PhD thesis in English

system of equations:4. Mean-field description of an interacting BEC][− 22M △ + V (r) + g n 0(r) + 2 g n th (r) ψ(r) = µ ψ(r) , (4.19)n th (r) = 1 (ζ ) λ 3 3/2 eβ(µ−2g(n th (r)+n 0 (r))−V (r)),T∫ ∫(4.20)N = N 0 + N th = d⃗r n 0 (r) + d⃗r n th (r) . (4.21)Here, as before, the condensate density is given by n 0 (r) = |ψ(r)| 2 . We denote theabove system of equations as GPSC model.Before considering the full GPSC model of a BEC, we will first consider a seriesof simpler models: the almost-ideal model (widely used in the analysis of the experimentaldata), the semi-ideal model [88, 89], and the Thomas-Fermi-semiclassicalmodel (TFSC), in which Eq. (4.19) is solved in the TF approximation. These modelsgradually add more details to the description of a BEC and become more complex,while at the same time they allow us to study properties of a BEC in a systematicway and to build new knowledge based on the previously obtained one.In the rest of this Chapter, we assume an isotropic harmonic trap V (r) =Mω 2 r 2 /2, with a typical length scale l = √ /Mω. For 87 Rb we use the valueof the scattering length of a = 5.4 nm in all presented numerical results.4.2.1 Almost-ideal modelThe crudest finite-temperature approximation of an interacting BEC takes into accountonly the effect of interactions on the condensate cloud, while it considersthermal atoms as an ideal gas. This is justified by the fact that, due to a spatialcompression of the condensate in the external trap, the density of this componentis high and effects of interaction play a prominent role. On the other hand, thermalcloud is more spread, with a lower density, which allows us to consider it as anideal gas. To simplify the model further, we describe the condensate within the TFapproximation. Thus, using Eqs. (4.14) and (1.13), we obtain:n 0 (r) = 1 (µ − V (r))θ(µ − V (r)) ,g(4.22)n th (r) = 1 (ζ ) λ 3 3/2 eβ(µ−V (r)).T(4.23)91