Energy spectrum and spin structure of harmonically trapped one-dimensional atoms with spin-orbit coupling

Ultracold atomic gases provide a novel platform with which to study spin-orbit coupling, a mechanism that plays a central role in the nuclear shell model, atomic fine structure and two-dimensional electron gases. We introduce a theoretical framework that allows for the efficient determination of the energy spectrum and spin structure of harmonically trapped atoms with zero-range interactions subject to an equal mixture of Rashba and Dresselhaus spin-orbit coupling created through Raman coupling of atomic hyperfine states. The spin structure of bosonic and fermonic two-particle systems with finite and infinite interaction strength $g$ is calculated. Taking advantage of the fact that the $N$-boson and $N$-fermion systems with infinitely large coupling strength $g$ are analytically solvable for vanishing spin-orbit coupling strength $k_{so}$ and vanishing Raman coupling strength $\Omega$, we develop an effective spin model that is accurate to second-order in $\Omega$ for any $k_{so}$ and infinite $g$. The three- and four-particle systems are considered explicitly. It is shown that the effective spin Hamiltonian describes the transitions that these systems undergo with the change of $k_{so}$ as a competition between independent spin dynamics and nearest-neighbor spin interactions.