the field of ultracold atomic gases which forms the backbone of much of this book. Relative newcomers to the field will likely find this a necessary prelude before moving on to the generally more advanced material that follows, and even relative sophisticates should find much of the material contained within this part to be a useful resource.
The areas of atomic, molecular and optical, and condensed-matter physics considered within this book are fortunate in typically having a close connection between their experimental and theoretical communities. The prominence given to experimental matters in Introductory Material, within what is largely a book on theoretical approaches and methodology, is in recognition of the importance of this interaction. Theoretical physicists should never forget the connection between their models and the real systems in real laboratories that these models are intended to describe!
We have subdivided Introductory Material into Quantum Gases: The Background (Chapter 1), Quantum Gases: Experimental Considerations (Chapters 2 and 3), and Quantum Gases: Background Key Theoretical Notions Chapter 4. Chapter 1, Quantum Gases: Setting the Scene, presents a brief history of the field of quantum degeneracy in interacting quantum many-body systems, setting it within an interdisciplinary context, and addressing both the successes to date and the remaining unresolved issues. Chapter 2, Ultracold Quantum Gases: Experiments with Many-Body Systems in Controlled Environments, gives an overview of selected key experiments within the field of ultracold and quantum degenerate atomic and molecular ensembles which has experienced such impressive progress since the first Bose–Einstein condensates were formed in 1995, with some emphasis here on experiments associated with finite-temperature and non-equilibrium settings which form the key theme of this book. Moreover, Chapter 3, Ultracold Quantum Gases: Key Experimental Techniques, attempts to give a well-referenced, comprehensive overview of the most essential methodologies used in such ultracold atom experiments.
Finally, Chapter 4, Introduction to Theoretical Modelling, gives a brief but, we hope, clear treatment of the essential aspects of the theoretical treatment of interacting identical quantum particles, upon which the theoretical treatments appearing in subsequent chapters are founded. This includes second quantisation, effective interactions, issues of broken symmetry, low-dimensional systems, issues specific to fermions, lattice systems, and Feshbach resonances and molecule formation. This is intended to be accessible to the beginning graduate student, as well as being a useful consolidation of essential material for the existing expert.
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PART I.A
Quantum Gases: The Background
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Chapter 1
Quantum Gases: Setting the Scene
Nick P. Proukakis
Joint Quantum Centre (JQC) Durham–Newcastle,
School of Mathematics and Statistics, Newcastle University,
Newcastle upon Tyne NE1 7RU, UK
Keith Burnett
Firth Court, Western Bank, Sheffield University, Sheffield S10 2TN, UK
We give a brief historical overview of the physical realisations of quantum degeneracy and Bose–Einstein condensation observed to date, with the aim of showing why quantum gases is a key rapidly evolving interdisciplinary field of physics. We motivate the need for developing more advanced theories to understand all the features observed in ultracold gases, and present some of the unresolved issues where the theories discussed in this book can play an important role.
1.1. Introduction: Background to Quantum Fluids and Gases
The ability to produce Bose–Einstein condensates in the laboratory has brought about a revolution in our study of ultracold matter and its properties. This includes an important theme of this book: condensed and superfluid systems far from equilibrium. The new opportunities that the experimental revolution has produced are driving a wide range of science and technology. In this chapter we give a brief presentation of this rapidly expanding research area. We do not aim to give a comprehensive overview, but rather we focus on our current views of important issues discussed in this book, with an emphasis on ultracold weakly interacting bosonic atomic gases. (For a broader discussion see the numerous review articles [1–14] and books [15–22]).
The study of Bose–Einstein condensation (BEC) starts, of course, with Bose [23] and Einstein [24, 25]. Their work made predictions about the presence of a macroscopic occupation of the lowest-energy state, that is a condensate, in what we now call a Bose gas at sufficiently low temperatures. This temperature can be loosely associated with the point at which the de Broglie waves of individual atoms overlap and the identical nature of the particles becomes crucial. It is interesting to note that Schrödinger, in his influential work on statistical mechanics, confidently predicted that no such object could be produced due to the quantum statistical effects being overwhelmed by more mundane interaction effects [26, 27]. There were others, such as Uhlenbeck [28–30], who doubted the validity of the predictions for production of a pure and distinct condensate.
For a long while the existence of a condensate in superfluid and superconducting sytems was inferred from other emergent properties related to its presence [31]. These macroscopic quantum phenomena, related to the presence of a condensate, i.e. macroscopic occupation of a wavefunction, have been the subject of an enormous amount of fascinating and important physics. In the case of liquid helium the presence of a condensate was inferred from the nature of the excitations observed [32], with neutron scattering playing a key diagnostic role. This inference is tricky, as the condensed fraction in a superfluid, like liquid helium, is rather modest due to the effects of interactions. The clear and direct observation of a condensation had to await the developments of laser cooling [33–35] combined with evaporative cooling [36] of atomic gases, in which the condensate fraction approaches 100%.
The fact that it recently proved possible to make a ‘pure’ condensate in the laboratory [37–40] relies on the fact that all weakly interacting atomic gaseous condensates are dilute and transient. Pure condensates are now produced in the laboratory using dilute atomic systems, whose equilibrium state at these extremely low temperatures is a rather uninteresting solid metallic one. So we are producing intrinsically non-equilibrium systems but in a near-steady state close to the local equilibrium, before the onset of cluster formation. For many of the experiments we are interested in, the lifetime of this condensate is sufficient for important issues to be studied.
It has even proved possible to study some aspects of strong interactions and correlations in atomic gases via control of their kinetic energy in an optical lattice [3, 12, ...
