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arXiv:cond-mat/0610117 · Birkhäuser-Verlag eBooks

The Mathematics of the Bose Gas and its Condensation

Élliott H. Lieb, Robert Seiringer, Jan Philip Solovej, Jakob Yngvason

Published Jan 1, 2005• 334 citations
arXivHTMLPDF

Abstract

This book surveys results about the quantum mechanical many-body problem of the Bose gas that have been obtained by the authors over the last seven years. These topics are relevant to current experiments on ultra-cold gases; they are also mathematically rigorous, using many analytic techniques developed over the years to handle such problems. Some of the topics treated are the ground state energy, the Gross-Pitaevskii equation, Bose-Einstein condensation, superfluidity, one-dimensional gases, and rotating gases. The book also provides a pedagogical entry into the field for graduate students and researchers.

DOI
10.1007/b137508

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•Mar 25, 2004•585 citations
This book surveys results about the quantum mechanical many-body problem of the Bose gas that have been obtained by the authors over the last seven years. These topics are relevant to current experiments on ultra-cold gases; they are also mathematically rigorous, using many analytic techniques developed over the years to handle such problems. Some of the topics treated are the ground state energy, the Gross-Pitaevskii equation, Bose-Einstein condensation, superfluidity, one-dimensional gases, and rotating gases. The book also provides a pedagogical entry into the field for graduate students and researchers.
L Pitaevskii and S Stringari Oxford: Oxford University Press (2003) £55.50 (hardback), ISBN 0-19-850719-4 The Gross--Pitaevskii equation, named after one of the authors of the book, and its large number of applications for describing the properties of Bose--Einstein condensation (BEC) in trapped weakly interacting atomic gases, is the main topic of this book. In total the monograph comprises 18 chapters and is divided into two parts. Part I introduces the notion of BEC and superfluidity in general terms. The most important properties of the ideal and the weakly interacting Bose gas are described and the effects of nonuniformity due to an external potential at zero temperature are studied. The first part is then concluded with a summary of the properties of superfluid 4^{4}4He. In Part II the authors describe the theoretical aspects of BEC in harmonically trapped weakly interacting atomic gases. A short and rather rudimentary chapter on collisions and trapping of atomic gases which seems to be included for completeness only is followed by a detailed analysis of the ground state,\newpage\noindent collective excitations, thermodynamics, and vortices as well as mixtures of BECs and the Josephson effect in BEC. Finally, the last three chapters deal with topics of more recent interest like BEC in optical lattices, low dimensional systems, and cold Fermi gases. The book is well written and in fact it provides numerous useful and important relations between the different properties of a BEC and covers most of the aspects of ultracold weakly interacting atomic gases from the point of view of condensed matter physics. The book contains a comprehensive introduction to BEC for physicists new to the field as well as a lot of detail and insight for those already familiar with this area. I therefore recommend it to everyone who is interested in BEC. Very clearly however, the intention of the book is not to provide prospects for applications of BEC in atomic physics, quantum optics or quantum state engineering and therefore the more practically oriented reader might sometimes wonder why exactly an equation is termed useful or important.
Bose-Einstein condensation is an exciting physical phenomenon. It was predicted in 1924 and experimentally produced in 1995 using powerful laser cooling techniques. Bose condensed gases become one of most emerging field of modern physics, embracing many area namely atomic and molecular physics, nuclear physics, astrophysics, quantum optics, and condensed matter physics. The advent of these quantum degenerate gases opens intriguing prospects for precision measurements, quantum information, and novel phases of matter. The book presents a new thorough theory treating Bose condensed gases at finite temperatures with extremely distinct background. The approach proved useful, efficient and successful as a theoretical mean to study all aspects of ultracold gases at finite temperatures. Equilibrium and dynamic properties, superfluidity and vortices, anomalous correlations, quantum mixtures and long range interactions are among the topics covered in this book. The book is written in a pedagogical style and features sufficient elementary background material. It can be therefore, accessible to all scientists from graduate students to senior researchers in cold atoms physics.
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.
We present an overview of mathematical results on the low temperature properties of dilute quantum gases, which have been obtained in the past few years. The presentation includes a discussion of Bose–Einstein condensation, the excitation spectrum for trapped gases and its relation to superfluidity, as well as the appearance of quantized vortices in rotating systems. All these properties are intensely being studied in current experiments on cold atomic gases. We will give a description of the mathematics involved in understanding these phenomena, starting from the underlying many-body Schrödinger equation.
Abstract The book covers three main strands, Bose-Einstein condensates (BEC), superfluid and superconductivity. The subjects are developed in a way which gradually builds up the key concepts and the necessary mathematical machinery. The book begins with the simplest of the three topics, BEC, and first thoroughly reviews the basics of the Bose-Einstein ideal gas before going into the techniques of magnetic trapping and cooling of atoms and the discovery of BEC in dilute atomic gases. Superfluidity in helium-4 is more difficult to understand, because it is a strongly interacting quantum liquid. The book introduces the main physical phenomena of superfluidity, and how it follows from the key concept of macroscopic quantum coherence and off-diagonal long ranged order. The theory of superconductivity is developed in stages, starting with the simpler London and Ginzburg Landau theories and their main applications, before developing the mathematical concepts of a quantum coherent state and the Bardeen Cooper Schrieffer (BCS) theory. The final chapter deals with the more advanced topics, including superfluid helium-3 and evidence for unconventional Cooper pairing in exotic superconductors. No prior knowledge of quantum many-body theory is assumed, and the necessary mathematical concepts are introduced in a self-contained manner where needed.