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+