Collective excitations across the BCS-BEC crossover induced by a synthetic Rashba spin-orbit coupling

Synthetic non-Abelian gauge fields in cold atom systems produce a Rashba spin-orbit interaction described by a vector $\blam = (\lambda_x, \lambda_y, \lambda_z)$. It was recently shown [Phys. Rev. B 84, 014512 (2011)] that on increasing $\lambda = |\blam|$, fermions at a finite density $\rho\approx\kf^3$ evolve to a BEC like state even in the presence of a weak attractive interaction (described by a scattering length $\as$). The BEC obtained at large spin-orbit coupling ($\lambda \gg k_F$) is a condensate of rashbons -- novel bosonic bound pairs of fermions whose properties are determined solely by the gauge field. Here we study the collective excitations of such superfluids by constructing a Gaussian theory using functional integral methods. We derive explicit expressions for superfluid phase stiffness, sound speed and mass of the Anderson-Higgs boson that are valid for any $\blam$ and scattering length. We find that at finite $\lambda$, the phase stiffness is always lower than that set by the density of particles, consistent with earlier work[arXiv:1110.3565] which attributed this to the lack of Galilean invariance of the system at finite $\lambda$. We show that there is an emergent Galilean invariance at large $\lambda$, and the phase stiffness is determined by the rashbon density and mass, consistent with Leggett's theorem. We further demonstrate that the rashbon BEC state is a superfluid of anisotropic rashbons interacting via a contact interaction characterized by a rashbon-rashbon scattering length $a_R$. We show that $a_R$ goes as $\lambda^{-1}$ and is essentially {\em independent} of the scattering length between the fermions as long as it is nonzero. Analytical results are presented for a rashbon BEC obtained in a spherical gauge field with $\lambda_x = \lambda_y = \lambda_z = \frac{\lambda}{\sqrt{3}}$.