PhD thesis in English

5. BEC excitation by modulation of scattering lengthu z (t)32.82.62.42.2Ω = 0.4, analyticsΩ = 0.4, numerics20 10 20 30 40 50 60 70 80tu z (t)2.552.52.452.42.35Ω = 1, analyticsΩ = 1, numerics0 10 20 30 40 50 60 70 80tu ρ (t)1.151.141.131.121.111.11.091.081.071.061.05Ω = 0.4, analyticsΩ = 0.4, numerics0 10 20 30 40 50 60 70 80tu ρ (t)1.161.141.121.11.081.061.04Ω = 1, analyticsΩ = 1, numerics0 5 10 15 20 25 30 35 40tFigure 5.13: Condensate dynamics within the Gaussian approximation for P = 1,Q = 0.2, λ z = 0.3 and two different driving frequencies Ω = 0.4 (left plot) and Ω = 1(right plot). We plot exact numerical solution of Eqs. (5.11) and (5.12) together withthe analytical second-order perturbative result, as explained in section IV.A.10 310 2 FT of u zFT of u10 1ρ 10 110 010 010 -110 -210 -110 -310 -210 -4-4 -2 0 2 410 -3ΩFT of u zω Qω B10 2 0.5 1 1.5 2FrequencyFrequencyFigure 5.14: Fourier transformed u ρ (t) and u z (t) for P = 1, Q = 0.2, λ z = 0.3, andΩ = 0.4. Left plot gives complete spectrum, while on the right plot we show part ofthe spectrum together with positions of prominent peaks.By inserting these expansions in Eqs. (5.11) and (5.12), and by performing additionalexpansions in Q, we obtain a system of linear differential equations of the generalform:ü ρn (t) + m 11 u ρn (t) + m 12 u zn (t) + f ρn (t) = 0, (5.25)m 21 u ρn (t) + ü zn (t) + m 22 u zn (t) + f zn (t) = 0, (5.26)127