0198506961.pdf

228 Magnetic trapping, evaporative cooling and Bose–Einstein condensation

and other alkalis can exist as atomic vapours at such low temperatures,

and indeed this was doubted by many people before it was achieved in

experiments. This is possible at very low densities because the processes

that lead to recombination into molecules, and also heat the sample,

occur slowly compared to the formation of a Bose–Einstein condensate.^16

(^16) In these systems the Bose-condensed
gas is metastable; however, conditions
can be reached where the condensate
has a lifetime of many minutes. The
ultra-cold molecules that form when
cold atoms recombine are interesting
to study in their own right. To
achieve long lifetimes, limited by the
recombination of atoms, the conden-
sate must be held in an ultra-high vac-
uum to reduce the rate of collisions with
molecules of the background gas in the
apparatus.

10.5 Bose–Einstein condensation in trapped atomic vapours

The usual textbook discussions of Bose–Einstein condensation consider

a gas with a uniform density throughout a box of fixed volume, i.e. a

homogeneous gas. However, the experiments with magnetically-trapped

atoms correspond to a Bose gas in a harmonic potential and here we look

at this inhomogeneous system. This section shows how to derive rough

values of the important quantities in a way that gives a good physical

understanding of the properties of the trapped Bose gas. A cloud of

thermal atoms (i.e. not Bose-condensed) in a harmonic potential with a

mean oscillation frequencyωhas a radiusrgiven by

1

2

Mω^2 r^2 

1

2

kBT. (10.16)

To the level of accuracy required we take the volume of the cloud as

V 4 r^3 (a reasonable approximation to the volume of a sphere 4πr^3 /3).

This gives the number density asnN/ 4 r^3 , which, when combined

with eqn 10.14, gives

N^1 /^3 

r

λdB

=

kBTC


ω

. (10.17)

This result comes from substituting forrfrom eqn 10.16 and forλdB

at the critical temperature,TC, from eqn 10.15.^17 Fortuitously, despite

(^17) Neglecting a factor of 10 1 / (^3) 2.
all our approximations, this expression lies within 10% of the value de-
rived more carefully for a Gaussian distribution of atoms (see books by
Pethick and Smith (2001) and Pitaevskii and Stringari (2003)). When
the trapping potential does not have spherical symmetry, this result can
be adapted by using the geometrical mean
ω=(ωxωyωz)^1 /^3. (10.18)
For a cloud ofN=4× 106 atoms we find^18
(^18) This number atTC has been cho-
sen because after further evaporation it
would lead to a condensate of roughly
106 atoms.
kBTC=
ωN^1 /^3 =
ω× 160. (10.19)
This result shows clearly that at the BEC transition the atoms occupy
many levels of the trap and that it is quantum statistics which causes
atoms to avalanche into the ground state.^19 A typical trap withω/ 2 π=
(^19) Once a few bosons have accumulated
in a particular state others want to
join them. This cooperative behaviour
arises because the constructive interfer-
ence for bosons leads to a rate of stim-
ulated transitions into a level propor-
tional to the number in that level.
100 Hz has a level spacing of
ω/kB≡5 nK (in temperature units). In
this trapTC 1 μK.^20 For 4× 106 sodium atoms in this trap, eqn 10.14
(^20) This number of atoms gives aTC
close to the recoil limit for sodium (and
TCvaries only slowly withN). Hence
the atoms have a de Broglie wavelength
comparable with that of laser cooling
light. Thiscoincidencegives a conve-
nient way of remembering the approx-
imate values. Note thatTCdepends
on the species of atom, since in a trap
with a given magnetic field, limited by
the maximum current through the coils,
the oscillation frequencies of atoms are
inversely proportional to
√
M.
gives as the density atTC
nC 40 μm−^3 ≡ 4 × 1013 cm−^3.