ar X iv : c on d - m at / 9 50 90 41 v 1 7 S ep 1 99 5 Bose - Einstein condensation in the harmonic oscillator potential

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 T c 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%. In the remarkable experiment[1] the rubidium vapor is magnetically trapped and evap-oratively cooled below 170 nK where a finite fraction of the particles is observed to occupy the ground state. When further lowering the temperature this fraction increases abruptly, signaling a Bose-Einstein condensation. The gas is very dilute and hence well approximated by an ideal gas. The confining magnetic field is a time-averaged, orbiting potential[2] whose effective potential to first order is a three-dimensional anisotropic harmonic oscillator. The Bose-Einstein condensation of an ideal gas in such a potential has been treated earlier in the semiclassical approximation[4]. We will first shortly review a part of this work using a slightly different approach leading to the same results.