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

Physical Review A

Observation of Bose-Einstein condensation in an atomic trap in terms of macroscopic thermodynamic parameters

Vı́ctor Romero-Rochı́n, R. F. Shiozaki, M. A. Caracanhas, E. A. L. Henn, K. M. F. Magalhães, G. Roati, Vanderlei Salvador Bagnato

Published Feb 24, 2012• 15 citations
Publisher

Abstract

To overcome the difficulties in defining pressure for a gas confined in an inhomogeneous trap, we define single macroscopic parameters that behave like pressure and volume. We measure the phase diagram of a ${}^{87}$Rb Bose gas in a harmonic trap in terms of those macroscopic parameters obtained from the spatial distribution of atoms. Considering the relevant variables such as the trap potential $\mathcal{V}={({\ensuremath{\omega}}_{x}{\ensuremath{\omega}}_{y}{\ensuremath{\omega}}_{z})}^{\ensuremath{-}1}$, number of atoms $N$, and temperature $T$, a parameter $\ensuremath{\Pi}=\ensuremath{\Pi}(N,\mathcal{V},T)$ is introduced to characterize the overall macroscopic pressure of the system. We construct the phase diagram ($\ensuremath{\Pi}$ vs $T$) identifying the main features related to the Bose-Einstein condensation (BEC) transition in a trapped gas. A thermodynamic description of the phase transition based on purely macroscopic parameters provides us with a description that does not need the local-density approximation. This procedure can be used to explore different aspects related to BEC such as the nature of the phase transition in a trapped gas.

DOI
10.1103/physreva.85.023632

Related papers

BEC phase diagram of a $^{87}$Rb trapped gas in terms of macroscopic thermodynamic parameters

Vı́ctor Romero-Rochı́n, R. F. Shiozaki, M. A. Caracanhas +4 more

We measure the phase diagram of a $^{87}$Rb Bose gas in a harmonic trap in terms of macroscopic parameters obtained from the spatial distribution of atoms. Considering the relevant variables as size of the cloud ${\cal V}$, number of atoms $N$ and temperature $T$, a novel parameter $Π= Π(N,{\cal V},T)$ is introduced to characterize the overall pressure of the system. We construct the phase diagram ($Π$ vs $T$) identifying new features related to Bose-Einstein condensation (BEC) transition in a trapped gas. A thermodynamic description of the phase transition based on purely macroscopic parameters, provide us with properties that do not need the local density approximation. An unexpected consequence of this analysis is the suggestion that BEC appears as a continuous third-order phase transition instead of being a second-order one.

arXiv:1102.4642•Feb 23, 2011•0 citations

Technical thermodynamics of an inhomogeneous gas around the Bose-Einstein transition using the global-variable method

Marcos Miotti

Thermodynamics of non-uniform systems has always been a challenging topic in the physical sciences.In the field of quantum gases, that subject is mainly studied with ultracold samples trapped in a harmonic potential, the first kind of confinement ever made to atomic fluids. 1 Nascimbène et al. 2 have already described a method to measure locally the equation of state and heat capacity of a harmonically trapped quantum gas.Nonetheless, local measurements cannot provide the volume-dependent susceptibilities of an inhomogeneous system, namely the thermal expansion and compressibility, meaning that the thermodynamic description of harmonically trapped gases is still incomplete.To solve that issue, Romero and Bagnato 3, 4 proposed canonical variables of work for a gas in a harmonic trap, which work like pressure and volume for a gas in a vessel and allow global measurements on the system.Therefore, we describe here our methodology to measure the thermodynamic susceptibilities of a harmonically trapped gas around the Bose-Einstein transition using the Romero-Bagnato formalism.Our experiments were divided in different phases, each having the harmonic trap with unique frequencies, related to the formalism's extensive variable.In each phase, we prepared individual samples of rubidium-87 gas in the trap and imaged them using the time-of-flight technique.Next, we fitted our images with the bimodal model and found the in situ density profiles of the samples with the standard regression procedures.Then, we determined the phase's equation of state as a function of temperature and number of atoms.For the thermodynamic analysis, we used an empirical model that we developed to fit the equation-of-state curves, allowing us to represent our data in terms of mathematical coefficients.In that way, we found the curves of internal energy, heat capacity, thermal expansion and compressibility, thus achieving a full thermodynamic description of a harmonically trapped gas.Our method and its results are unprecedented in the literature and might contribute to the further understanding of non-uniform systems, as well as the future development of quantum thermal engines.

