PhD thesis in English

5. BEC excitation by modulation of scattering length10 3-40 -30 -20 -10 0 10 20 30 4010 210 110 010 -110 -210 -310 -410 -510 -6Ω = 2Frequency10 210 110 0ω - Ω10 -110 -20.05 0.06 0.07 0.08Frequency10 2 Ω + ω10 0 2Ω 2ω10 -210 -44 4.05 4.1 4.15 4.2Frequency10 210 1 Ω ω10 010 -110 -21.9 2 2.1 2.2Frequency10 110 0 2Ω + ω Ω + 2ω10 -110 -2 3Ω3ω10 -36 6.05 6.1 6.15 6.2FrequencyFigure 5.9: Fourier transform of u(t) for P = 0.4, Q = 0.1, and Ω = 2. First plotpresents the complete spectrum on a semi-log scale, while the subsequent plots focuson regions of interest in the spectrum.for instance, in the context of the Paul trap [130].To examine such excited modes directly, we look at the Fourier transform of thecondensate width u(t). To this end, we numerically solve Eq. (5.15) and find theFourier transform of its solution using the MATHEMATICA software package [54].An example of such an excitation spectrum for P = 0.4, Q = 0.1, and Ω = 2 isgiven in Fig. 5.9. The spectrum contains two prominent modes: a breathing modeof frequency ω (close, but not equal to ω 0 ), and a mode that corresponds to thedriving frequency Ω, along with many higher-order harmonics of the general formmΩ + nω, where m and n are integers.In Fig. 5.10 we juxtapose two zoomed-in Fourier spectra for two different drivingfrequencies for P = 0.4 and Q = 0.2. On the left plot, we show a spectrum forΩ = 1. The vertical solid line corresponds to ω 0 and we find the peak in the120