Harmonically trapped two-atom systems: Interplay of short-range<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>s</mml:mi></mml:math>-wave interaction and spin-orbit coupling

Abstract

The coupling between the spin degrees of freedom and the orbital angular momentum has a profound effect on the properties of nuclei, atoms, and condensed-matter systems. Recently, synthetic gauge fields have been realized experimentally in neutral cold-atom systems, giving rise to a spin-orbit coupling term with ``strength'' ${k}_{\text{so}}$. This paper investigates 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. Specifically, we consider two different harmonically trapped two-atom systems. The first system consists of an atom with spin-orbit coupling that interacts with a structureless particle through a short-range two-body potential. The second system consists of two atoms that both feel the spin-orbit coupling term and that interact through a short-range two-body potential. Treating the spin-orbit term perturbatively, we determine the correction to the ground-state energy for various generic parameter combinations. Selected excited states are also treated. An important aspect of our study is that the perturbative treatment is not limited to small $s$-wave scattering lengths but provides insights into the system behavior over a wide range of scattering lengths, including the strongly interacting unitary regime. We find that the interplay between the spin-orbit coupling term and the $s$-wave interaction generically enters, depending on the exact parameter combinations of the $s$-wave scattering lengths, at order ${k}_{\text{so}}^{2}$ or ${k}_{\text{so}}^{4}$ for the ground state and leads to a shift of the energy of either sign. While the absence of a term proportional to ${k}_{\text{so}}$ follows straightforwardly from the functional form of the spin-orbit coupling term, the absence of a term proportional to ${k}_{\text{so}}^{2}$ for certain parameter combinations is unexpected. The well-known fact that the spin-orbit coupling term couples the relative and center-of-mass degrees of freedom has interesting consequences for the trapped two-particle systems. For example, we find that, for certain parameter combinations, the spin-orbit coupling term turns sharp crossings into avoided crossings with an energy splitting proportional to ${k}_{\text{so}}$. 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.