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

is thus at best equal during the transfer from the magnetic to the optical trap. In a
one-dimensional system without phase-space compression, a transfer efficiency ηi can
therefore be defined as
In the three-dimensional case
thus defines the transfer efficiency.
ηi = N′
N = σ′ i
σi
η =
σ ′ i
σi
iǫ{x,y,z}
T ′
i
Ti
T ′
i
Ti
. (5.5)
(5.6)
Neglecting the one photon recoil, a maximally efficient single-photon cooling se-
quence compresses the phase-space along the vertical (z) direction completely. In the
quadrupole trap, the spatial distribution of atoms is anisotropic, σx = 2σy ≡ σ, but the
atoms are in thermal equilibrium, Tx = Ty ≡ T. The maximum transfer efficiency is
thus given by
η =
σ ′ i
σi
iǫ{x,y}
′ T i
Ti = (σ′
x T ′
x)(σ ′
y T ′
y)
1/2σ2 . (5.7)
T
Any transfer efficiencies below this value would be indicative of an incomplete phase-
space compression along the vertical dimension or of an additional heating mechanism.
Equation 5.7 can be simplified even further by exploiting the linear dependence
of σ on the temperature of the magnetic trap. This relationship is shown in figure
5.10. A linear regression in the relevant region of temperature leads to a fit for σ of
σ = (25.8+5.5T µK −1 )µm.
The expected transfer efficiency η, determined by the model, can now be com-
pared with the measured transfer efficiency. The red line in figure 5.11 shows the upper
limit of the transfer efficiency as given by the model. Experimentally measured transfer
efficiencies are shown in black. The temperature and radius of the optical trough are
given by T ′ x = 9.5 µK, T ′ y = 5.2 µK, σ ′ x = 63 µm and σ ′ y = 56 µm. The atoms in the
optical trough do not have time to equilibrate, and the temperatures thus represent the
strength of the trapping potential along the x and y direction respectively. In addition,
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