PhD thesis in English

4. Mean-field description of an interacting BECn 0 (r)n th (r)3⋅10 19 0 10 20 30 40 502⋅10 191⋅10 190r[µm]Figure 4.4: Density profiles of different components (m −3 ) versus the radial coordinater within the TFSC model for the parameters N = 874403, T = 60 nK,ω = 100 Hz. Red solid curves are analytic solutions obtained using the Robinsonformula.This is too crude approximation from the experimental point of view and, in orderto understand how the properties of a BEC phase transition are modified in thepresence of interactions, we need to develop a better approximation.4.2.3 TFSC modelIn this model, we describe the ground state using the TF approximation, whileexcited states are treated within the semiclassical approximation (4.20), withoutfurther simplifications. Hence, we arrive at the following system of equations:n 0 (r) = 1 g (µ − V (r) − 2gn th(r)), (4.26)n th (r) = 1 (ζ ) λ 3 3/2 e−β|µ−V (r)−2gn th (r)|, (4.27)Twhich we call the Thomas-Fermi-semiclassical model (TFSC). A typical numericalsolution of Eqs. (4.26) and (4.27) is given in Fig. 4.4. The obtained density profilesare similar to those of the semi-ideal model. However, in this model certain unphysicaldiscontinuities appear close to the condensate boundary, where we observejumps both in the density of the thermal cloud and of the condensate component.In order to verify the obtained numerical results, we will use an analytic approx-94