PhD thesis in English

5. BEC excitation by modulation of scattering lengtha method for excitation of collective oscillation modes was proposed in Refs. [113,114, 115, 116], but it was experimentally realized only recently in Ref. [15]. In themean-field approximation at T = 0, the time-dependent interaction leads to a timedependentnonlinearity g(t) in the GP equation. Depending on the closeness of theexternal modulation frequency Ω to one of condensate’s eigenmodes, a qualitativelydifferent dynamical behaviors emerge. In the non-resonant case, we have smallamplitudeoscillations of the condensate size around the equilibrium widths, and weare in the regime of linear response. However, as Ω approaches an eigenmode, weexpect a resonant behavior which is characterized by large amplitude oscillations.In this case it is clear that a linear response analysis does not provide a qualitativelygood description of the system dynamics.Motivated by the experimental study, described in Ref. [15], in this Chapter weconsider dynamical features induced by harmonic modulation of the s-wave scatteringlength. Our study is a step beyond the linear regime, toward the resonantbehavior, and it is suited for the parametric region where low-lying collective modescan still be defined as in the linear regime, but their properties are modified bynonlinear effects. Obtained results are relevant for the proper interpretation ofexperimental data, and for understanding of near-resonant properties of nonlinearsystems.In the following, we first review variational description of low-lying modes in thelinear regime. Then, we turn to the recent experiment, published in Ref. [15], thathas achieved harmonic modulation of the s-wave scattering length and briefly explainexperimental procedure and results. Finally, we study the nonlinear dynamicalregime induced by harmonic modulation of the s-wave scattering length, first for aspherically symmetric BEC, and afterwards for an axially-symmetric BEC. In bothcases, we obtain excitation spectra as Fourier transforms of the time-dependent condensatesizes and from here we identify nonlinear features. In addition, we developperturbation theory based on the Poincaré-Lindstedt method which successfully explainsthe observed nonlinear effects.5.1 Variational description of low-lying modesOur analytical method of choice for studying nonlinear BEC dynamics is variationalapproach introduced in Refs. [104, 105]. For completeness and for instructivereasons, we first present the method and main results on low-lying collective BEC107