Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Cambridge University Press eBooks

Overview and introduction

Allan Griffin, Tetsuro Nikuni, E. Zaremba

Published Feb 19, 2009• 2 citations
Publisher

Abstract

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.

DOI
10.1017/cbo9780511575150.002

Related papers

Condensate dynamics at T = 0

Allan Griffin, Tetsuro Nikuni, E. Zaremba

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.

Cambridge University Press eBooks•Feb 19, 2009•0 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-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.

Journal of Physics A Mathematical and General•Sep 2, 2003•1,020 citations

Coupled equations for the condensate and thermal cloud

Allan Griffin, Tetsuro Nikuni, E. Zaremba

In contrast with Chapter 2, in this chapter we include the dynamics of the thermal cloud. As noted in Chapter 1, we treat the noncondensate atoms using the simplest microscopic model approximation that captures the important physics. In particular, we consider only temperatures high enough ( T ≥ 0.4 T BEC ) that the noncondensate atoms can be described by a particle-like Hartree–Fock (HF) spectrum. To extend the analysis to very low temperatures is in principle straightforward (see Chapter 7). However, the details are more complicated since the excitations of the thermal cloud take on a collective aspect (i.e. a Bogoliubov-type quasiparticle spectrum must be used). In trapped Bose gases, the HF single-particle spectrum gives a good approximation down to much lower temperatures than in the case of uniform Bose gases, as first emphasized by Giorgini et al . (1997). In Section 3.1, we derive a generalized form of the Gross–Pitaevskii equation for the Bose order parameter Φ( r , t ) that is valid at finite temperatures. It involves terms that are coupled to the noncondensate component (the thermal cloud) and thus its solution in general requires one to know the equations of motion for the dynamics of the noncondensate atoms. In Section 3.2 we restrict ourselves to finite temperatures high enough that the noncondensate atoms can be described by a quantum kinetic equation for the single-particle distribution function f ( p, r , t ). A detailed microscopic derivation of this kind of kinetic equation is given in Chapters 6 and 7 using the Kadanoff–Baym Green's function formalism. A characteristic feature of a Bose-condensed gas is that the kinetic equation governing f ( p, r , t ) involves a collision integral C 12 [ f , Φ] describing collisions between condensate and noncondensate atoms.

Cambridge University Press eBooks•Feb 19, 2009•0 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. $\textit{Full-text embargoed until: 2026-01-01}$

•Dec 9, 2025•0 citations

Termodinamica di bosoni confinati e interagenti

Matteo Ferraretto

The Bose-Einstein condensation (BEC) is a quantum phenomenon which was theorized in 1920s, but it was realized for the first time in the JILA Laboratory (University of Colorado, Boulder) by Eric Cornell and Carl Wieman only in 1995. Since 1995, lots of experiments on BEC have been realized using different gases and various conditions. One can give a correct interpretation of the results of these works only by considering the role of interaction between particles on relevant physical quantities, such as critical temperature and condensed fraction. The purpose of our work is deducing the main thermodynamic properties of trapped and interacting Bose gases using the tools of Quantum Mechanics, avoiding the introduction of (which often recurs in litterature). The first chapter shows a derivation of the famous Gross-Pitaevskii equation, starting from the many-bodies Dirac action and applying the principle of least action. Moreover, two approximate solutions of this equation are discussed as well: the gaussian variational ansatz for weakly interacting particles, and the Thomas-Fermi approximation, for strongly interacting particles. In chapter 2 we derive the Bogoliubov dispersion relation for excited states (or quasiparticle energy spectrum) with the tools of Hamiltonian Mechanics and semiclassical approximation. Furthermore, we discuss some approximations for this formula (especially the Hartree-Fock spectrum for weakly interacting bosons) and we use the Bose-Einstein distribution to find the thermal cloud density. In chapter 3 we use the previous results to calculate explicit expressions for critical temperature and condensed fraction (where possible) in four particular cases: ideal free and trapped Bose gas, interacting free and trapped Bose gas. We also discuss the main results obtained by Giorgini, Pitaevskii and Stringari, who managed to explain accurately Cornell and Wieman's experimental plots. Finally we conclude that traditional Quantum Mechanics is quite efficient to derive the main equations of BEC and that introducing second quantization is not essential. Nevertheless, some issues remain unresolved in this context: for example the phenomenon of quantum depletion and the generalisation of the Gross-Pitaevskii equation at non-zero temperature.

