Multipolar Fermi surface deformation in a Rydberg-dressed Fermi gas with long-range anisotropic interactions.

We study theoretically the deformation of the Fermi surface (FS) of a three dimensional gas of Rydberg-dressed $^6$Li atoms. The laser dressing to high-lying Rydberg $D$-states results in angle-dependent soft-core shaped interactions whose anisotropy is described by multiple spherical harmonics. We show that this can drastically modify the shape of the FS and that its deformation depends on the interplay between the Fermi momentum $k_F$ and the reciprocal momentum $\bar{k}$ corresponding to the characteristic soft-core radius of the dressing-induced potential. When $k_F< \bar{k}$, the dressed interaction stretches a spherical FS into an ellipsoid. When $k_F\gtrsim \bar{k}$, complex deformations are encountered which exhibit multipolar characteristics. We analyze the formation of Cooper pairs around the deformed FS, and show that they occupy large orbital angular momentum states ($p$-, $f$- and $h$-wave) coherently. Our study demonstrates that Rydberg-dressing to high angular momentum states may pave a route towards the investigation of unconventional Fermi gases, and multiwave superconductivity.