PhD thesis in English

5. BEC excitation by modulation of scattering length[97, 98, 99, 100, 101]. For instance, in the early experiment reported in Ref. [97]collective modes were excited by applying a small time-dependent perturbation of agiven frequency to the transversal component of the trap-potential and the real-timedynamics in terms of shape oscillations of the condensate was observed. Based onthese measurements, two low-lying eigenmodes of different symmetry were identified.The same procedure was repeated at higher temperatures, when the condensateis not present, and, as expected, the thermal cloud has produced only a responsecorresponding to the excitations of a normal noninteracting gas. In the experimentalstudy published in the same year Ref. [98], the condensate was excited by thetime-dependent modulation of the trapping potential, which additionally includeda spatial displacement of the potential minimum. In this case, shape oscillationscoupled with the center of mass motion were observed. A typical experimental observationis given in Fig. 5.1. In order to make a detailed comparison with theoreticalmodels, the eigenmode frequencies were measured for different numbers of trappedatoms and for different trap strengths.Figure 5.1: An experimental measurement of the collective BEC modes. A figureobtained by Stamper-Kurn and Ketterle, taken from Ref. [5]. The time evolutionof a BEC cloud is presented by a series of density profiles. Shape oscillations arecoupled with the center of mass motion. The field of view in the vertical directionis about 620 µm and time step is 5 ms per frame.The original derivation of the Bogoliubov excitation spectrum was performed fora translationally invariant system. However, in cold-atom experiments, bosons arealways confined by the external trap and we have to deal with an inhomogeneoussystem. In order to be able to keep up with the experimental advances, new theoreticalapproaches for the description of collective modes of a trapped system were105