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

Bose–Einstein Condensation and Superfluidity

Kerson Huang

Published Apr 6, 1995• 28 citations
Publisher

Abstract

We review generally accepted definitions of Bose–Einstein condensation and superfluidity, emphasizing that they are independent concepts. These ideas are illustrated in a dilute hard-sphere Bose gas, which is relevant to experiments on excitons and spin-aligned atomic hydrogen. We then discuss superfluid He in porous media, as simulated by different models in different regimes. At low coverage, we model it by a dilute hard-sphere Bose gas in random potentials, and show that superfluidity is destroyed through the pinning of the Bose condensate by the external potentials. At full coverage, we model the random medium by an ohmic network of random resistors, and argue that the superfluid transition is a percolation transition in d = 3, with critical exponent 1.7. This book is devoted to the phenomenon of Bose–Einstein condensation [1, 2] and inevitably, its relevance to superfluidity [3]. To provide some background for other articles in this volume, I would like to summarize some commonly accepted views on these phenomena, and illustrate them in the context of a dilute hard-sphere Bose gas, a model in which we have some control over the approximations made. I will also describe some recent work on the effect of randomness on the Bose condensate, which shows that Bose–Einstein condensation does not automatically give rise to superfluidity.

DOI
10.1017/cbo9780511524240.005

Related papers

Hard-sphere Bose gas in random external potentials

Kerson Huang, Hsin‐Fei Meng

We consider a dilute hard-sphere Bose gas in random external potentials at low temperatures, in D=3, using the technique of pseudopotentials and the Bogoliubov transformation. At absolute zero, the random potentials can deplete the Bose condensate, though not completely. On the other hand, they generate an amount of normal fluid equal to 4/3 the condensate depletion. This is a localization effect that can destroy superfluidity at absolute zero. General features of the superfluid density in the neighborhood of this transition point agree qualitatively with experimental results on helium in porous media.

Physical Review Letters•Jul 27, 1992•194 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

Bose-Einstein condensation in the alkali gases: Some fundamental concepts

Anthony J. Leggett

The author presents a tutorial review of some ideas that are basic to our current understanding of the phenomenon of Bose-Einstein condensation (BEC) in the dilute atomic alkali gases, with special emphasis on the case of two or more coexisting hyperfine species. Topics covered include the definition of and conditions for BEC in an interacting system, the replacement of the true interatomic potential by a zero-range pseudopotential, the time-independent and time-dependent Gross-Pitaevskii equations, superfluidity and rotational properties, the Josephson effect and related phenomena, and the Bogoliubov approximation.

Reviews of Modern Physics•Apr 24, 2001•1,960 citations

Bose-Einstein Condensation

This is the first book devoted to Bose–Einstein condensation (BEC) as an interdisciplinary subject, covering atomic and molecular physics, laser physics, low temperature physics and astrophysics. It contains 18 authoritative review articles on experimental and theoretical research in BEC and associated phenomena. Bose–Einstein condensation is a phase transition in which a macroscopic number of particles all go into the same quantum state. It has been known for some time that this phenomenon gives rise to superfluidity in liquid helium but recent research has focused on the search for BEC in other condensed matter systems, such as excitons, spin-polarised hydrogen, laser-cooled atoms, high-temperature superconductors and subatomic matter. This unique book gives an in-depth report on progress in this field and suggests promising research topics for the future. It will be of interest to graduate students and research workers in condensed matter, low temperature, atomic and laser physics.

Cambridge University Press eBooks•Apr 6, 1995•284 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

Bose-Einstein condensation and superfluidity of a dilute Bose gas in a random potential

Michikazu Kobayashi, Makoto Tsubota

There is a growing interest in the relation between Bose-Einstein condensation (BEC) and the superfluidity. A Bose system confined in random media such as porous glass is suitable for studying this relation because BEC and superfluidity can be suppressed and controlled in such a disordered environment. However, it is not clear how this relation is affected by disorder and there are few theoretical studies that can be quantitatively tested by experiment. In this work we develop a dilute Bose gas model with a random potential that takes into account the pore-size dependence of porous glass. Then we compare our model with the measured low-temperature specific heat, condensate density, and superfluid density of ${}^{4}\mathrm{He}$ in Vycor glass. This comparison uses no free parameters. We predict phenomena at low temperatures: First, the random potential causes a T-linear specific heat instead of the ${T}^{3}$ dependence that is usually caused by phonons. Second, the BEC can remain even when the superfluidity disappears at low densities. Third, the system makes a reentrant transition at low densities; that is, the superfluid phase changes to the normal phase again as the temperature is reduced. This reentrant transition is more likely to be observed when the strength of the random potential is increased.

