• Bose-Einstein condensation

Therecentprogressinultracold-atomexperimentshasprovidedaverypowerfultooltotestnewtheories. The possibility to realize isotopic mixtures of bosonic and fermionic atoms at very low temperatures and with tunable interactions, has naturally lead to the search for a fermionic excitation due to the breaking of the symmetry which allows to exchange bosons and fermions, analogously to the Supersymmetry in elementary particle physics. In the context of ultracold atoms an excitation with quadratic spectrum should arise: the goldstino. Its spectrum has been studied in the case of weakly interacting bosons coupled to a polarized Fermi gas through a density-density interaction, but when bosons are not condensed. This work aims to study the goldstino spectrum when dealing with Bose-Einstein condensation, after a broad overview of different limit cases regarding the single-particle spectrum for bosons and fermions. The thesis is structured as follows: first the standard theory of Bose-Einstein Condensation is briefly reviewed, then the formalism that is needed to face the problem is introduced. At this point, several explicit effects of renormalization on the boson speed of sound and fermion gap and effective mass are shown and discussed; finally the goldstino excitation is presented and related conclusions are drawn.