The name is absent

weakly interacting Bose gases under harmonic confinement, mean-field Gross-Pitaevskii
(GP) theory has been proven capable of accounting for most of the relevant experimentally
measured quantities in Bose-Einstein condensed gases such as density profiles, collective
oscillations, and vortex structures. The GP equation for a dilute condensed Bose gas in a
trapping potential V_e×t(r) is written as:

∂        ( S²V²                  A

m-Ψ(r, r) = —— + V_ext(r) + ⅛∣Ψ(r, r)∣² Ψ(r, O ∙            (1.1)

Ot        ∖ 2m                    )

where Ψ(r, t) is the order parameter and V_ext(r) is the external trapping potential, usually
harmonic in real experimental situations. In chapter 2, we introduce an efficient split-step
Fourier scheme which not only can be used to solve the GP equation above but also can
be generalized to solve some revised GP equations with much more complicated terms
involved. One example as we will show in chapter 2 is the GP equation describing the
scalar dipolar condensate.

Compared to dilute Bose gases, the mean-field theory for interacting Fermi gas is more
compUcated. An analogy to GP theory as in BECs is not available for the Fermi gas along
the BCS-BEC Crossover. Because the focus of this thesis is about the numerical aspects,
here we skip the detailed introductions to the various basic theories about the ultracold
Fermi gases. A thorough review on can be found in [1] and references therein. In chapter
3, we study the trapped population imbalanced Fermi gases in the BCS-BEC crossover
using the mean-field Bogoliubov-de Gennes (BdG) calculation.

Therefore this short thesis is organized into two parts. Firstly in chapter 2, we will
explain the basic idea of the split-step Fourier scheme using the basic GP equation as the

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