Martin Teichmann Atomes de lithium-6 ultra froids dans la ... - TEL

CHAPTER 4. DATA ANALYSIS
In order to determine the temperature of the gas, we release the
atoms from their trap and study the momentum distribution. Since the
gas is in the trap, this distribution will not have the shape of a Fermi
distribution – the chemical potential varies in the trap. Instead, we
obtain a bell-shaped distribution. At higher temperatures the width of
this curve gives the temperature, while at low temperatures the width
stays constant. This is an effect known as the Fermi pressure [14, 15].
We determine the temperature of a fermionic cloud by fitting an image
of the cloud to a theoretical function, which we want to develop in this
section.
One of our former group members, Julien Cubizolles [71], has developed
a fit based on the Sommerfeld approximation of the Fermi
function. This fitting routine was useful, while limited, since he used
approximations only valid in a certain temperature range, leading to a
divergence of the fit once out of this range. A variation of the method we
present here has been used in the group at Duke University. They claim
that the fitting function developed here can also be used with a strongly
interacting gas if one introduces and additional calibration parameter
[31]. They developed a new technique to determine this parameter:
they released the gas from the trap for a short while, recapturing it
afterwards. The gas heats during this time in a controlled way. This
way they created a reference scale using which they could determine
the calibration parameter for their fitting routine.
Via absorption imaging, one measures the density of atoms integrated
over the line of sight. Traditionally we integrate over a second direction
to get a resulting curve which can then be fitted to a function. In case
of a non-interacting Fermi gas, the density can be easily calculated.
For fermions, the occupation probability follows the well-known
Fermi-Dirac distribution
f(E) =
1
e β(E−µ) + 1
with β = 1
kBT
(4.1)
µ is the chemical potential. Using the local density approximation, the
phase space density w(r, k) in a harmonic trap is given by
w(r, p) = (2π¯h) −3
�
e β
�
p2 2m + 1 �
2 m j ω2 j r2 j −µ
�
+ 1
� −1
(4.2)
As we want to use these functions to fit our data, we need to integrate
over four or five dimensions, depending on whether we want to do a twoor
one-dimensional fit. One should remember that we do not normally
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