Universal relations for spin-orbit coupled Fermi gas near an s-wave resonance

The synthetic spin-orbit coupled quantum gases is widely studied both experimentally and theoretically in recent years. As previous studies show, this modification of single-body dispersion will in general couple different partial waves and thus distort the wave-function of bound states which determines the short-distance behavior of many-body wave function. In this work, we focus on the two-component Fermi gas with one-dimensional or three-dimensional spin-orbit coupling near an s-wave resonance. Using the method of effective field theory and the operator product expansion, we derive universal relations for both systems, and obtain the momentum distribution matrix $\left<\psi^\dagger_a(\mathbf{q})\psi_b(\mathbf{q})\right>$ at large $\mathbf{q}$ ($a,b$ are spin index), which shows anisotropic features. We also discuss the experimental implication of these results depending on the realization of the spin-orbit coupling.