PhD thesis in English

5. BEC excitation by modulation of scattering lengthis given in Fig. 5.16. Our analytical perturbative result for the shifted quadrupolemode frequency contains poles at ω Q0 , 2ω Q0 , ω B0 −ω Q0 , ω Q0 +ω B0 and ω B0 . Similarly,for the shifted frequency of the breathing mode poles in the perturbative solutionare found at ω Q0 , ω B0 , 2ω B0 , ω B0 − ω Q0 and ω Q0 + ω B0 . In both figures we seeexcellent agreement of the perturbatively obtained results with the exact numerics.In the experiment from Ref. [15], excitations of a highly elongated and stronglyrepulsive BEC were considered with the system parameters given in Eq. (5.9). Forthat case, according to Eq. (5.10) we get ω Q0 ≪ ω B0 , and the driving frequencywas chosen in the range (0, 3ω Q0 ). Good agreement of real-time dynamics obtainedfrom the variational approximation with the exact solution of the time-dependentGP simulation occurs even for long propagation times, as can be seen in Fig. 5.5,which implies a good accuracy of the Gaussian approximation for calculating thefrequencies of the excited modes. From the real-time dynamics shown in Fig. 5.5, weobserve the excitation of the slow quadrupole mode as an out-of phase oscillation inthe axial and in the radial direction. In addition, in the radial direction we observefast breathing mode oscillations. This is typical for highly elongated condensates[133] and our analysis for the experimental parameters shows a strong excitationof the quadrupole mode, but also a significant excitation of the breathing modein the radial direction. Due to the large modulation amplitude Q, many higherorder harmonics are excited, and, most importantly, we find frequency shifts of thequadrupole mode of about 10% in Fig 5.17. From the same figure we notice that,due to the chosen frequency range for Ω, only resonances located at ω Q and 2ω Qare observed. The presence of nonlinear effects is already mentioned in Ref. [15].However, we conclude that frequency shifts calculated here have to be taken intoaccount for extracting the resonance curves from the underlying experimental data.To achieve more clear-cut experimental observation of the nonlinearity-inducedfrequency shifts calculated in this paper, we suggest a different trap geometry fromthe one used in Ref. [15]. Measurements of stable BEC modes can be performedfor about 1 s, and in order to extract precise values of the excited frequencies inthe Fourier analysis, several oscillation periods should be captured within this timeinterval. A higher frequency of the quadrupole mode, that can be realized by usinga larger trap aspect ratio λ z , in combination with a higher modulation frequencywould fulfill this condition. According to the results presented in Ref. [15], resonantdriving may lead to condensate fragmentation. However, our numerical resultsindicate frequency shifts of 10 % even outside the resonant regions according to131