PhD thesis in English

4. Mean-field description of an interacting BEC2⋅10 19 0 10 20 30 40 50 60 70 801⋅10 19 26 27 28 29 30 31 321.5⋅10 198⋅10 18n th (r)1⋅10 195⋅10 180semi-idealTFSCGPSCr[µm]Figure 4.5: The density profile of thermal atoms n th (m −3 ) versus the radial coordinater, calculated using the three different approximations for the values ofparameters N = 10 7 , T = 100 nK, ω = 100 Hz. We see a good agreement betweenthe TFSC and GPSC calculations, due to a large numbers of atoms in the trap.for the condensate cloud is inserted into the algebraic Eq. (4.20), which is thensolved for n th (r). After integration of n th (r) over r, a new value of the number ofatoms in the condensate N 0 is calculated as N −N th , and the procedure is repeatedby solving Eq. (4.19), but with the new values of N 0 and n th (r), calculated in theprevious step. This procedure is repeated until the desired convergence of the resultsis achieved. Of course, many modifications of the described procedure are possible,and practically one would like to identify the iteration procedure which leads to thefastest convergence toward the final result. The obtained density profiles are shownin Fig. 4.5. We stress that the discontinuities present in the TFSC model are absentin the GPSC model. Obviously, the presence of the kinetic energy term in Eq. (4.19)leads to smooth density distributions and cures the discontinuities problem of theTFSC model. For a large number of particles, TFSC and GPSC solutions are veryclose to each other, while for the smaller number of particles the TFSC solution iscloser to the semi-ideal model result.To study onset of Bose-Einstein condensation, let us consider the following scenario:we fix the temperature T, and then change the total number of particles N inthe trap. As the number of particles increases, the occupation of the ground statebecomes higher and higher, and, above a certain threshold, condensation sets in. Illustrativenumerical results are given in Fig. 4.6. A normalized density 4πn(r)r 2 /N96