PhD thesis in English

4. Mean-field description of an interacting BECcontains terms up to the 4 th order in δψ that we should integrate over. Withoutfurther approximations, however, this problem is equivalent to the original functionalintegral. Numerous approximation techniques were developed to treat terms of the3 rd and 4 th order in different approximative ways [19, 82]. In the Hartree-Fock-Bogoliubov approach, we approximate the 4 th order term asδψ ∗ δψ δψ ∗ δψ ≈4〈δψ ∗ δψ〉δψ ∗ δψ + 〈δψ ∗ δψ ∗ 〉δψδψ + 〈δψδψ〉δψ ∗ δψ ∗−2〈δψ ∗ δψ〉〈δψ ∗ δψ〉 − 〈δψδψ〉〈δψ ∗ δψ ∗ 〉. (4.7)In accordance with the previous decomposition, we introduce auxiliary functionsh(⃗r, τ;⃗r, τ) = 〈δψ ∗ (⃗r, τ)δψ(⃗r, τ)〉 ,f(⃗r, τ;⃗r ′ , τ ′ ) = 〈δψ ∗ (⃗r, τ)δψ(⃗r ′ , τ ′ )〉 ,b(⃗r, τ;⃗r ′ , τ ′ ) = 〈δψ(⃗r, τ)δψ(⃗r ′ , τ ′ )〉 ,which are denoted as Hartree, Fock and Bogoliubov term, respectively. At themoment, the introduced average values are purely formal, but can be later definedso as to make the complete procedure self-consistent. In the case of the contactinteraction (1.21), the Hartree and Fock terms yield equal contributions, hence afactor of 4 in front of the corresponding term in Eq. (4.7).After applying the mean-field approximation (4.7), the action A E becomes quadraticin δψ, and now the functional integrations of the Gaussian integrals can be explicitlyperformed. The final result for the partition function can be written in theformZ(β) = e −β Γ eff[ψ,ψ ∗ ,h,f,b] , (4.8)where Γ eff is the effective action, defined as a functional of five arguments: ψ(⃗r, τ),ψ ∗ (⃗r, τ), h(⃗r, τ;⃗r, τ), f(⃗r, τ;⃗r ′ , τ ′ ) and b(⃗r, τ;⃗r ′ , τ ′ ). They are determined by extremizingthe effective action Γ eff [ψ, ψ ∗ , h, f, b] with respect to each of them:δΓ effδψ = 0, δΓ effδψ ∗= 0, δΓ effδh = 0, δΓ effδf = 0, δΓ effδb= 0.In the quest for the simplest mean-field description of an inhomogeneous BEC,we will make another simplification by neglecting Bogoliubov terms, i.e. anomalouscorrelations b(⃗r, τ;⃗r ′ , τ ′ ), as discussed in Ref. [83]. This assumption is justified in86