PhD thesis in English

3. Rotating ideal BECpronounced if we decrease the particle number to 10 4 , which is quite typical for manyBEC experiments. In that case, semiclassical results already have an error of theorder of 20 %. For large anharmonicity the rotation effect is not so important, as wecan see from insets on both graphs in Fig. 3.10. However, for the particle number 10 4a numerical treatment is indispensable, since the errors of the semiclassical resultsamount up to 5 %.3.4 Local properties of rotating BECsLocal properties of ultra-cold quantum gases are ubiquitously used to observe andstudy the phenomenon of Bose-Einstein condensation. The prominent peak in TOFabsorption pictures, which appears suddenly when the temperature is decreasedbelow T c , is a clear signature for the occurrence of a BEC phase transition. It isexperimentally used to measure the thermodynamic properties of the condensate.In this section we will show how the presented numerical approach can be appliedto calculate both the density profiles and the TOF absorption imaging profiles.3.4.1 Density profilesFor the ideal Bose gas, the density profiles of the condensate and of the gas phaseare given by Eqs. (1.11) and (1.12), respectively. Having at our disposal numericallycalculated energy eigenvalues and eigenfunctions, we can calculate the density profileof the condensate. In order to do so, we first have to obtain the ground-stateoccupancy number N 0 using the approach described in the previous section. Oncethis is done, Eq. (1.12) allows to calculate the density profile. In view of a comparisonwith absorption imaging, which always produces two-dimensional profiles, we have tointegrate our numerically determined three-dimensional particle density n(⃗r) alongthe imaging axis. Fig. 3.11 presents typical results for the resulting density profilesof Bose-Einstein condensates for both the non-rotating and the critically-rotatingcase. Obviously, a rotation of the condensate leads to an effective spreading due tothe appearance of a centrifugal potential.Although this approach is sufficient for treating the low-temperature regime,where the condensate is present, we emphasize that the same method can also beused to deal with the thermal regime, when the temperature is increased above T c .For even higher temperatures, when the number of energy eigenstates, that need to77