PhD thesis in English

3. Rotating ideal BEC Figure 3.1: Images of a rotating BEC along the rotation direction for differentrotation frequencies Ω/2π. The linear size of each image is 306 µm. The results aretaken from Ref. [12].To probe the system, the TOF absorption imaging is performed. Typical resultsof the measurements for different rotation frequencies are presented in Fig. 3.1. Itis obvious that up to Ω = 2π × 68 Hz the radius of a trapped cloud increaseswith increase in Ω. For Ω = 2π × 66 Hz and Ω = 2π × 67 Hz the radial densityprofile of the cloud follows the Mexican-hat shape of the potential, however numberof observed vortices in this case is smaller than expected for such a large rotationfrequency. Several possible schemes are discussed as a possible explanation of theobserved features.In order to contribute to the understanding of the experimental results, we studythe BEC phase transition of an ideal Bose gas in the trapping potential (3.4), modifiedby the presence of a quartic term and a rotation with respect to the commonharmonic trap (1.6). Depending on the value of the rotation frequency, the shape ofthe potential changes from convex with a single minimum to the Mexican-hat shape,which significantly influences the properties of a condensate. To study these effects,in the rest of this Chapter we calculate the condensation temperature, condensatefraction and density profiles of the cloud in the trap (3.4) and also simulate a freeexpansion of the condensate, corresponding to the TOF imaging.As long as we approximately describe the system with the ideal Bose gas, allof its many-body properties in the grand-canonical ensemble can be derived purelyfrom single-particle states. When considering the thermodynamic limit, usually thesemiclassical approximation is applied, where the single-particle ground state E 0is retained and treated quantum mechanically, while all other excited states aretreated as a continuum [19, 65]. The validity of the semiclassical approximationcan be rigorously justified in some regimes of the relevant parameters. As brieflymentioned in Chapter 1, the semiclassical approximation is justified in the hightemperaturelimit, when the thermal energy is larger than the typical spacing ofenergy levels, k B T ≥ (E n+1 − E n ). Also, it remains reasonable good irrespective59