10.2 Magnetic trapping 221(a) (b) Fig. 10.2(a) A cross-section through
the magnetic potential (eqn 10.8) in a
radial direction, e.g. along thex-ory-
axis. The cusp at the bottom of the
conical potential leads to non-adiabatic
transitions of the trapped atoms. (b) A
bias field along thez-direction rounds
the bottom of the trap to give a har-
monic potential near the axis (in the
region where the radial field is smaller
than the axial bias field).This force confines atoms in a low-field-seeking state, i.e. one withgFMF
0, so that the magnetic energy decreases as the atom moves into a
lower field (see Fig. 6.10, for example). However, a quadrupole field has
a fundamental problem—the atoms congregate near the centre where
B= 0 and the Zeeman sub-levels (|IJFMF〉states) have a very small
energy separation. In this region of very low magnetic field the states
with different magnetic quantum numbers mix together and atoms can
transfer from one value ofMFto another (e.g. because of perturbations
caused by noise or fluctuation in the field). These non-adiabatic tran-
sitions allow the atoms to escape and reduce the lifetime of atoms in
the trap. The behaviour of atoms in this magnetic trap with a leak at
the bottom resembles that of a conical funnel filled with water—a large
volume of fluid takes a considerable time to pass through a funnel with
a small outlet at its apex, but clearly it is desirable to plug the leak at
the bottom of the trap.^33 These losses prevent the spherical
quadrupole field configuration of the
two coils in the MOT being used di-
rectly as a magnetic trap—the MOT
operates with gradients of 0.1 T m−^1 so
thirty times more current-turns are re-
quired in any case. We do not dis-
cuss the addition of a time-varying field
to this configuration that leads to the
TOPtrapusedinthefirstexperimental
observation of Bose–Einstein condensa-
tion in the dilute alkali vapours (Ander-
sonet al.1995).
The loss by non-adiabatic transitions cannot be prevented by the addi-
tion of a uniform field in thex-ory-directions, since this simply displaces
the line whereB= 0 , to give the same situation as described above (at
a different location). A fieldB 0 =B 0 ̂ezalong thez-axis, however, has
the desired effect and the magnitude of the field in eqn 10.7 becomes
|B|={
B 02 +(b′r)^2} 1 / 2
B 0 +
b′^2 r^2
2 B 0. (10.10)
This approximation works for smallrwhereb′rB 0. The bias field
along zrounds the point of the conical potential, as illustrated in
Fig. 10.2(b), so that near the axis the atoms of massMsee a harmonic
potential. From eqn 10.2 we find
V(r)=V 0 +1
2
Mωr^2 r^2. (10.11)The radial oscillation has an angular frequency given by
ωr=√
gFμBMF
MB 0×b′. (10.12)10.2.2 Confinement in the axial direction
TheIoffetrap, shown in Fig. 10.3, uses the combination of a linear
magnetic quadrupole and an axial bias field described above to give