Experiments to Control Atom Number and Phase-Space Density in ...

The magnetic field gradients introduce a position-dependent scattering force on
the atoms, confining the atoms within the magneto-optical trap radius. For simplicity
consider a one-dimensional setup and an atom with the ground state J = 0 and an
excited state J = 1. In the presence of the quadrupole magnetic field, the Zeeman effect
splits the excited state into three states, denoted by m = +1, m = 0 and m = −1, see
figure 2.16. The transition frequency between the ground and the excited states m±1
is thus varying along the quadrupole field. The laser beams are red-detuned from the
transition frequency (ω0) at zero magnetic field by an amount ∆ . At the position z = z ′
the transition frequency from the ground state to the excited state with m = −1 is then
resonant with the laser frequency, ω(z = z ′ ) = ω(z = 0)−∆. This transition is driven by
σ − circularly polarized light, the beam impinging from the right hand side. Because the
detuning for the transition between the ground state and the m = +1 state is 2∆, the
atom is more likely to scatter photons from the laser beams coming from the right than
from the beam coming from the left hand side having σ + polarization. This imbalance in
scattering events moves the atom back towards the center of the quadrupole field. If the
atom is displaced along the other direction, the selection rules for driving a transition
now imply that the atom will scatter more photons from the beam coming from the left
than from the beam coming from the right, again moving the atom back towards the
center of the trap. This scheme only works as long as atoms return to the same ground
state. This type of transition is then called a cycling transition.
equation 2.30.
Quantitatively the restoring force is described by adding the Zeeman shift into
where β = gµB
FMOT = F σ+
scatt(ω −(ω0 +βz)−kv)−F σ−
scatt(ω −(ω0 +βz)+kv), (2.32)
dB.
Near the trap center it follows that
dz
FMOT ≈ −2 ∂F ∂F
kv +2 βz (2.33)
∂ω ∂ω0
= −2 ∂F
(kv +βz) (2.34)
∂ω
= −αv − αβ
z. (2.35)
k
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