•Oct 13, 2021•2 citations

Isothermal compressibility determination across Bose-Einstein condensation

Freddy Jackson Poveda-Cuevas, P. C. M. Castilho, E. D. Mercado-Gutiérrez +5 more

We apply the global thermodynamic variables approach to experimentally determine the isothermal compressibility parameter ${\ensuremath{\kappa}}_{T}$ of a trapped Bose gas across the phase transition. We demonstrate the behavior of ${\ensuremath{\kappa}}_{T}$ around the critical pressure, revealing the second-order nature of the phase transition. Compressibility is the most important susceptibility to characterize the system. The use of global variables shows advantages with respect to the usual local density approximation method and can be applied to a broad range of situations.

arXiv:1607.04897•Physical Review A•Jul 31, 2015•16 citations

Critical exponents and fluctuations at BEC in a 2D harmonically trapped ideal gas

M. I. Morales-Amador, Vı́ctor Romero-Rochı́n, R. Paredes

Abstract The critical properties displayed by an ideal 2D Bose gas trapped in a harmonic potential are determined and characterized in an exact numerical fashion. Beyond thermodynamics, addressed in terms of the global pressure and volume which are the appropriate variables of a fluid confined in a non-uniform harmonic potential, the density-density correlation function is also calculated and the corresponding correlation length is found. Evaluation of all these quantities as Bose–Einstein condensation (BEC) is approached manifest its critical continuous phase transition character. The divergence of the correlation length as the critical temperature is reached, unveils the expected spatial scale invariance proper of a critical transition. The logarithmic singularities of this transition are traced back to the non-analytic behavior of the thermodynamic variables at vanishing chemical potential, which is the onset of BEC. The critical exponents associated with the ideal BEC transition in the 2D inhomogeneous fluid reveals its own universality class.

arXiv:2308.02744•Journal of Physics B Atomic Molecular and Optical Physics•Feb 12, 2024•2 citations

Phase transition of a Bose gas in a harmonic potential

K Damle, T. Senthil, Satya N. Majumdar +1 more

We consider a Bose gas confined by a harmonic potential. We define an appropriate thermodynamic limit and analyze the properties of the phases and phase transition in this limit. Critical properties in the presence of the potential are found to be different from, though simply related to, those in the usual translationally invariant case. We argue that the properties of magnetically trapped rubidium and sodium gases (in which Bose-Einstein condensation has recently been observed) are well approximated by our thermodynamic limit except in a narrow window of temperature around the critical temperature. We also consider the effect of the confining potential on the non-equilibrium dynamics following a rapid quench to the ordered side and give a scaling description of the late-time universal dynamics.

arXiv:cond-mat/9604037•Europhysics Letters (EPL)•Oct 1, 1996•47 citations

The Ideal Bose Gas in the Harmonic Trap

Лев П. Питаевский, S. Stringari

Abstract This chapter describes the main properties of the ideal Bose gas confined by a harmonic potential. It calculates the temperature dependence of the condensate density and identify the value of the critical temperature in terms of the oscillator frequency and the number of atoms. This identification provides an important temperature scale for experiments with ultracold atomic gases. The chapter also discusses the behaviour of the expanding cloud after the release of the trap and how the anisotropy of the gas evolves in time, reflecting the initial anisotropy of the momentum distribution. It finally discusses the possibility of getting Bose–Einstein condensation through adiabatic transformations, working with non-harmonic trapping potentials or with mixtures of two atomic species trapped by different harmonic potentials.

