We present a generalization of the fluid-dynamical scheme developed for nuclear physics to the case of two trapped fermion species with pairing interactions. To establish a macroscopic description of the mass and momentum conservation laws, we adopt a generalization of the usual Thomas-Fermi approach that includes the pairing energy. We analyze the equilibrium density and gap profiles for an equal population mixture of harmonically trapped
6Li atoms for different choices of the local equation of state. We examine slight departures from equilibrium within our formulation, finding that density oscillations can propagate as first sound coupled to pairing vibrations, that in a homogeneous fermion system exhibit a Bogoliubov-like quasiparticle spectrum. In this case, the dispersion relation for the coupled modes displays a rich scenario of stable, unstable, and damped regimes.
A theory that describes the dynamic behavior of a system containing an arbitrary number of different interacting Bose species is developed. The excited eigenstates of the system are generated by the superposition of plane-wave creation- and annihilation-operator products ${{\ensuremath{\mu}}_{k}}^{\ifmmode\dagger\else\textdagger\fi{}}=\ensuremath{\Sigma}{\ensuremath{\gamma}}^{}\ensuremath{\Sigma}{p}^{}g(p,k,\ensuremath{\gamma}){{a}_{p+k}}^{\ifmmode\dagger\else\textdagger\fi{}}(\ensuremath{\gamma}){a}_{p}(\ensuremath{\gamma})$, where the
c-number coefficients $g(p,k,\ensuremath{\gamma})$ are determined by solving the equation of motion for ${{\ensuremath{\mu}}_{k}}^{\ifmmode\dagger\else\textdagger\fi{}}$ in an approximate way. The sum over $\ensuremath{\gamma}$ extends over all different species of the system. The equation of motion is solved by a method similar to that originally derived by Pines and Bohm for the electron gas. Results are obtained for the energy spectrum of the elementary excitations in the system. In particular, explicit results are obtained for the energy spectrum of a two-species system---first for a weakly coupled gas, and secondly for the low-momentum collective modes of a strongly coupled fluid. The low-momentum excitations of the strongly coupled fluid are completely analogous to the "zero sound" modes commonly associated with Fermi liquids. The theory is then applied to a dilute mixture of
He3 in
He4 and the phonon velocity is determined as a function of
He3 concentration. The probability per unit time $P(\mathbf{k},{\ensuremath{\omega}}_{k})$ that a particle transfers momentum k and energy ${\ensuremath{\omega}}_{k}$ to the system by scattering is also calculated in terms of the dynamic form factor ${S}_{\ensuremath{\gamma}}(\mathbf{k},\ensuremath{\omega})$.