Scientific Report - BEC

Rotating quantum gases 36vortices through the avarage vorticity, within this description it is impossible to discussthe effects of vortex-vortex interaction in the lattice dynamics. Such phenomena areat the basis of Tkachenko modes, corresponding to the shear distorsions of the lattice.Their frequency, being driven by the elastic forces of the vortex array, is typically muchsmaller than the frequency of the hydrodynamic modes. The Tkachenko oscillationshave been already studied in the case of BEC, both theoretically and experimentally.The usually adopted theoretical framework is given by the elasto-hydrodynamic equations,which generalize the rotational hydrodynamic equations by the inclusion of anelastic energy term. The averaged density and velocity (including the rotational termcorresponding to average vorticity) are hence coupled to a displacement field, accountingfor the average displacement of vortices from the equilibrium position in the lattice.The study of Tkachenko modes in Fermi gases is particularly appealing due to thepossibility of generating very large vortex arrays in these systems. The number ofvortices can be almost an order of magnitude larger than in the bosonic case, due tothe larger size of the trapped configurations. This offers the possibility of investigatingTkachenko modes at relatively small rotation rates, where the oscillation frequencyis significantly enhanced with respect to the high rotation regime and experimentaldetection can be more accurate. In [8] the Tkachenko oscillations were investigated foran axisymmetric configuration of a Fermi gas at unitarity.Bose-Einstein condensate in ring geometryIn the recent years, many laboratories started developing ring traps for ultra-cold atoms.The interest lies not only in the possibility of providing a direct manifestation of superfluidity,but also in the potential applications as high precision gyroscopes, basedon matter-wave rather than light-wave interferometry. The predicted better precisionand stability are especially interesting for the aerospace industry.In the framework of a collaboration with the experimental group of Philippe Bouyerand Alain Aspect, Sylvain Schwartz, under the supervision of Iacopo Carusotto andChiara Menotti, has devoted an important part of the PhD thesis to the theoreticalstudy of tight closed-loop waveguides for atoms. Using a Born-Oppenheimer point ofview, we have reduced the full 3D experimental situation in the presence of strongtransverse confinement and rotation to an effective 1D problem [9]. We obtain extraeffective potential terms with respect to the ones previously known in the literature,which arise due to the spatial variation of the curvature and the transverse confinement.We have checked the validity of our analytical results comparing successfully with 2Dnumerical simulations (see Fig. 3). This provides us with a simple but accurate theory,