We consider cigar-shaped Fermi superfluid in the Bose-Einstein condensation (BEC)-BCS crossover. Using the polytropic form of equation of state, we derive low energy multibranch bosonic excitations and the corresponding density fluctuations in three different regimes along the crossover, namely weak-coupling BCS, unitarity, and molecular BEC regimes. Bragg spectroscopy can be used to probe the multibranch nature of the low-energy bosonic excitations by measuring the dynamic structure factor. Therefore we calculate the dynamic structure factor in those three different regimes. In Bragg spectroscopy, an actual observable is momentum imparted to the superfluid due to the Bragg potential. We also present results of the momentum imparted to the superfluid due to the Bragg pulses.
We investigate the superfluid phase transition in a gas of Fermi atoms loaded on a three-dimensional optical lattice. When the lattice potential is strong, this system can be well described by an attractive Hubbard model. In this model, we calculate the superfluid phase transition temperature
Tc, including both superfluid and (spin and charge) density fluctuations within the self-consistent
t-matrix theory and fluctuation exchange approximation, respectively. Since we treat these fluctuations in a consistent manner, our theory satisfies the required particle-hole symmetry over the entire BCS--Bose-Einstein-condensation (BEC) crossover region. We show that charge density fluctuations compete against superfluid fluctuations near the half-filling, leading to the suppression of
Tc. As a result, the maximum
Tc is obtained away from the half-filling. Since the strong density fluctuations originate from the nesting property of the Fermi surface at the half-filling (which is absent in a uniform gas with no lattice potential), our results would be useful in considering lattice effects on strong-coupling superfluidity.