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arXiv:cond-mat/9508040 · Physical Review Letters

Ground-State Properties of Magnetically Trapped Bose-Condensed Rubidium Gas

Gordon Baym, C. J. Pethick

Published Jan 1, 1996• 696 citations
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Abstract

In light of the recent experimental observation of Bose-Einstein condensation in dilute ${}^{87}\mathrm{Rb}$ gas cooled to nanokelvin-scale temperatures, we give a quantitative account of the ground-state properties of magnetically trapped Bose gases. Using simple scaling arguments, we show that at large particle number the kinetic energy is a small perturbation, and find a spatial structure of the cloud of atoms and its momentum distribution dependent in an essential way on particle interactions. We also estimate the superfluid coherence length and the critical angular velocity at which vortex lines become energetically favorable.

DOI
10.1103/physrevlett.76.6

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In light of the recent experimental observation of Bose-Einstein condensation in dilute 87Rb{}^{87}\mathrm{Rb}87Rb gas cooled to nanokelvin-scale temperatures, we give a quantitative account of the ground-state properties of magnetically trapped Bose gases. Using simple scaling arguments, we show that at large particle number the kinetic energy is a small perturbation, and find a spatial structure of the cloud of atoms and its momentum distribution dependent in an essential way on particle interactions. We also estimate the superfluid coherence length and the critical angular velocity at which vortex lines become energetically favorable.
We consider the leading corrections to the Thomas-Fermi approach to the description of the properties of a magnetically trapped, Bose-condensed cloud of atoms. Simple analytical expressions are derived for the kinetic energy in terms of an effective cutoff length, which we calculate numerically by considering the one-dimensional problem for a linear-ramp potential. We also determine the lowest angular velocity at which it is energetically favorable for a vortex to enter the cloud. For large clouds, our results are in excellent agreement with available numerical calculations.
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.
We show that recent treatments of magnetically trapped Bose-Einstein condensates (BECs) at temperature T >0 fit a simple picture: the condensate wavefunction is that of a T = 0 BEC with the appropriate (depleted) number of condensate atoms N 0 , and the surrounding thermal cloud is described by the statistical mechanics of an ideal gas in the combined potentials of the magnetic trap and the cloud-condensate interaction. Even a simple classical description of the cloud reproduces its essential outline, and facilitates easy approximate calculations of BEC properties for T >0.
A Bose-Einstein condensate was produced in a vapor of rubidium-87 atoms that was confined by magnetic fields and evaporatively cooled. The condensate fraction first appeared near a temperature of 170 nanokelvin and a number density of 2.5 x 10(12) per cubic centimeter and could be preserved for more than 15 seconds. Three primary signatures of Bose-Einstein condensation were seen. (i) On top of a broad thermal velocity distribution, a narrow peak appeared that was centered at zero velocity. (ii) The fraction of the atoms that were in this low-velocity peak increased abruptly as the sample temperature was lowered. (iii) The peak exhibited a nonthermal, anisotropic velocity distribution expected of the minimum-energy quantum state of the magnetic trap in contrast to the isotropic, thermal velocity distribution observed in the broad uncondensed fraction.
The Bose-Einstein condensation of a dilute gas of rubidium-87 atoms was achieved by cooling a small number of atoms in a magnetic trap. The effective potential of the trap is to lowest order harmonic and under these conditions we estimate the critical temperature and the heat capacity of an ideal gas of Bose particles. Previously this has been estimated within the semiclassical approximation. However, because of the small number of atoms, the resulting TcT_cTc​ deviates slightly from our result obtained by exact numerical calculations. For the gap of the heat capacity at the transition point the deviation is 20%20\%20%.
For Bose–Einstein condensation of neutral atoms in anisotropic traps at zero temperature, we present simple analytical methods for computing the properties of ground state and single vortex of Bose–Einstein condensates, and compare those results to extensive numerical simulations. The critical angular velocity for production of vortices is calculated for both positive and negative scattering lengths a, and find an analytical expression for the large- N limit of the vortex critical angular velocity for a >0, and the critical number for condensate population approaches the point of collapse for a <0, by using approximate variational method.