  The present work aims at providing a systematic analysis of the current
density versus momentum characteristics for a fermionic superfluid throughout
the BCS-BEC crossover, even in the fully homogeneous case. At low temperatures,
where pairing fluctuations are not strong enough to invalidate a quasi-particle
approach, a sharp threshold for the inception of a back-flow current is found,
which sets the onset of dissipation and identifies the critical momentum
according to Landau. This momentum is seen to smoothly evolve from the BCS to
the BEC regimes, whereby a single expression for the single-particle current
density that includes pairing fluctuations enables us to incorporate on equal
footing two quite distinct dissipative mechanisms, namely, pair-breaking and
phonon excitations in the two sides of the BCS-BEC crossover, respectively. At
finite temperature, where thermal fluctuations broaden the excitation spectrum
and make the dissipative (kinetic and thermal) mechanisms intertwined with each
other, an alternative criterion due to Bardeen is instead employed to signal
the loss of superfluid behavior. In this way, detailed comparison with
available experimental data in linear and annular geometries is significantly
improved with respect to previous approaches, thereby demonstrating the crucial
role played by quantum fluctuations in renormalizing the single-particle
excitation spectrum.
