Harmonically trapped two-atom systems: Interplay of short-range $s$-wave interaction and spin-orbit coupling

We investigate the interplay between the single-particle spin-orbit coupling term of Rashba type and the short-range two-body $s$-wave interaction for cold atoms under external confinement. Treating the spin-orbit term with strength $k_{so}$ perturbatively, we determine the correction to the ground state energy for various parameter combinations. We find that the interplay between the spin-orbit coupling term and the $s$-wave interaction enters, depending on the exact parameter combinations of the $s$-wave scattering lengths, at order $k_{so}^2$ or $k_{so}^4$ for the ground state and leads to a shift of the energy of either sign. Additionally, we find that, for certain parameter combinations, the spin-orbit coupling term turns sharp crossings into avoided crossings with an energy splitting proportional to $k_{so}$. Our perturbative results are confirmed by numerical calculations that expand the eigenfunctions of the two-particle Hamiltonian in terms of basis functions that contain explicitly correlated Gaussians.