PhD thesis in English

5. BEC excitation by modulation of scattering lengththe first-order correction ω 1 vanishes, leading to a frequency shift which is quadraticin Q:ω = ω 0 +Q212u 200 ω3 0where the polynomial P(Ω) is given byP(Ω)(Ω 2 − ω 2 0 )2 (Ω 2 − 4ω 2 0 ) + . . . , (5.21)P(Ω)=Ω 4 [ −240Pu 5 0 + 36u6 0 (−4 + 3u4 0 ω2 0 )]+Ω 2 [ −1100P 2 − 30Pu 0 (44 − 65u 4 0 ω2 0 ) + 9u2 0 (−44 + 127u4 0 ω2 0 − 44u8 0 ω4 0 )]+5600P 2 ω 2 0 + 840Pu 0 ω 2 0(8 − 3u 4 0ω 2 0) + 36u 2 0ω 2 0(56 − 39u 4 0ω 2 0 + 8u 8 0ω 4 0). (5.22)MATHEMATICA notebook which implements this analytical calculation is availableat our web site [60].5.4.2 Results and discussionThe result given by Eq. (5.21) is the main achievement of our analytical analysis inthe previous section. It is obtained within a second-order perturbative approach inQ and it describes the breathing mode frequency dependence on the driving Ω andthe modulation amplitude Q as a result of nonlinear effects. Due to the underlyingperturbative expansion, we do not expect Eq. (5.21) to be meaningful at the preciseposition of the resonances. However, by comparison with numerical results basedon the variational equation, we find that Eq. (5.21) represents quite a reasonableapproximation even close to the resonant region.To illustrate this, in Fig. 5.11 we show two such comparisons. In the upperpanel we consider the parameter set P = 0.4 and Q = 0.1, and observe significantfrequency shifts only in the narrow resonant regions. We notice an excellent agreementof numerical values with the analytical result given by Eq. (5.21). In the lowerpanel we consider the parameter set P = 1 and Q = 0.8, with much stronger modulationamplitude. In this case we observe significant frequency shifts for the broaderrange of modulation frequencies Ω. In spite of a strong modulation, we still seea qualitatively good agreement of numerical results with the analytical predictiongiven by Eq. (5.21). In principle, better agreement can be achieved using higherorderperturbative approximation. The dashed line on both figures represents Ω/2,given as a guide to the eye. It also serves as a crude description of what we observenumerically in the range Ω ≈ 2ω 0 .123