The stability of persistent currents in a two-component Bose-Einstein condensate in a toroidal trap is studied both in the miscible and immiscible regimes. In the miscible regime we show that superflow decay is related to linear instabilities of the spin-density Bogoliubov mode. We find a region of partial stability, where the flow is stable in the majority component while it decays in the minority component. We also characterize the dynamical instability appearing for a large relative velocity between the two components. In the immiscible regime the stability criterion is modified and depends on the specific density distribution of the two components. The effect of a coherent coupling between the two components is also discussed.

RBniCS (freely available under the GNU LGPL, version 3, at http://mathlab.sissa.it/rbnics) is a python-based library, developed on top of FEniCS (http://fenicsproject.org), aimed at the developement of reduced order models in the FEniCS environment. In particular, reduced order techniques such as the certified reduced basis method and proper orthogonal decomposition-Galerkin methods are implemented. The FEniCS project allows RBniCS to take advantage of the high-level (e.g., human readable) code used for the automated solution of partial differential equations. Thanks to the features of FEniCS the final user needs to prepare a short code (around 100 lines) to carry out a reduced order simulation. It is ideally suited for novice users willing to learn reduced basis methods and reduced order modelling, thanks to an object-oriented approach and an intuitive and versatile python interface. Indeed, it is a companion of the introductory reduced basis handbook [Hesthaven, Rozza, Stamm. Certified Reduced Basis Methods for Parametrized Partial Differential Equations. SpringerBriefs in Mathematics, Springer International Publishing, 2015], and has been already used in doctoral classes within the "Mathematical Analysis, Modelling, and Applications" PhD course at SISSA, as well as for courses within the "Master in High Performance Computing" held by SISSA and International Centre for Theoretical Physics (ICTP). RBniCS can also be used as a basis for more advanced projects that would like to assess the capability of reduced order models in their existing FEniCS-based software, thanks to the availability of several reduced order methods and algorithms in the library.

We study the static and the dynamic response of coherently coupled two component Bose-Einstein condensates due to a spin-dipole perturbation. The static dipole susceptibility is determined and is shown to be a key quantity to identify the second order ferromagnetic transition occurring at large inter-species interaction. The dynamics, which is obtained by quenching the spin-dipole perturbation, is very much affected by the system being paramagnetic or ferromagnetic and by the correlation between the motional and the internal degrees of freedom. In the paramagnetic phase the gas exhibits well defined out-of-phase dipole oscillations, whose frequency can be related to the susceptibility of the system using a sum rule approach. In particular in the interaction SU(2) symmetric case, i.e., all the two-body interactions are the same, the external dipole oscillation coincides with the internal Rabi flipping frequency. In the ferromagnetic case, where linear response theory in not applicable, the system shows highly non-linear dynamics. In particular we observe phenomena related to ground state selection: the gas, initially trapped in a domain wall configuration, reaches a final state corresponding to the magnetic ground state plus small density ripples. Interestingly, the time during which the gas is unable to escape from its initial configuration is found to be proportional to the square root of the wall surface tension.

We study the collective modes of the minority component of a highly unbalanced Bose-Bose mixtures. In the miscible case the minority component feels an effective external potential and we derive an analytical expression for the mode frequencies. The latter is independent of the minority component interaction strength. In the immiscible case we find that the ground state can be a two-domain walls soliton. Although the mode frequencies are continuous at the transition, their behaviour is very different with respect to the miscible case. The dynamical behaviour of the solitonic structure and the frequency dependence on the inter-and intra-species interaction is numerically studied using coupled Gross-Pitaevskii equations. p-6

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