0198506961.pdf

190 Laser cooling and trapping

This function ofx=− 2 δ/Γ has a minimum atδ=ω−ω 0 =−Γ/2of

kBTD=

Γ

2

. (9.28)

This key result is theDoppler cooling limit. It gives the lowest temper-

ature expected in the optical molasses technique. On general grounds,

we expect a limit of this magnitude for processes in a two-level atom

(^22) Einstein pointed out in his funda- since
Γ=
/τrepresents the smallest energy scale in the system. (^22) For
mental work on radiative absorption
and emission processes of atoms in a
thermal radiation field that the momen-
tum exchange between light and mat-
ter would bring the atoms into ther-
mal equilibrium with the surroundings
(Einstein 1917). When the radiation
has a spectral distribution correspond-
ing to 0 K (monochromatic light) the
atom would be expected to approach
this temperature.
sodiumTD= 240μK, which corresponds to a most probable velocity of
0 .5ms−^1. This velocity can be written as
vD

(

Γ

M

) 1 / 2

=

(

k

M

·

Γ

k

) 1 / 2

=(vrvc)^1 /^2 , (9.29)

wherevcΓ/kgives an estimate (to within a factor of 2) of the capture

velocity for the optical molasses technique, i.e. the velocity range over

whichFscatthas a significant value. For sodiumvr=0.03 m s−^1 and

vc=6ms−^1 , and the above treatment of the optical molasses technique

(^23) Narrow transitions with
Γ <Er is valid for velocities within this range. 23
give rise to a different behaviour, that
has some similarities to the discussion
of cooling using narrow Raman transi-
tions in Section 9.8.
The theory as presented so far was initially thought to describe the
optical molasses technique until experimental measurements found much
lower temperatures under certain conditions, in particular when the
Earth’s magnetic field was cancelled out. The two-level model of an
atom cannot explain thissub-Doppler cooling. Realalkaliatomshave
degenerate energy levels (|IJFMF〉states). Remarkably, this does not
just complicate the situation but actually allows new cooling mecha-
nisms to occur, as described in Section 9.7. This is a rare example in
which things turned out to be much better than expected. The fact
that Doppler cooling theory does not accurately describe the optical
molasses experiments with alkali metal atoms gives an excuse for the
rather cavalier treatment of saturation in this section.

9.4 The magneto-optical trap

In the optical molasses technique cold atoms accumulate in the region

where the three orthogonal pairs of laser beams intersect because it

takes a considerable time for atoms to diffuse out, e.g. several seconds

for beams of 1 cm radius. With the correct choice of polarizations for

the laser beams, this configuration can be turned into a trap by the

addition of a magnetic field gradient, as illustrated in Figs 9.8 and 9.9;

the two coils with currents in opposite directions produce a quadrupole

magnetic field. This magnetic field is much weaker than in the purely

magnetic traps described in Chapter 10 anddoes notconfine atoms by

itself. In themagneto-optical trap(MOT) the quadrupole magnetic field

causes an imbalance in the scattering forces of the laser beams and it

is the radiation force that strongly confines the atoms.^24 The principle

(^24) The principal idea of magneto-
optical trapping was suggested by Jean
Dalibard and demonstrated at Bell
Laboratories, USA in collaboration
with a group from MIT.
of the MOT is illustrated in Fig. 9.9(a) for a simpleJ=0toJ=1
transition. At the point in the middle of the coils the magnetic fields