•Jan 1, 2016•0 citations
To overcome the difficulties in defining pressure for a gas confined in an inhomogeneous trap, we define single macroscopic parameters that behave like pressure and volume. We measure the phase diagram of a 87{}^{87}87Rb Bose gas in a harmonic trap in terms of those macroscopic parameters obtained from the spatial distribution of atoms. Considering the relevant variables such as the trap potential V=(\ensuremathωx\ensuremathωy\ensuremathωz)\ensuremath−1\mathcal{V}={({\ensuremath{\omega}}_{x}{\ensuremath{\omega}}_{y}{\ensuremath{\omega}}_{z})}^{\ensuremath{-}1}V=(\ensuremathωx​\ensuremathωy​\ensuremathωz​)\ensuremath−1, number of atoms NNN, and temperature TTT, a parameter \ensuremathΠ=\ensuremathΠ(N,V,T)\ensuremath{\Pi}=\ensuremath{\Pi}(N,\mathcal{V},T)\ensuremathΠ=\ensuremathΠ(N,V,T) is introduced to characterize the overall macroscopic pressure of the system. We construct the phase diagram (\ensuremathΠ\ensuremath{\Pi}\ensuremathΠ vs TTT) identifying the main features related to the Bose-Einstein condensation (BEC) transition in a trapped gas. A thermodynamic description of the phase transition based on purely macroscopic parameters provides us with a description that does not need the local-density approximation. This procedure can be used to explore different aspects related to BEC such as the nature of the phase transition in a trapped gas.
BEC phase diagram of a 87^{87}87Rb trapped gas in terms of macroscopic thermodynamic parameters
We measure the phase diagram of a 87^{87}87Rb Bose gas in a harmonic trap in terms of macroscopic parameters obtained from the spatial distribution of atoms. Considering the relevant variables as size of the cloud V{\cal V}V, number of atoms NNN and temperature TTT, a novel parameter Π=Π(N,V,T)Π= Π(N,{\cal V},T)Π=Π(N,V,T) is introduced to characterize the overall pressure of the system. We construct the phase diagram (ΠΠΠ vs TTT) identifying new features related to Bose-Einstein condensation (BEC) transition in a trapped gas. A thermodynamic description of the phase transition based on purely macroscopic parameters, provide us with properties that do not need the local density approximation. An unexpected consequence of this analysis is the suggestion that BEC appears as a continuous third-order phase transition instead of being a second-order one.
Thermodynamics of non-uniform systems has always been a challenging topic in the physical sciences.In the field of quantum gases, that subject is mainly studied with ultracold samples trapped in a harmonic potential, the first kind of confinement ever made to atomic fluids. 1 Nascimbène et al. 2 have already described a method to measure locally the equation of state and heat capacity of a harmonically trapped quantum gas.Nonetheless, local measurements cannot provide the volume-dependent susceptibilities of an inhomogeneous system, namely the thermal expansion and compressibility, meaning that the thermodynamic description of harmonically trapped gases is still incomplete.To solve that issue, Romero and Bagnato 3, 4 proposed canonical variables of work for a gas in a harmonic trap, which work like pressure and volume for a gas in a vessel and allow global measurements on the system.Therefore, we describe here our methodology to measure the thermodynamic susceptibilities of a harmonically trapped gas around the Bose-Einstein transition using the Romero-Bagnato formalism.Our experiments were divided in different phases, each having the harmonic trap with unique frequencies, related to the formalism's extensive variable.In each phase, we prepared individual samples of rubidium-87 gas in the trap and imaged them using the time-of-flight technique.Next, we fitted our images with the bimodal model and found the in situ density profiles of the samples with the standard regression procedures.Then, we determined the phase's equation of state as a function of temperature and number of atoms.For the thermodynamic analysis, we used an empirical model that we developed to fit the equation-of-state curves, allowing us to represent our data in terms of mathematical coefficients.In that way, we found the curves of internal energy, heat capacity, thermal expansion and compressibility, thus achieving a full thermodynamic description of a harmonically trapped gas.Our method and its results are unprecedented in the literature and might contribute to the further understanding of non-uniform systems, as well as the future development of quantum thermal engines.
We apply the global thermodynamic variables approach to experimentally determine the isothermal compressibility parameter ${\ensuremath{\kappa}}_{T}$ of a trapped Bose gas across the phase transition. We demonstrate the behavior of ${\ensuremath{\kappa}}_{T}$ around the critical pressure, revealing the second-order nature of the phase transition. Compressibility is the most important susceptibility to characterize the system. The use of global variables shows advantages with respect to the usual local density approximation method and can be applied to a broad range of situations.
Abstract The critical properties displayed by an ideal 2D Bose gas trapped in a harmonic potential are determined and characterized in an exact numerical fashion. Beyond thermodynamics, addressed in terms of the global pressure and volume which are the appropriate variables of a fluid confined in a non-uniform harmonic potential, the density-density correlation function is also calculated and the corresponding correlation length is found. Evaluation of all these quantities as Bose–Einstein condensation (BEC) is approached manifest its critical continuous phase transition character. The divergence of the correlation length as the critical temperature is reached, unveils the expected spatial scale invariance proper of a critical transition. The logarithmic singularities of this transition are traced back to the non-analytic behavior of the thermodynamic variables at vanishing chemical potential, which is the onset of BEC. The critical exponents associated with the ideal BEC transition in the 2D inhomogeneous fluid reveals its own universality class.
We consider a Bose gas confined by a harmonic potential. We define an appropriate thermodynamic limit and analyze the properties of the phases and phase transition in this limit. Critical properties in the presence of the potential are found to be different from, though simply related to, those in the usual translationally invariant case. We argue that the properties of magnetically trapped rubidium and sodium gases (in which Bose-Einstein condensation has recently been observed) are well approximated by our thermodynamic limit except in a narrow window of temperature around the critical temperature. We also consider the effect of the confining potential on the non-equilibrium dynamics following a rapid quench to the ordered side and give a scaling description of the late-time universal dynamics.
Abstract This chapter describes the main properties of the ideal Bose gas confined by a harmonic potential. It calculates the temperature dependence of the condensate density and identify the value of the critical temperature in terms of the oscillator frequency and the number of atoms. This identification provides an important temperature scale for experiments with ultracold atomic gases. The chapter also discusses the behaviour of the expanding cloud after the release of the trap and how the anisotropy of the gas evolves in time, reflecting the initial anisotropy of the momentum distribution. It finally discusses the possibility of getting Bose–Einstein condensation through adiabatic transformations, working with non-harmonic trapping potentials or with mixtures of two atomic species trapped by different harmonic potentials.