PhD thesis in English

4. Mean-field description of an interacting BECAs one of the chief outcomes of that study, the mean-field interaction-induced shiftin the condensation temperature, Eq. (4.34), was experimentally confirmed.Finally, a new series of experiments [80, 81] in 2011 addressed the same topic byutilizing the Feshbach resonance technique of 39 K in combination with absorptionTOF imaging. Several advantageous aspects of a Feshbach resonance were used.First of all, a wide range of interaction strengths was covered. Also, referencemeasurements for the same number of atoms and different interaction strengthswere performed. Furthermore, a ballistic expansion was carefully engineered bytuning the value of the scattering length to zero during the TOF expansion. Forthe weakly interacting gas, the mean-field correction result (4.34) was found to bein good agreement with the experimental data. In addition, for the case of strongerinteractions, beyond-mean-field effects were clearly observed for the first time.4.4 Conclusions and outlookIn this Chapter we have reviewed the mean-field description of harmonically trappedBECs. In the zero-temperature limit we have presented widely used Gross-Pitaevskiiequation, which we will further exploit in Chapter 5. The relevance of the finitetemperaturemean-field models stems from the fact that they are regularly used inthe interpretation of experimental data and extraction of density profiles for thecondensate and for the thermal cloud. We have considered four different finitetemperaturemean-field models. By comparing the predictions of all the consideredmodels, we have concluded that the TFSC model provides the most appropriateand consistent choice from the practical point of view. It is quite easy for numericalimplementation, and is capable to reasonably describe the BEC phase transition.Within this approach we have derived the interaction-induced shift of the condensationtemperature. The model, however, has a major drawback in that it leads to theunphysical discontinuities in the density profiles. The semi-ideal and GPSC modelyield continuous density profiles, but do not capture thermodynamic properties ofthe system properly. Some of the mentioned issues are even more serious in thecase of a more complex trap geometry or in the rotating systems [95]. The resultsobtained so far point to the necessity of using improved semiclassical approximationsfor the thermal atoms, which are better suited to the quantum-mechanicaldescription of the ground-state. Now that experiments have succeeded in achievingthe clear observation of beyond-mean-field effects, the improvements in widely used102