PhD thesis in English

5. BEC excitation by modulation of scattering lengthwhere we expand ω around ω 0 and introduce the frequency shifts ω 1 , ω 2 , . . ., for eachorder in the expansion in Q. By inserting the above expansions into the Eq. (5.18)and collecting terms of the same order in Q, we obtain a hierarchical system of lineardifferential equations. To the third order, we find:ω 2 0ü1(s) + ω 2 0 u 1(s) = 1 u 4 0cos Ωsω ,ω0ü2(s) 2 + ω0 2 u 2(s) = −2ω 0 ω 1 ü 1 (s) − 4 uu 5 1 (s) cos Ωs0 ω + αu 1(s) 2 ,ω0ü3(s) 2 + ω0 2 u 3(s) = −2ω 0 ω 2 ü 1 (s) − 2βu 1 (s) 3 + 2αu 1 (s)u 2 (s) − ω1ü1(s)2+ 10 uu 6 1 (s) 2 cos Ωs0 ω − 4 uu 5 2 (s) cos Ωs0 ω − 2ω 0 ω 1 ü 2 (s),where α = 10 P/u 6 0 + 6/u5 0 and β = 10 P/u7 0 + 5/u6 0 .These equations disentangle in a natural way: we solve the first one for u 1 (s), anduse that solution to solve the second one for u 2 (s), and so on. At the n-th level of theperturbative expansion (n ≥ 1) we use the initial conditions u n (0) = 0, ˙u n (0) = 0.As is well known, the presence of the term cos s on the right-hand side of some ofthe previous equations would yield a solution that contains the secular term s sin s.Such a secular term grows linearly in time, which makes it the dominant term inthe expansion (5.19) that otherwise contains only periodic functions in s. In orderto ensure a regular behavior of the perturbative expansion, the respective frequencyshifts ω 1 , ω 2 , . . . are determined by imposing the cancellation of secular terms.This analytical procedure is implemented up to the third order in the modulationamplitude Q by using the software package MATHEMATICA [54]. Although thecalculation is straightforward, it easily becomes tedious for higher orders of perturbationtheory. Note that it is necessary to perform the calculation to at least thirdorder since it turns out to be the lowest-order solution where secular terms appearand where the nontrivial frequency shift can be calculated. We have solved explicitlyequations for u 1 (s), u 2 (s), and u 3 (s) and in Fig. 5.6 we show an excellent agreementof our analytical solutions with a respective numerical solution of Eq. (5.15). Fromthe first-order solution u 1 (t) we read off only the two basic modes ω 0 and Ω, whilethe second-order harmonics 2ω 0 , ω 0 − Ω, ω 0 + Ω and 2Ω appear in u 2 (t). In thethird order of perturbation theory, higher-order harmonics ω − 2Ω, 2ω − Ω, 2ω + Ω,ω + 2Ω, 3ω, and 3Ω are also present. Concerning the cancellation of secular terms,122