9.4 The magneto-optical trap 191CoilCoilFig. 9.8A pair of coils with cur-
rents in opposite directions produces a
quadrupole magnetic field. The field is
zero at the centre of the coils and its
magnitude increases linearly in every
direction for small displacements from
the zero point.produced by the coils cancel out, so thatB= 0. Close to this zero of the
field there is a uniform field gradient that perturbs the atomic energy
levels; the Zeeman effect causes the energy of the three sub-levels (with
MJ=0,±1) of theJ= 1 level to vary linearly with the atom’s posi-
tion, as shown for thez-axis in Fig. 9.9(a).^25 The counter-propagating^25 The energy levels also vary in the
other directions. The Maxwell equation
divB= 0 implies that
dBx
dx
=dBy
dy
=−^1
2
dBz
dz
,so the gradient in any radial direction
is half of that along thez-direction.laser beams have circular polarization as shown in Fig. 9.9(b) and a fre-
quency slightly less than the atomic resonance frequency. The Zeeman
shift causes an imbalance in the radiation force in the following way.
Consider an atom displaced from the centre of the trap along thez-axis
withz>0, so the ∆MJ=−1 transition moves closer to resonance with
the laser frequency—the laser has a frequency below the atomic reso-
nance in zero field to give damping by the optical molasses mechanism.^2626 The MOT requires three orthogonal
pairs ofσ+–σ−beams, but the opti-
cal molasses technique works with other
polarization states, e.g. the Sisyphus
cooling in Section 9.7 uses linearly-
polarized beams.
The selection rules lead to absorption of photons from the beam that
excites theσ−transition and this gives a scattering force that pushes
the atom back towards the trap centre. A similar process occurs for a
displacement in the opposite direction (z<0); in this case the Zeeman
shift of the transition frequency and selection rules favour absorption
from the beam propagating in the positivez-direction that pushes the
atom back towardsz= 0. Note that these beam polarizations and the
quantisation axis of the atom have been defined relative to a fixed direc-
tion in space, i.e. thez-direction in this one-dimensional example. For
z>0 this is the same as the direction of the magnetic field, but forz< 0
the magnetic field points the opposite way; hence theMJ=−1 state
lies above +1 in this region, as shown in Fig. 9.9(a). Strictly speaking,
σ+andσ−refer to transitions of the atom and labelling the radiation
asσ+is shorthand for circularly-polarized radiation of the handedness