arXiv:cond-mat/9612192 · Physical Review Letters

Scaling and Thermodynamics of a Trapped Bose-Condensed Gas

S. Giorgini, Лев П. Питаевский, S. Stringari

Published May 26, 1997• 86 citations

Abstract

We investigate the thermodynamics of a Bose gas interacting with repulsive forces and confined in a harmonic trap. We show that the relevant parameters of the system (temperature, number $N$ of atoms, harmonic oscillator length, deformation of the trap, $s$-wave scattering length) fix its thermodynamic behavior in the large $N$ limit through two dimensionless scaling parameters. These are the reduced temperature $t\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}T/{T}_{c}^{0}$ and the ratio $\ensuremath{\eta}$ between the $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ value of the chemical potential, evaluated in the Thomas-Fermi limit, and the critical temperature ${T}_{c}^{0}$ of the noninteracting model. The scaling functions relative to the condensate fraction, energy, chemical potential, and moment of inertia are calculated within the Popov approximation.

DOI: 10.1103/physrevlett.78.3987

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Condensate fraction and critical temperature of a trapped interacting Bose gas

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arXiv:cond-mat/9612192 · Physical Review Letters

Scaling and Thermodynamics of a Trapped Bose-Condensed Gas

S. Giorgini, Лев П. Питаевский, S. Stringari

Published May 26, 1997• 86 citations

arXiv HTML PDF

Abstract

We investigate the thermodynamics of a Bose gas interacting with repulsive forces and confined in a harmonic trap. We show that the relevant parameters of the system (temperature, number

N

of atoms, harmonic oscillator length, deformation of the trap,

s

-wave scattering length) fix its thermodynamic behavior in the large

N

limit through two dimensionless scaling parameters. These are the reduced temperature

t\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}T/{T}_{c}^{0}

and the ratio

\ensuremath{\eta}

between the

T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0

value of the chemical potential, evaluated in the Thomas-Fermi limit, and the critical temperature

{T}_{c}^{0}

of the noninteracting model. The scaling functions relative to the condensate fraction, energy, chemical potential, and moment of inertia are calculated within the Popov approximation.

DOI: 10.1103/physrevlett.78.3987

Related papers

Condensate fraction and critical temperature of a trapped interacting Bose gas

S. Giorgini, Лев П. Питаевский, S. Stringari

By using a mean-field approach, based on the Popov approximation, we calculate the temperature dependence of the condensate fraction of an interacting Bose gas of N atoms confined in an anisotropic harmonic trap. For systems interacting with repulsive forces we find a significant decrease of the condensate fraction and of the critical temperature with respect to the predictions of the noninteracting model with the same value of N. An analytic result for the shift of the critical temperature holding to first order in the scattering length is also derived. \textcopyright{} 1996 The American Physical Society.By using a mean-field approach, based on the Popov approximation, we calculate the temperature dependence of the condensate fraction of an interacting Bose gas of N atoms confined in an anisotropic harmonic trap. For systems interacting with repulsive forces we find a significant decrease of the condensate fraction and of the critical temperature with respect to the predictions of the noninteracting model with the same value of N. An analytic result for the shift of the critical temperature holding to first order in the scattering length is also derived. \textcopyright{} 1996 The American Physical Society.

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Hui Hu, Xia-Ji Liu

By using the Hartree-Fock-Bogoliubov equations within the Popov approximation, we investigate the thermodynamic properties of a dilute binary Bose-Fermi mixture confined in an isotropic harmonic trap. For mixtures with an attractive Bose-Fermi interaction, we find a sizable enhancement of the condensate fraction and of the critical temperature of Bose-Einstein condensation with respect to the predictions for a pure interacting Bose gas. Conversely, the influence of the repulsive Bose-Fermi interaction is less pronounced. The possible relevance of our results in current experiments on trapped ${}^{87}\mathrm{Rb}{\ensuremath{-}}^{40}\mathrm{K}$ mixtures is discussed.By using the Hartree-Fock-Bogoliubov equations within the Popov approximation, we investigate the thermodynamic properties of a dilute binary Bose-Fermi mixture confined in an isotropic harmonic trap. For mixtures with an attractive Bose-Fermi interaction, we find a sizable enhancement of the condensate fraction and of the critical temperature of Bose-Einstein condensation with respect to the predictions for a pure interacting Bose gas. Conversely, the influence of the repulsive Bose-Fermi interaction is less pronounced. The possible relevance of our results in current experiments on trapped ${}^{87}\mathrm{Rb}{\ensuremath{-}}^{40}\mathrm{K}$ mixtures is discussed.

arXiv:cond-mat/0302570Physical Review AAug 26, 200322 citations

We investigate the thermodynamics of a Bose gas interacting with repulsive forces and confined in a harmonic trap. We show that the relevant parameters of the system (temperature, number

N

of atoms, harmonic oscillator length, deformation of the trap,

s

-wave scattering length) fix its thermodynamic behavior in the large

N

limit through two dimensionless scaling parameters. These are the reduced temperature

t\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}T/{T}_{c}^{0}

and the ratio

\ensuremath{\eta}

between the

T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0

value of the chemical potential, evaluated in the Thomas-Fermi limit, and the critical temperature

{T}_{c}^{0}

of the noninteracting model. The scaling functions relative to the condensate fraction, energy, chemical potential, and moment of inertia are calculated within the Popov approximation.

s

-wave scattering length. We have computed temperature (

T

) dependence of the chemical potential, specific heat, condensate fraction, entropy, pressure, and the average energy per particle of a system containing a large number (

A

) of

{}^{87}

Rb atoms in the Joint Institute for Laboratory Astrophysics (JILA) trap. The repulsion among the interacting bosons results in a small but measurable drop of condensate fraction and critical temperature (

{T}_{c}

), compared to those of a noninteracting condensate. These are in agreement with the experiment. Although all thermodynamic quantities have a strong dependence on

A

and to a smaller extent on the interatomic interaction, our numerical calculation appears to show that a thermodynamic quantity per particle follows a universal behavior as a function of

T/{T}_{c}

. This shows the importance of

{T}_{c}

for all thermodynamic properties of the condensate. As expected, for

T>{T}_{c}

, these properties follow those of a trapped noncondensed Bose gas.

{\omega}

, the temperature T, and the particle number N are related by

{N \sim (T / \omega)^{3} \to\infty}

while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap. We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose–Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer.