Two-body problem of impurity atoms in dipolar Fermi gas

The polarized dipolar Fermi gas shows exotic properties at low temperatures, characterized by an axially-deformed Fermi surface and anisotropic single-particle energy, due to the long-range and anisotropic nature of dipole-dipole interaction. In cold-atom experiments such a system has been realized, e.g., in degenerate gas of Er and Dy atoms. In the case that non-dipolar impurity atoms are introduced in such system, they undergoes an induced interaction mediated by the density fluctuations of the background dipolar Fermi gas. We derive the induced interaction potential to the single-loop order of fluctuations and show that it becomes indeed an anisotropic Ruderman-Kittel-Kasuya-Yosida-type potential which preserves the axial symmetry around the polarization axis. We then solve the two-body problem of impurity atoms interacting via the anisotropic potential and figure out the dependence of bound state and scattering properties on the parameters of dipolar Fermi gas.