Density profiles and collective excitations of a trapped two-component Fermi vapour

We discuss the ground state and the small-amplitude excitations of a degene rate vapour of fermionic atoms placed in two hyperfine states inside a sphe rical harmonic trap. An equations-of-motion approach is set up to discuss t he hydrodynamic dissipation processes from the interactions between the two components of the fluid beyond mean-field theory and to emphasize analogie s with spin dynamics and spin diffusion in a homogeneous Fermi liquid. The conditions for the establishment of a collisional regime via scattering aga inst cold-atom impurities are analyzed. The equilibrium density profiles ar e then calculated for a two-component vapour of K-40 atoms: they are little modified by the interactions for presently relevant values of the system p arameters, but spatial separation of the two components will spontaneously arise as the number of atoms in the trap is increased. The eigenmodes of co llective oscillation in both the total particle number density and the conc entration density are evaluated analytically in the special case of a symme tric two-component vapour in the collisional regime. The dispersion relatio n of the surface modes for the total particle density reduces in this case to that of a one-component Fermi vapour, whereas the frequencies of all oth er modes are shifted by the interactions.