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arXiv:cond-mat/9612176 · Physical Review Letters

Quantum Monte Carlo Calculations for a Large Number of Bosons in a Harmonic Trap

Werner Krauth

Published Oct 28, 1996• 163 citations
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Abstract

In this Letter, I present a precise quantum Monte Carlo calculation at finite temperature for a very large number (many thousands) of bosons in a harmonic trap, which may be anisotropic. The calculation applies directly to the recent experiments of Bose-Einstein condensation of atomic vapors in magnetic traps. I show that the critical temperature of the system decreases with the interaction. I also present profiles for the overall density and the one of condensed particles, and obtain excellent agreement with solutions of the Gross-Pitaevskii equation.

DOI
10.1103/physrevlett.77.3695

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In this Letter, I present a precise quantum Monte Carlo calculation at finite temperature for a very large number (many thousands) of bosons in a harmonic trap, which may be anisotropic. The calculation applies directly to the recent experiments of Bose-Einstein condensation of atomic vapors in magnetic traps. I show that the critical temperature of the system decreases with the interaction. I also present profiles for the overall density and the one of condensed particles, and obtain excellent agreement with solutions of the Gross-Pitaevskii equation.
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The mean-field properties of finite-temperature Bose-Einstein gases confined in spherically symmetric harmonic traps are surveyed numerically. The solutions of the Gross-Pitaevskii (GP) and Hartree-Fock-Bogoliubov (HFB) equations for the condensate and low-lying quasiparticle excitations are calculated self-consistently using the discrete variable representation, while the most high-lying states are obtained with a local-density approximation. Consistency of the theory for temperatures through the Bose condensation point Tc{T}_{c}Tc​ requires that the thermodynamic chemical potential differ from the eigenvalue of the GP equation; the appropriate modifications lead to results that are continuous as a function of the particle interactions. The HFB equations are made gapless either by invoking the Popov approximation or by renormalizing the particle interactions. The latter approach effectively reduces the strength of the effective scattering length asc,{a}_{\mathrm{sc}},asc​, increases the number of condensate atoms at each temperature, and raises the value of Tc{T}_{c}Tc​ relative to the Popov approximation. The renormalization effect increases approximately with the log of the atom number, and is most pronounced at temperatures near Tc.{T}_{c}.Tc​. Comparisons with the results of quantum Monte Carlo calculations and various local-density approximations are presented, and experimental consequences are discussed.
We solve the Gross-Pitaevskii equations for a dilute atomic gas in a magnetic trap, modeled by an anisotropic harmonic potential. We evaluate the wave function and the energy of the Bose-Einstein condensate as a function of the particle number, both for positive and negative scattering length. The results for the transverse and the vertical size of the cloud of atoms, as well as for the kinetic and potential energy per particle, are compared with the predictions of approximated models. We also compare the aspect ratio of the velocity distribution with experimental estimates available for 87Rb^{87}\mathrm{Rb}87Rb. Vortex states are considered and the critical angular velocity for production of vortices is calculated. We show that the presence of vortices significantly increases the stability of the condensate in the case of attractive interactions. \textcopyright{} 1996 The American Physical Society.
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We study thermal properties of a trapped Bose-Bose mixture in a dilute regime using quantum Monte Carlo methods. Our main aim is to investigate the dependence of the superfluid density and the condensate fraction on temperature for the mixed and separated phases. To this end we use the diffusion Monte Carlo method in the zero-temperature limit and the path-integral Monte Carlo method for finite temperatures. The results obtained are compared with solutions of the coupled Gross-Pitaevskii equations for the mixture at zero temperature. We notice the existence of an anisotropic superfluid density in some phase-separated mixtures. Our results also show that the temperature evolution of the superfluid density and condensate fraction is slightly different, showing noteworthy situations where the superfluid fraction is smaller than the condensate fraction.
We introduce a family of correlated trial wave functions for the NNN-particle ground state of an interacting Bose gas in a harmonic trap. For large NNN, the correlations lead to a relative energy decrease of a fraction 3/5N3/5N3/5N, compared to mean field Gross-Pitaevskii theory. The kinetic energy in the weakly confining direction turns out to be most sensitive to our correlations and, remarkably, is higher by as much as a few per cent for condensates with atom numbers of a few thousand. Thus, the predicted deviations from Gross-Pitaevskii theory originating from ground state correlations might be observed in momentum distribution measurements of small condensates.