PhD thesis in English

5. BEC excitation by modulation of scattering lengthdeveloped. In the mean-field framework, linearization of the GP equation (4.15)around the ground-state [102] is analogous to the Bogoliubov approach for the inhomogeneouscase. Of course, the procedure is not analytically tractable and has to beimplemented numerically. Most widespreadly used analytical approximative results,which also exploit GP equation as a starting point, are Stringari’s results obtained inthe Thomas-Fermi limit [103], and variational results presented in Refs. [104, 105].A very good agreement between the experimental values of the frequencies of lowlyingmodes and theoretical results based on the linearized mean-field equations wasestablished [102, 103, 104] and validity of the mean-field description, given by theGP equation (4.15), was fully confirmed.All results mentioned until now have assumed small-amplitude oscillations andthus rely on a linearized regime of the initial GP equation. Once this regime wasexplored to some extent, more complex, physically content-rich, and more interestingdynamical features became the focus of experimental studies. We will mention themost prominent examples. As a first notable example of nonlinear collective BECexcitations we mention localized solutions, usually called solitons [19, 85, 86]. Asalready pointed out, the main equation that describes the dynamics of a BEC atT = 0, the GP equation (4.15), is a nonlinear Schrödinger equation. This typeof equations is studied extensively in the field of nonlinear optics and from thiscontext it is well-known that it admits nonlinear localized solutions. Several recentexperiments [106, 107] have studied creation and interaction of solitons in the atomicBEC. Another subject of wide interest is pattern formation in a driven system, inparticular formation of Faraday patterns [108, 109, 110]. An observation of this typeof dynamics in a BEC was given in Ref. [111], where a density wave in the axialdirection was produced by a strong modulation of the strength of the radial trappingpotential. A further experimental research includes study of a quantum turbulentregime in a BEC [112] by a combination of rotation, strong modulation of a trapstrength and a trap displacement. In essence, quantum turbulence is a superfluidturbulence characterized by the presence of tangled vortices. It was initially studiedin the superfluid Helium, but now it can be studied in a more controlled way in aBEC setup, as shown in Ref. [112].In addition to already mentioned research avenues, an exciting possibility forreaching a non-trivial nonlinear dynamical regime in a BEC cloud is given by areal-time tuning of the interaction strength via a Feshbach resonance mechanismintroduced in Chapter 1. Harmonic modulation of the s-wave scattering length as106