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

arXiv:2301.10997 · Physical Review A

Lee-Yang theory of Bose-Einstein condensation

Fredrik Brange, Tuomas Pyhäranta, Eppu Heinonen, Kay Brandner, Christian Flindt

Published Mar 23, 2023• 10 citations
arXivHTMLPDF

Abstract

Bose gases form into Bose-Einstein condensates at a certain transition temperature, at which a large fraction of bosons in the gas become cool enough to occupy the lowest quantum state together. Here, taking inspiration from recent advances in Lee-Yang theories of phase transitions, the authors demonstrate a method to predict this condensation temperature for an ideal gas of billions of bosons using the energy fluctuations of less than a hundred bosons.

DOI
10.1103/physreva.107.033324

Related papers

Bose–Einstein Condensation and Superfluidity

William J. Mullin

Bose–Einstein condensation is a phenomenon, first proved by Einstein, in which a gas of bosons condenses into the lowest quantum state below a certain critical temperature. It occurs due to the symmetry, under particle exchange, of the boson wave function. The chapter discusses temperature, the Kelvin scale, the concept of absolute zero, and how laser cooling can reach ultralow temperatures. The property of bosons in which they often prefer to be in the same state is seen to apply to lasers, superfluid liquid helium, and to dilute nearly ideal gases. The latter are trapped in magnetic or optical potential wells where the condensation can be observed directly. These large condensate matter waves show quantum interference in a variety of possible experiments.

Oxford University Press eBooks•Mar 16, 2017•233 citations

Ideal Bose–Einstein system

Allen L. Wasserman

From a certain temperature on, the molecules “condense” without attractive forces; that is, they accumulate at zero velocity. The theory is pretty, but is there also some truth to it? Albert Einstein, Letter to Ehrenfest (Dec. 1924). Abraham Pais, Subtle Is the Lord: The Science and the Life of Albert Einstein , Oxford University Press, New York (1982) Introduction For over 50 years the low-temperature liquid state of uncharged, spinless He 4 was the only system in which a Bose–Einstein (BE) condensation was considered experimentally realized. In that case, cold (<2.19K) liquid He 4 passes into an extraordinary phase of matter called a superfluid , in which the liquid's viscosity and entropy become zero. With advances in atomic cooling (to ≈ 10 –9 K) the number of Bose systems which demonstrably pass into a condensate has considerably increased. These include several isotopes of alkali gas atoms as well as fermionic atoms that pair into integerspin (boson) composites. Although an ideal Bose gas does exhibit a low-temperature critical instability, the ideal BE gas theory is not, on its own, able to describe the BE condensate wave state. In order for a theory of bosons to account for a condensate wave state, interactions between the bosons must be included. Nevertheless, considerable interesting physics is contained in the ideal Bose gas model.

Cambridge University Press eBooks•Oct 27, 2011•0 citations

Critical nature of ideal Bose-Einstein condensation: Similarity with Yang-Lee theory of phase transition

Xianzhi Wang, Jai Sam Kim

It is found that the singularity of the thermodynamic potential of an ideal Bose gas is connected with the physical root ${z}_{0}=1$ of the inverse of the grand partition function. The critical nature is determined solely by the behavior of the root distribution of the inverse of the grand partition function near ${z}_{0}=1$. This is quite similar to the situation of phase transition described by Yang-Lee theory.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jan 1, 1999•6 citations

Bose-Einstein condensation: what, how and beyond

Mittar, Vishwa

The piling up of a macroscopic fraction of noninteracting bosons in the lowest energy state of a system at very low temperatures is known as Bose-Einstein condensation. It took nearly 70 years to observe the condensate after their theoretical prediction. A brief history of the relevant developments, essentials of the basic theory, physics of the steps involved in producing the condensate in a gas of alkali atoms together with the pertinent theory, and some important features of the research work carried out in the last about 25 years have been dealt with. An effort has been made to present the material in a manner that it can be easily followed by undergraduate students as well as non-specialists and may even be used for classroom teaching.

arXiv:2203.07038•Mar 5, 2022•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

Bose-Einstein condensation

C. G. Townsend, Wolfgang Ketterle, S. Stringari

In 1924 the Indian physicist Satyendra Nath Bose sent Einstein a paper in which he derived the Planck law for black-body radiation by treating the photons as a gas of identical particles. Einstein generalized Bose's theory to an ideal gas of identical atoms or molecules for which the number of particles is conserved and, in the same year, predicted that at sufficiently low temperatures the particles would become locked together in the lowest quantum state of the system. We now know that this phenomenon, called Bose-Einstein condensation (BEC), only happens for “bosons” – particles with a total spin that is an integer multiple of h, the Planck constant divided by 2π.

