Oxford University Press eBooks

Basics on Bose‐Einstein condensation

David Guéry-Odelin, Thierry Lahaye

Published May 5, 2011• 1 citation

Abstract

This chapter introduces basic notions about Bose-Einstein condensates (BECs). One first studies ideal Bose gases, putting an emphasis on the role of the trapping geometry and of the dimensionality. One also introduces the correlation functions characterizing coherence. The second section deals with weakly-interacting BECs, which are described, in a mean-field approach, by the so-called Gross-Pitaevskii equation. This equation is derived and applied to a variety of experimentally relevant situations. The third section deals with beyond-mean-field effects that appear when the interactions are stronger, and introduces in particular the Bogolubov approximation. The last section is devoted to the study of BECs in double-well potentials, a situation inspired from condensed matter systems where beyond-mean-field effects can appear quite easily, and which remains simple enough to be studied theoretically in details.

DOI: 10.1093/acprof:oso/9780199603657.003.0001

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Oxford University Press eBooks

Basics on Bose‐Einstein condensation

David Guéry-Odelin, Thierry Lahaye

Published May 5, 2011• 1 citation

Publisher

Abstract

DOI: 10.1093/acprof:oso/9780199603657.003.0001

Related papers

Bose-Einstein Condensation

Dieter Jaksch

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 $^{4}$ He. 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.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 $^{4}$He. 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.

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This chapter introduces the fundamental concepts of Bose–Einstein condensates (BECs), providing a comprehensive theoretical foundation for subsequent investigation on their localization and numerical simulation. It begins with a historical overview of BECs, tracing their theoretical origins and experimental realization. The chapter then focuses on the theoretical perspective of scalar BECs and how it can be modelled by a single-component Gross–Pitaevskii (GP) equation in the mean-field description, a variable-coefficient nonlinear Schrödinger (NLS) equation encompassing a trapping potential and interatomic interaction. The mean-field description is then extended to govern the dynamics of vectorial condensates described by a coupled GP equation with intra and inter-species (atomic) interactions. Subsequently, the chapter discusses the origin of fermionic condensates, highlighting their distinct pairing mechanisms. Beyond mean-field approaches are then introduced to address the limitations of the GP framework, incorporating quantum fluctuations and correlations. The chapter concludes with an outline of the textbook, setting the stage for analytical and numerical investigations of ultracold atoms in ensuing chapters.This chapter introduces the fundamental concepts of Bose–Einstein condensates (BECs), providing a comprehensive theoretical foundation for subsequent investigation on their localization and numerical simulation. It begins with a historical overview of BECs, tracing their theoretical origins and experimental realization. The chapter then focuses on the theoretical perspective of scalar BECs and how it can be modelled by a single-component Gross–Pitaevskii (GP) equation in the mean-field description, a variable-coefficient nonlinear Schrödinger (NLS) equation encompassing a trapping potential and interatomic interaction. The mean-field description is then extended to govern the dynamics of vectorial condensates described by a coupled GP equation with intra and inter-species (atomic) interactions. Subsequently, the chapter discusses the origin of fermionic condensates, highlighting their distinct pairing mechanisms. Beyond mean-field approaches are then introduced to address the limitations of the GP framework, incorporating quantum fluctuations and correlations. The chapter concludes with an outline of the textbook, setting the stage for analytical and numerical investigations of ultracold atoms in ensuing chapters.

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Condensate dynamics at T = 0

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Trapped Bose-condensed atomic gases are remarkable because, in spite of the fact that these are very dilute systems, they exhibit robust coherent dynamic behaviour when perturbed. These quantum “wisps of matter” are a new phase of highly coherent matter. While binary collisions are very infrequent, the large coherent mean field associated with the Bose condensate ensures that interactions play a crucial role in determining the collective response of these trapped superfluid gases. In our discussion of the theory of collective oscillations of atomic condensates, the macroscopic Bose wavefunction Φ( r , t ) plays a central role. This wavefunction is the BEC order parameter. As discussed in Chapter 1, the initial attempts at defining this order parameter began with the pioneering work of London (1938a), were further developed by Bogoliubov (1947) and finally extended to deal with any Bose superfluid using the general quantum field theoretic formalism developed by Beliaev (1958a). Almost all this early theoretical work was limited to T = 0 where, in a dilute weakly interacting Bose gas, all the atoms are in the condensate. The first extension of these ideas to nonuniform Bose condensates was by Pitaevskii (1961) and, independently, by Gross (1961), which led to the now famous Gross–Pitaevskii (GP) equation of motion for Φ( r , t ). Before the discovery of BEC in trapped gases, the time-dependent GP equation was mainly used to study vortices in Bose superfluids, which involve a spatially nonuniform ground state. Apart from this application, the GP equation was largely unknown. The situation changed overnight in 1995 with the creation of trapped nonuniform Bose condensates in atomic gases.Trapped Bose-condensed atomic gases are remarkable because, in spite of the fact that these are very dilute systems, they exhibit robust coherent dynamic behaviour when perturbed. These quantum “wisps of matter” are a new phase of highly coherent matter. While binary collisions are very infrequent, the large coherent mean field associated with the Bose condensate ensures that interactions play a crucial role in determining the collective response of these trapped superfluid gases. In our discussion of the theory of collective oscillations of atomic condensates, the macroscopic Bose wavefunction Φ( r , t ) plays a central role. This wavefunction is the BEC order parameter. As discussed in Chapter 1, the initial attempts at defining this order parameter began with the pioneering work of London (1938a), were further developed by Bogoliubov (1947) and finally extended to deal with any Bose superfluid using the general quantum field theoretic formalism developed by Beliaev (1958a). Almost all this early theoretical work was limited to T = 0 where, in a dilute weakly interacting Bose gas, all the atoms are in the condensate. The first extension of these ideas to nonuniform Bose condensates was by Pitaevskii (1961) and, independently, by Gross (1961), which led to the now famous Gross–Pitaevskii (GP) equation of motion for Φ( r , t ). Before the discovery of BEC in trapped gases, the time-dependent GP equation was mainly used to study vortices in Bose superfluids, which involve a spatially nonuniform ground state. Apart from this application, the GP equation was largely unknown. The situation changed overnight in 1995 with the creation of trapped nonuniform Bose condensates in atomic gases.