•Sep 1, 2016•0 citations
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.
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.
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.
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.
In contrast with Chapter 2, in this chapter we include the dynamics of the thermal cloud. As noted in Chapter 1, we treat the noncondensate atoms using the simplest microscopic model approximation that captures the important physics. In particular, we consider only temperatures high enough ( T ≥ 0.4 T BEC ) that the noncondensate atoms can be described by a particle-like Hartree–Fock (HF) spectrum. To extend the analysis to very low temperatures is in principle straightforward (see Chapter 7). However, the details are more complicated since the excitations of the thermal cloud take on a collective aspect (i.e. a Bogoliubov-type quasiparticle spectrum must be used). In trapped Bose gases, the HF single-particle spectrum gives a good approximation down to much lower temperatures than in the case of uniform Bose gases, as first emphasized by Giorgini et al . (1997). In Section 3.1, we derive a generalized form of the Gross–Pitaevskii equation for the Bose order parameter Φ( r , t ) that is valid at finite temperatures. It involves terms that are coupled to the noncondensate component (the thermal cloud) and thus its solution in general requires one to know the equations of motion for the dynamics of the noncondensate atoms. In Section 3.2 we restrict ourselves to finite temperatures high enough that the noncondensate atoms can be described by a quantum kinetic equation for the single-particle distribution function f ( p, r , t ). A detailed microscopic derivation of this kind of kinetic equation is given in Chapters 6 and 7 using the Kadanoff–Baym Green's function formalism. A characteristic feature of a Bose-condensed gas is that the kinetic equation governing f ( p, r , t ) involves a collision integral C 12 [ f , Φ] describing collisions between condensate and noncondensate atoms.
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 L2L^2L2-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. Full-text embargoed until: 2026-01-01\textit{Full-text embargoed until: 2026-01-01}Full-text embargoed until: 2026-01-01
The Bose-Einstein condensation (BEC) is a quantum phenomenon which was theorized in 1920s, but it was realized for the first time in the JILA Laboratory (University of Colorado, Boulder) by Eric Cornell and Carl Wieman only in 1995. Since 1995, lots of experiments on BEC have been realized using different gases and various conditions. One can give a correct interpretation of the results of these works only by considering the role of interaction between particles on relevant physical quantities, such as critical temperature and condensed fraction. The purpose of our work is deducing the main thermodynamic properties of trapped and interacting Bose gases using the tools of Quantum Mechanics, avoiding the introduction of (which often recurs in litterature). The first chapter shows a derivation of the famous Gross-Pitaevskii equation, starting from the many-bodies Dirac action and applying the principle of least action. Moreover, two approximate solutions of this equation are discussed as well: the gaussian variational ansatz for weakly interacting particles, and the Thomas-Fermi approximation, for strongly interacting particles. In chapter 2 we derive the Bogoliubov dispersion relation for excited states (or quasiparticle energy spectrum) with the tools of Hamiltonian Mechanics and semiclassical approximation. Furthermore, we discuss some approximations for this formula (especially the Hartree-Fock spectrum for weakly interacting bosons) and we use the Bose-Einstein distribution to find the thermal cloud density. In chapter 3 we use the previous results to calculate explicit expressions for critical temperature and condensed fraction (where possible) in four particular cases: ideal free and trapped Bose gas, interacting free and trapped Bose gas. We also discuss the main results obtained by Giorgini, Pitaevskii and Stringari, who managed to explain accurately Cornell and Wieman's experimental plots. Finally we conclude that traditional Quantum Mechanics is quite efficient to derive the main equations of BEC and that introducing second quantization is not essential. Nevertheless, some issues remain unresolved in this context: for example the phenomenon of quantum depletion and the generalisation of the Gross-Pitaevskii equation at non-zero temperature.