Physics World•Mar 1, 1997•2,623 citations
Bose gases form into Bose-Einstein condensates at a certain transition temperature, at which a large fraction of bosons in the gas become cool enough to occupy the lowest quantum state together. Here, taking inspiration from recent advances in Lee-Yang theories of phase transitions, the authors demonstrate a method to predict this condensation temperature for an ideal gas of billions of bosons using the energy fluctuations of less than a hundred bosons.
Bose–Einstein condensation is a phenomenon, first proved by Einstein, in which a gas of bosons condenses into the lowest quantum state below a certain critical temperature. It occurs due to the symmetry, under particle exchange, of the boson wave function. The chapter discusses temperature, the Kelvin scale, the concept of absolute zero, and how laser cooling can reach ultralow temperatures. The property of bosons in which they often prefer to be in the same state is seen to apply to lasers, superfluid liquid helium, and to dilute nearly ideal gases. The latter are trapped in magnetic or optical potential wells where the condensation can be observed directly. These large condensate matter waves show quantum interference in a variety of possible experiments.
From a certain temperature on, the molecules “condense” without attractive forces; that is, they accumulate at zero velocity. The theory is pretty, but is there also some truth to it? Albert Einstein, Letter to Ehrenfest (Dec. 1924). Abraham Pais, Subtle Is the Lord: The Science and the Life of Albert Einstein , Oxford University Press, New York (1982) Introduction For over 50 years the low-temperature liquid state of uncharged, spinless He 4 was the only system in which a Bose–Einstein (BE) condensation was considered experimentally realized. In that case, cold (<2.19K) liquid He 4 passes into an extraordinary phase of matter called a superfluid , in which the liquid's viscosity and entropy become zero. With advances in atomic cooling (to ≈ 10 –9 K) the number of Bose systems which demonstrably pass into a condensate has considerably increased. These include several isotopes of alkali gas atoms as well as fermionic atoms that pair into integerspin (boson) composites. Although an ideal Bose gas does exhibit a low-temperature critical instability, the ideal BE gas theory is not, on its own, able to describe the BE condensate wave state. In order for a theory of bosons to account for a condensate wave state, interactions between the bosons must be included. Nevertheless, considerable interesting physics is contained in the ideal Bose gas model.
It is found that the singularity of the thermodynamic potential of an ideal Bose gas is connected with the physical root z0=1{z}_{0}=1z0​=1 of the inverse of the grand partition function. The critical nature is determined solely by the behavior of the root distribution of the inverse of the grand partition function near z0=1{z}_{0}=1z0​=1. This is quite similar to the situation of phase transition described by Yang-Lee theory.
The piling up of a macroscopic fraction of noninteracting bosons in the lowest energy state of a system at very low temperatures is known as Bose-Einstein condensation. It took nearly 70 years to observe the condensate after their theoretical prediction. A brief history of the relevant developments, essentials of the basic theory, physics of the steps involved in producing the condensate in a gas of alkali atoms together with the pertinent theory, and some important features of the research work carried out in the last about 25 years have been dealt with. An effort has been made to present the material in a manner that it can be easily followed by undergraduate students as well as non-specialists and may even be used for classroom teaching.
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
In 1924 the Indian physicist Satyendra Nath Bose sent Einstein a paper in which he derived the Planck law for black-body radiation by treating the photons as a gas of identical particles. Einstein generalized Bose's theory to an ideal gas of identical atoms or molecules for which the number of particles is conserved and, in the same year, predicted that at sufficiently low temperatures the particles would become locked together in the lowest quantum state of the system. We now know that this phenomenon, called Bose-Einstein condensation (BEC), only happens for “bosons” – particles with a total spin that is an integer multiple of h, the Planck constant divided by 2π.