PhD thesis in English

ter 3 we have studied the effects of the shape of trapping potentials to the propertiesof a Bose-Einstein condensate of an ideal gas. A rotation gives rise to the appearanceof a deconfining harmonic potential. Particularly, in the interesting fast-rotatingregime, when the rotation frequency approaches the confining frequency of the harmonictrap, the gas becomes unconfined and the condensate would disperse. Oneway to mitigate the deconfining effect is to use an additional, quartic term in thepotential. As a result, the total external potential in the co-rotating frame acquiresdifferent shapes, depending on the rotation frequency. It is important to understandhow this affects the properties of a BEC for the interpretation of data obtained inexperiments with fast rotating BECs. By employing the method of exact diagonalizationof the time-evolution operator from Chapter 2, we have obtained numericallyexact energy spectra of the harmonic plus quartic trapping potential for differentvalues of the rotation frequency. Using this, we have calculated the condensationtemperature and found that it decreases with an increase of the rotation frequency.We have also presented density profiles of the condensate and thermal cloud at differenttemperatures and have simulated the time-of-flight imaging procedure for thissetup. Interesting expansion dynamics has been found for the external potential inthe shape of a Mexican hat, in the over-critical rotation regime. In the initial stage ofthe expansion the gas expands inwards, into the previously unoccupied inner space,and only after that the common free expansion starts. This leads to an increase inthe typical time scales for the expansion of about one order of magnitude.Chapter 4 is dedicated to the review of the mean-field description of a weakly interactingBEC. We have presented several widely used approximation techniques. Inthe zero temperature limit, we have introduced nonlinear mean-field Gross-Pitaevskiiequation. In order to study BEC at finite temperature and to explore the BEC phasediagram, we have used Hartree-Fock framework in the form in which higher, thermallyexcited states are treated within the semiclassical approximation. Within thismean-field picture, a two-component model of a BEC naturally arises. In an approximativeway, the condensate and thermal component are introduced enabling us tokeep the intuition built on a noninteracting model. Depending on whether the interactionwithin each component is taken into account and how their mutual interactionis considered, several approximation schemes come into play. We have comparedtheir properties, and emphasized the identified drawbacks. The interaction-inducedshift of the condensation temperature has been re-derived. With these results, wehave shown how the noninteracting BEC picture is modified in the presence of weak135