PhD thesis in English

4. Mean-field description of an interacting BECimation of the TFSC model. To this end, we rewrite the semiclassical expression(4.27) using the Robinson formula [63]:ζ ν (e z ) = Γ(1 − ν)(−z) ν−1 +∞∑k=0z kk! ζ ν−k, (4.28)where, as before, Γ(z) is the Gamma function, and ζ ν is the Riemann zeta function.Close to the condensate boundary, the value of µ − V (r) − 2gn th (r) is small and wecan rely on the approximationζ 3/2 (e z ) ≈ Γ(−1/2)(−z) 1/2 + ζ 3/2 + zζ 1/2 , (4.29)to obtain an implicit equation for n th (r):n th (r) ≈ 1 [Γ(−1/2) β 1/2 |µ − V (r) − 2gnλ 3 th (r)| 1/2T]+ζ 3/2 − ζ 1/2 β |µ − V (r) − 2gn th (r)| . (4.30)When solving Eq. (4.30), we encounter four different possible branches for n th (r),due to the two quadratic equations. However, only two branches are physicallymeaningful and can be identified easily. Results for the density of the thermalcomponent obtained in this way are also shown in Fig. 4.4 by red solid lines. Wesee good agreement of the approximation (4.30) with numerically calculated valuesbased on the full TFSC model. Furthermore, this analytical approach confirms thepresence of discontinuities in the density profiles close to the condensate boundary.Therefore, the discontinuities are not an artifact of numerical simulations, but theproblem of the model itself. This issue was noticed already in the early Ref. [33],and it is a consequence of the TF approximation. However, the model is able topredict an approximate phase diagram of a BEC, which is in a very good agreementwith experimental data, as we discuss later in this Chapter.4.2.4 GPSC modelFinally, we consider the full set of Eqs. (4.19)-(4.21), i.e. the GPSC model. We solvethe equations for a fixed number of particles N using an iterative method. Oneway to start the iterations is to assume that all atoms are in the condensate and tosolve Eq. (4.19) without the thermal cloud. In the next step, the previous solution95