• local field factor
• Fano Feshbach resonance
• dielectric theory
• beyond RPA
• boson-fermion model
• BEC-BCS crossover
• beyond mean-field

The aim of the present thesis is to explore the crossover in a quantum atomic gas of attractively-interacting fermions from a Bardeen-Cooper-Schrieffer (BCS) superfluid state (BCS) [1] to a Bose-Einstein condensate (BEC) of composite bosons. The
BCS-BEC crossover is a powerful concept, in that while the attraction strength is progressively increased and all the fundamental quantities smoothly interpolate between the two limiting regimes, the ground-state firmly maintains the same kind of spontaneous symmetry-breaking. BCS superfluidity is a state of matter in which a whatever weak attractive interaction produces pairing of fermions in momentum-space, i.e. of fermions located at opposite sides in the Fermi surface with different possible symmetries of the spin-part of the wave-function. These (Cooper) pairs have large size on the scale of the average inter-particle distance and strongly overlapping in real-space, while occupying the lowest quantum state.
A minimum threshold energy is required to break them up, that shows up as an energy gap in the collective excitation spectrum. On the other side, BEC superfluidity is a quantum degenerate state in which all the fermion pairs can be considered as very small-size objects, say point-like composite bosons with given binding energy, that then may Bose-Einstein condense in the lowest quantum state.
In the crossover, the Pauli principle is thus preserved while the pairing evolves from occurring in momentum to real space, and the chemical potential correspondingly evolves from the Fermi-energy value in the BCS limit to half-the binding energy
of the composite boson in the BEC regime. In all cases, this process leads to the formation of a coherent state characterised by well-established phase coherence of the many-body wave function: this produces at the same time the macroscopic occupation of the lowest quantum state, that is the condensate fraction, and a new stiffness against transverse perturbations, that is the superfluid fraction, the former mainly influenced by the interaction strength and the latter by the system geometry.