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15

cylindrical symmetry of the system and discretize the p-z plane on a triangular mesh using
a finite element scheme, which reduces the system to 2D effectively.

3.1.2 BdG Formalism

Let us consider a Fermi gas distributed in two hyperfine spin states. The Fermi system
across a broad Feshbach resonance, which is realized in 6Li or 40K atoms, can be well

described by the single-channel Hamiltonian as following:

fH =



ψl(r)Ho-ψa(r) + g^J(r)^J(r)^(r)^τ(r)],


(3.1)


with the creation and annihilation operators of fermions, ≠τr(r) and ≠σ-(r).

Following the procedure in Reference [22], we can obtain the Bogoliubov-de Gennes

(BdG) equations [21] which take the form
where the single-particle Hamiltonian is given by

H↑ ∆(r)

∆*(r) -H1


ui(r)           ui(r)

= Ei
v,(r)             v,∙(r)


(3.2)


Hσ = -n2V2∕(2m) + Veχt(r) - μσ(cr =↑, J,)                  (3.3)

The quasiparticle energies E7 take both positive and negative values. The order parameter
and the densities are given by Δ = g X, uiv*f(Ei), n↑ = X1 ∣u,∣2∕(E,), and ni = Xl∙ ∣v,∣2∕(-E,∙)
where f(E) = [exp(E∕kT) + 1]~1 is the Fermi distribution function and the densities must
be constrained as N↑,i = ʃ dr n↑j(r), g is the bare coupling constant which will be replaced
by the s-wave scattering length as via the regularization prescription: (4πħ2as∕m~)~l = l∕g+



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