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

which maintains a constant η = U/kBT during the evaporation process. Near the
Feshbach resonance, in the unitary regime, the scattering cross section becomes energy-
dependent. The ideal shape of the trap depth lowering curve is then described by [41]
U(t)
U0
2.11 Degenerate Fermi Gas
=
1− t
2(η−3)/(η−6) . (2.37)
τ
In the quantum regime the properties of bosons and fermions are very different.
While the wave-function of bosons, like 87 Rb, is symmetric under the exchange of two
particles, the wavefunction of fermions, like 6 Li, is anti-symmetric under the exchange. In
addition, fermions obey the Pauli-exclusion principle and two indistinguishable fermions
cannot occupy the same energy level. These differences lead to bosons following Bose-
Einstein statistics and fermions obeying Fermi-Dirac statistics. In the thermal regime
the difference between bosons and fermions can be neglected, both following Maxwell-
Boltzmann statistics. However, as the temperature of the gas falls below a transition
temperature and degeneracy is approached, exciting new physics can be discovered. This
section focuses on properties of degenerate non-interacting fermions.
For non-interaction fermions the Fermi-Dirac distribution is given by
f(ǫν) =
exp
1
ǫν−µ
kBT
, (2.38)
+1
where µ is the chemical potential [42]. At zero temperature the energy distribution is
f(ǫ) = 1 below the Fermi energy EF = µ(T = 0,N) and f(ǫ) = 0 above. For higher
temperatures the distribution is smeared out around EF. In a harmonic trap the density
of states is given by [43]
g(ǫ) =
ǫ 2
2 3 ωxωyωz
= ǫ2
2 3 ¯ω 3,
(2.39)
where ωi are the trapping frequencies and ¯ω 3 ≡ ωxωyωz. The total number of particles
is therefore determined by
N =
dǫf(ǫ)g(ǫ). (2.40)
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