0198506961.pdf

9.2 Slowinganatomicbeam 183

field; on the other side of the equation, the Doppler shift adds to the
laser frequencyω. Hence we find from eqn 9.9 that the required magnetic
field profile is

B(z)=B 0

(

1 −

z

L 0

) 1 / 2

+Bbias (9.11)

for 0
z
L 0 ,where

B 0 =

hv 0

λμB

. (9.12)

IfμBBbias
ω−
ω 0 then the atoms come to a complete stop at the end
of the tapered solenoid; generally, it is more useful to leave the atoms
with a small velocity so that they travel out of the tapered solenoid
to a region where experiments, or further cooling, can be performed.
Figure 9.3(a) shows the field profile forωω 0 andBbias0, so that the
maximum field at the entrance to the solenoid is aboutB 0 ; Fig. 9.3(b)
shows the field profile for a different choice ofBbiasthat requires a lower
magnitude of the field.

Example 9.1 For a transition between a state with quantum num-
bersF andMF andanexcitedstatewithquantumnumbersF′ and
MF′, the Zeeman effect causes an (angular) frequency shift of (gF′MF′−

(b)

(a)

(c)

Fig. 9.3The magnetic field along the

tapered solenoid in the first Zeeman

slowing experiment (see Fig. 9.2) var-

ied with position as shown in (a). This

magnetic field is described by eqn 9.11.

Nowadays, some experiments use the

variantshownin(b)wherethefield

drops to zero and then reverses; this

gives the same decrease in velocity for

a given change in fieldB 0 , but has the

following three advantages: (i) the field

has a lower maximum value so that the

coils need less current-turns; (ii) the ‘re-

verse slower’ produces less field at po-

sitions downstream,z>L 0 , because

the contributions from the coils with

currents in opposite directions tend to

cancel out; and (iii) the abrupt change

in the field at the exit helps the atoms

to leave the solenoid cleanly (the atoms

see a sudden increase in the frequency

detuning of the light from resonance

that cuts off the radiation force). In

a real solenoid the field changes gradu-

ally, as illustrated in (c). This smooth-

ing has little influence on the overall

length required because it does not sig-

nificantly affect the early part of the

slowing process.