arXiv:cond-mat/0202364•Physical review. B, Condensed matter•Nov 18, 2002•66 citations
We review generally accepted definitions of Bose–Einstein condensation and superfluidity, emphasizing that they are independent concepts. These ideas are illustrated in a dilute hard-sphere Bose gas, which is relevant to experiments on excitons and spin-aligned atomic hydrogen. We then discuss superfluid He in porous media, as simulated by different models in different regimes. At low coverage, we model it by a dilute hard-sphere Bose gas in random potentials, and show that superfluidity is destroyed through the pinning of the Bose condensate by the external potentials. At full coverage, we model the random medium by an ohmic network of random resistors, and argue that the superfluid transition is a percolation transition in d = 3, with critical exponent 1.7. This book is devoted to the phenomenon of Bose–Einstein condensation [1, 2] and inevitably, its relevance to superfluidity [3]. To provide some background for other articles in this volume, I would like to summarize some commonly accepted views on these phenomena, and illustrate them in the context of a dilute hard-sphere Bose gas, a model in which we have some control over the approximations made. I will also describe some recent work on the effect of randomness on the Bose condensate, which shows that Bose–Einstein condensation does not automatically give rise to superfluidity.
We consider a dilute hard-sphere Bose gas in random external potentials at low temperatures, in D=3, using the technique of pseudopotentials and the Bogoliubov transformation. At absolute zero, the random potentials can deplete the Bose condensate, though not completely. On the other hand, they generate an amount of normal fluid equal to 4/3 the condensate depletion. This is a localization effect that can destroy superfluidity at absolute zero. General features of the superfluid density in the neighborhood of this transition point agree qualitatively with experimental results on helium in porous media.
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 author presents a tutorial review of some ideas that are basic to our current understanding of the phenomenon of Bose-Einstein condensation (BEC) in the dilute atomic alkali gases, with special emphasis on the case of two or more coexisting hyperfine species. Topics covered include the definition of and conditions for BEC in an interacting system, the replacement of the true interatomic potential by a zero-range pseudopotential, the time-independent and time-dependent Gross-Pitaevskii equations, superfluidity and rotational properties, the Josephson effect and related phenomena, and the Bogoliubov approximation.
This is the first book devoted to Bose–Einstein condensation (BEC) as an interdisciplinary subject, covering atomic and molecular physics, laser physics, low temperature physics and astrophysics. It contains 18 authoritative review articles on experimental and theoretical research in BEC and associated phenomena. Bose–Einstein condensation is a phase transition in which a macroscopic number of particles all go into the same quantum state. It has been known for some time that this phenomenon gives rise to superfluidity in liquid helium but recent research has focused on the search for BEC in other condensed matter systems, such as excitons, spin-polarised hydrogen, laser-cooled atoms, high-temperature superconductors and subatomic matter. This unique book gives an in-depth report on progress in this field and suggests promising research topics for the future. It will be of interest to graduate students and research workers in condensed matter, low temperature, atomic and laser 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.
There is a growing interest in the relation between Bose-Einstein condensation (BEC) and the superfluidity. A Bose system confined in random media such as porous glass is suitable for studying this relation because BEC and superfluidity can be suppressed and controlled in such a disordered environment. However, it is not clear how this relation is affected by disorder and there are few theoretical studies that can be quantitatively tested by experiment. In this work we develop a dilute Bose gas model with a random potential that takes into account the pore-size dependence of porous glass. Then we compare our model with the measured low-temperature specific heat, condensate density, and superfluid density of 4He{}^{4}\mathrm{He}4He in Vycor glass. This comparison uses no free parameters. We predict phenomena at low temperatures: First, the random potential causes a T-linear specific heat instead of the T3{T}^{3}T3 dependence that is usually caused by phonons. Second, the BEC can remain even when the superfluidity disappears at low densities. Third, the system makes a reentrant transition at low densities; that is, the superfluid phase changes to the normal phase again as the temperature is reduced. This reentrant transition is more likely to be observed when the strength of the random potential is increased.