10.6 A Bose–Einstein condensate 237(a)(b)Fig. 10.11 In the Thomas–Fermi
regime the condensate has the same
shape as the confining potential. (a)
A harmonic potential. (b) The density
of the atoms in a harmonic trap has
an inverted-parabolic shape (along all
three axes).The critical temperature and density at the onset of Bose condensation
are not properties of the condensate, but are calculated for a cloud
of 4× 106 atoms (from eqn 10.19); this would lead to a condensate
of roughlyN 0 106 atoms after evaporative cooling toT/TC 0. 5
(where most atoms are in the condensate, see eqn F.16). Bothμand
TChave a weak dependence onN(eqns 10.19 and 10.42) and a similar
(but not the same) dependence onω, so these quantities have a similar
relative magnitude for many cases. Note, however, thatμdepends on
(a) (b) (c)Fig. 10.12This sequence of images shows a Bose–Einstein condensate being born out of a cloud of evaporatively-cooled atoms
in a magnetic trap. Each image was taken after a time-of-flight expansion. The cloud changes its size and shape as it undergoes
a phase transition: (a) a thermal cloud just above the critical temperatureTChas a spherical shape (isotropic expansion); (b) a
cloud of atoms at 0. 9 TChas a Bose-condensed fraction in the centre surrounded by a halo of thermal atoms; and (c) well below
the critical temperature (< 0. 5 TC) most of the atoms are in the condensate (lowest energy state of the trap). These images
come from a system that does not have the same aspect ratio as an Ioffe trap, but they illustrate the anisotropic expansion
of the condensate wavefunction. See Fig. 10.13 for dimensions and other details. Data provided by Nathan Smith, Physics
department, University of Oxford.