Spin structure of harmonically trapped one-dimensional atoms with spin-orbit coupling

TL;DR

This paper develops a theoretical method to analyze the spin structure of one-dimensional harmonically trapped atoms with spin-orbit coupling, considering both bosonic and fermionic systems with varying interaction strengths.

Contribution

It introduces an effective spin model accurate to second-order in Raman coupling, capturing the transition dynamics in few-particle systems with spin-orbit coupling.

Findings

Effective spin Hamiltonian includes Heisenberg and Dzyaloshinskii-Moriya terms.

Model accurately describes spin transitions with changing spin-orbit coupling.

Explicit analysis of three- and four-particle systems.

Abstract

We introduce a theoretical approach to determine the spin structure of harmonically trapped atoms with two-body 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 fermionic two-particle systems with finite and infinite two-body 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, which contains a Heisenberg exchange term and an anisotropic Dzyaloshinskii-Moriya exchange term, 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.