Phonon spectrum and damping of a dilute quantum degenerate Bose-Fermi mixture at finite temperature

In this Letter we investigated Bose-Fermi gas mixture at finite temperature by using Green's function method. First for spin-polarized system the temperature effects on the phonon spectrum have been taken into account. Second we consider a dilute mixture of a Bose gas and a two components Fermi gas when both bosons and fermions have undergone superfluid transitions. In this case the interaction between two hyperfine Fermi components is important. In addition we calculate the dispersion of phonon excitations, including correction to the sound velocity and Landau damping. The damping rate varies as T2 in low temperatures limit. There is also damping at absolute zero temperature due to the interaction between bosons and fermions.