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Since the dramatic discovery of Bose–Einstein condensation (BEC) in trapped atomic gases in 1995 (Anderson et al. , 1995), there has been an explosion of theoretical and experimental research on the properties of Bose-condensed dilute gases. The first phase of this research was discussed in the influential review article by Dalfovo et al. (1999) and in the proceedings of the 1998 Varenna Summer School on BEC (Inguscio et al. , 1999). More recently, this research has been well documented in two monographs, by Pethick and Smith (2008, second edition) and by Pitaevskii and Stringari (2003). Most of this research, both experimental and theoretical, has concentrated on the case of low temperatures (well below the BEC transition temperature, T BEC ), where one is effectively dealing with a pure Bose condensate. The total fraction of noncondensate atoms in such experiments can be as small as 10% of the total number of atoms and, equally importantly, this low-density cloud of thermally excited atoms is spread over a much larger spatial region compared with the high-density condensate, which is localized at the centre of the trapping potential. Thus most studies of Bose-condensed gases at low temperatures have concentrated entirely on the condensate degree of freedom and its response to various perturbations. This region is well described by the famous Gross–Pitaevskii (GP) equation of motion for the condensate order parameter Φ( r , t ). As shown by research since 1995, this pure condensate domain is very rich in physics. The main goal of the present book, in contrast, is to describe the dynamics of dilute trapped atomic gases at finite temperatures such that the noncondensate atoms also play an important role.Since the dramatic discovery of Bose–Einstein condensation (BEC) in trapped atomic gases in 1995 (Anderson et al. , 1995), there has been an explosion of theoretical and experimental research on the properties of Bose-condensed dilute gases. The first phase of this research was discussed in the influential review article by Dalfovo et al. (1999) and in the proceedings of the 1998 Varenna Summer School on BEC (Inguscio et al. , 1999). More recently, this research has been well documented in two monographs, by Pethick and Smith (2008, second edition) and by Pitaevskii and Stringari (2003). Most of this research, both experimental and theoretical, has concentrated on the case of low temperatures (well below the BEC transition temperature, T BEC ), where one is effectively dealing with a pure Bose condensate. The total fraction of noncondensate atoms in such experiments can be as small as 10% of the total number of atoms and, equally importantly, this low-density cloud of thermally excited atoms is spread over a much larger spatial region compared with the high-density condensate, which is localized at the centre of the trapping potential. Thus most studies of Bose-condensed gases at low temperatures have concentrated entirely on the condensate degree of freedom and its response to various perturbations. This region is well described by the famous Gross–Pitaevskii (GP) equation of motion for the condensate order parameter Φ( r , t ). As shown by research since 1995, this pure condensate domain is very rich in physics. The main goal of the present book, in contrast, is to describe the dynamics of dilute trapped atomic gases at finite temperatures such that the noncondensate atoms also play an important role.

Cambridge University Press eBooksFeb 19, 20092 citations

Equilibrium and Non-Equilibrium Properties of Dilute Bose Gases

Oldenburg, Jakob Felix

Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum eﬀects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an eﬀective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in diﬀerent, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in $L^2$ -norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for suﬃciently low densities as predicted in the physics literature first by Wu and subsequently others. $\textit{Full-text embargoed until: 2026-01-01}$ Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum eﬀects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an eﬀective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in diﬀerent, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in $L^2$-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for suﬃciently low densities as predicted in the physics literature first by Wu and subsequently others. $\textit{Full-text embargoed until: 2026-01-01}$

Dec 9, 20250 citations

Bose–Einstein condensates with long-range interactions

Paulsamy Muruganandam, Ramaswamy Radha

Institute of Physics eBooksNov 1, 20250 citations

^{4}

He. 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.

L^2

-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for suﬃciently low densities as predicted in the physics literature first by Wu and subsequently others.

\textit{Full-text embargoed until: 2026-01-01}