PhD thesis in English

4. Mean-field description of an interacting BEC4 π n(r) r 2 /N [m -1 ]35000300002500020000150001000050000N = 4 × 10 7N = 3 × 10 7N = 2 × 10 7N = 1.46 × 10 7N = 1.09 × 10 70 50 100 150 200r [µm]ω = 125.664 HzT = 200 nKFigure 4.6: Plot of 4πn(r)r 2 /N versus the radial coordinate r within the GPSCmodel for the parameters T = 200 nK, ω = 125.664 Hz.versus the radial coordinate r within the GPSC model is shown in the graph. Fora very large number of atoms we see a single dominant peak corresponding to thecondensate. As the number of atoms decreases, the thermal component becomesvisible, and another broader peak emerges. Finally, below the critical value of thetotal number of atoms, the condensate component disappears completely.High accuracy of the GPSC model is confirmed by direct comparison with thenumerical results obtained from Monte Carlo simulations [76]. However, GPSCmodel fails precisely in describing the phase transition properly. It turns out thatthe system of Eqs. (4.19)-(4.21) does not admit a solution in the vicinity of thephase transition, when it is approached from the condensate side. For all physicalquantities to be well-behaved, it is crucial that the value of the chemical potentialis lower than the value of the classical energy of excited states, µ < H(⃗r, ⃗p). Unfortunately,for small values of n 0 (r), this condition is not satisfied. The inconsistencyarises since we treat the ground state quantum-mechanically and all the other statessemiclassically. For this reason, in the next subsection, we calculate thermodynamicproperties of interacting bosons within the TFSC model.4.2.5 Calculation of the condensation temperatureA representative phase diagram of ultra-cold gas of bosons in the harmonic trapis displayed in Fig. 4.7. Calculations are performed within the TFSC model. We97