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Physical Review A

Effects of interaction on thermodynamics of a repulsive Bose-Einstein condensate

Satadal Bhattacharyya, Tapan Kumar Das, Barnali Chakrabarti

Published Nov 13, 2013• 12 citations
Publisher

Abstract

We report the effects of interaction on thermodynamic properties of a repulsive Bose-Einstein condensate confined in a harmonic trap by using the correlated potential harmonics expansion method. This many-body technique permits the use of a realistic interactomic interaction, which gives rise to the effective long-range interaction of the condensate in terms of the $s$-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.

DOI
10.1103/physreva.88.053614

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We report the effects of interaction on thermodynamic properties of a repulsive Bose-Einstein condensate confined in a harmonic trap by using the correlated potential harmonics expansion method. This many-body technique permits the use of a realistic interactomic interaction, which gives rise to the effective long-range interaction of the condensate in terms of the sss-wave scattering length. We have computed temperature (TTT) dependence of the chemical potential, specific heat, condensate fraction, entropy, pressure, and the average energy per particle of a system containing a large number (AAA) of 87{}^{87}87Rb 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 (Tc{T}_{c}Tc​), compared to those of a noninteracting condensate. These are in agreement with the experiment. Although all thermodynamic quantities have a strong dependence on AAA 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/TcT/{T}_{c}T/Tc​. This shows the importance of Tc{T}_{c}Tc​ for all thermodynamic properties of the condensate. As expected, for T>TcT>{T}_{c}T>Tc​, these properties follow those of a trapped noncondensed Bose gas.
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 NNN of atoms, harmonic oscillator length, deformation of the trap, sss-wave scattering length) fix its thermodynamic behavior in the large NNN limit through two dimensionless scaling parameters. These are the reduced temperature t=T/Tc0t\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}T/{T}_{c}^{0}t=T/Tc0​ and the ratio $\ensuremath{\eta}$ between the T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 value of the chemical potential, evaluated in the Thomas-Fermi limit, and the critical temperature Tc0{T}_{c}^{0}Tc0​ 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.
In this paper the thermodynamical parameters of a condensed Boson gas are calculated from the partial derivative of the grand potential. In particular, the analytical expressions for some important parameters, such as the condensed fraction, specific heat, critical temperature, effective size, and release energy are investigated. The mean effects which can be altered the ideal Bose gas, such as finite size, highly anisotropic of the external potential and interatomic interaction effects are considered simultaneously. Some new characteristics of the trapped interacting Bose gases in a highly anisotropic trap are revealed. The calculated result for the condensed fraction is compared with the ongoing Stuttgart experiment for 52Cr (Griesmaier et al., Phys. Rev. Lett. 94, 160401(2005)) directly. Good agreement between both the theoretical and experimental data are obtained.
We present a correlated many-body approach to calculate the distribution function and fluctuations for a Bose-Einstein condensate with NNN interacting atoms in the harmonic confinement. The present formulation uses the recursion relation for the canonical ensemble partition function (Z)(Z)(Z). ZZZ is calculated from the energy spectrum of the many-body effective potential, which keeps all possible two-body correlations and uses the realistic interatomic interaction. The condensate statistics are in very good agreement with earlier results of an ideal gas for which exact statistical moments for all temperature are known. We also present the numerical results of condensate statistics for real experimental situations. The calculated moments nicely exhibit the mesoscopic effect for a few hundred atoms, whereas the sharp fall in the variance for the large condensate near the critical temperature shows the possibility of phase transition. We also calculate the critical temperature for the mesoscopic regime. Our present calculation mimics the JILA experiment with 87Rb^{87}\mathrm{Rb}87Rb atoms.
We investigate Bose-Einstein condensates (BEC) containing a large number of bosonic atoms interacting via a finite-range semirealistic interatomic interaction. Ground state properties for an increasing number of atoms in the condensate have been calculated using a modification of the potential harmonic expansion method (including a short range correlation function in the expansion basis) to solve the many-body Schr\"odinger equation. An improved numerical algorithm for the calculation of the potential matrix elements permits us to have up to 14 000 atoms in the condensate. Although our approach is approximate and justified for dilute condensates, our results agree well with available diffusion Monte Carlo results for the same case. The ground state energies also agree well with those by the Gross-Pitaevskii equation method for up to 100 particles in the trap and become gradually larger than the latter (up to 5% for 14 000 atoms). The difference is attributed to the effects of finite range interatomic interaction and two-body correlations. Our approach presents a clear physical picture of the condensate, being computationally economical at the same time.
The zero-temperature ground state properties of experimental 87Rb condensate are studied in a harmonic plus quartic trap [ V(r) = ½mω2r2 + λr4 ]. The anharmonic parameter (λ) is slowly tuned from harmonic to anharmonic. For each choice of λ, the many-particle Schrödinger equation is solved using the potential harmonic expansion method and determines the lowest effective many-body potential. We utilize the correlated two-body basis function, which keeps all possible two-body correlations. The use of van der Waals interaction gives realistic pictures. We calculate kinetic energy, trapping potential energy, interaction energy, and total ground state energy of the condensate in this confining potential, modelled experimentally. The motivation of the present study is to investigate the crucial dependency of the properties of an interacting quantum many-body system on λ. The average size of the condensate has also been calculated to observe how the stability of repulsive condensate depends on anharmonicity. In particular, our calculation presents a clear physical picture of the repulsive condensate in an anharmonic trap.
We investigate the interaction effect between atoms and the finite size effect of a Bose-Einstein gas at finite temperature. Using a mean field approach, we derive the thermodynamic potential on finite systems and obtain the condensate fraction, the chemical potential and the internal energy with the interaction and the finite size effects taken into account. In this approach, we can treat thermodynamic quantities as those of the ideal Bose gas. It is found that the interaction effect shifts the transition temperature by a few percent toward the lower temperature. Above the transition temperature, these quantities agree with the noninteracting Bose gas.