PhD thesis in English

4. Mean-field description of an interacting BECthe vicinity of the BEC phase transition. We focus on harmonically trapped bosonsfor two reasons: the trapping is always present in the experiments, and also thecalculation of the condensation temperature of the homogenous system turned out tobe a notorious problem, debated in many ways for several decades [79]. We scrutinyand compare several widely used implementations of the HF approximation with theemphasis on their advantages and drawbacks. The interest in the topic increases asnew experiments have reported the observation of beyond mean-field effects [80, 81],which should be incorporated into the existing models.The partition function of the system in the grand canonical ensemble is given byZ(β) = Tr e −β(Ĥ−µ ˆN) , (4.2)and can be rewritten as a bosonic functional integral in the imaginary time [32]:∮Z(β) =∮DΨDΨ ∗ e −A E[Ψ(⃗r,τ),Ψ ∗ (⃗r,τ)]/ , (4.3)where Ψ(⃗r, τ) and Ψ ∗ (⃗r, τ) are periodic functions, with the period β:Ψ(⃗r, τ) = Ψ(⃗r, τ + β) , Ψ ∗ (⃗r, τ) = Ψ ∗ (⃗r, τ + β) . (4.4)For the Hamiltonian (4.1) the Euclidean action A E is given byA E [Ψ(⃗r, τ), Ψ ∗ (⃗r, τ)] =∫ β0+ g 2∫dτ∫ β0(d⃗r Ψ ∗ (⃗r, τ) ∂)∂τ − 22M △ + V (⃗r) − µ Ψ(⃗r, τ)∫d⃗r Ψ ∗ (⃗r, τ)Ψ(⃗r, τ)Ψ ∗ (⃗r, τ)Ψ(⃗r, τ) . (4.5)dτA presence of the interacting Ψ 4 term in the action makes the calculation of the partitionfunction analytically intractable, and to proceed further we apply the standardmean-field approach. In order to study Bose-Einstein condensation, according tothe Bogoliubov prescription (1.27), we first decompose the field Ψ into the orderparameter ψ(⃗r, τ), which corresponds to the macroscopic condensate wave-function,and fluctuations δψ(⃗r, τ):Ψ(⃗r, τ) = ψ(⃗r, τ) + δψ(⃗r, τ) . (4.6)In the functional formalism this represents a change of variables, and the action now85