# Spin-orbit coupled bosons interacting in a two-dimensional harmonic trap

## TL;DR

This paper investigates the ground state and low-energy spectra of two bosons with Rashba spin-orbit coupling in a 2D harmonic trap, revealing energy-level crossings and structural changes due to interactions.

## Contribution

It provides an analytic and numerical analysis of the two-boson system, including beyond-approximation results and insights into energy-level crossings caused by spin-orbit coupling and interactions.

## Key findings

- Energy-level crossing between states of different $J_z$ in the ground state.
- Discontinuities in energy contributions and spatial density at strong interactions.
- Relation between spin-orbit expectation value and $raket{\sigma_z raket{L_z}$ for eigenstates.

## Abstract

A system of bosons in a two-dimensional harmonic trap in the presence of Rashba-type spin-orbit coupling is investigated. An analytic treatment of the ground state of a single atom in the weak-coupling regime is presented and used as a basis for a perturbation theory in the interacting two-boson system. The numerical diagonalization of both the single-particle and the two-boson Hamiltonian matrices allows us to go beyond those approximations and obtain not only the ground state, but also the low-energy spectra and the different energy contributions separately. We show that the expectation value of the spin-orbit term is related to the expectation value of $\hat{\sigma}_z \hat{L}_z$ for the eigenstates of the system, regardless of the trapping potential. The low-energy states of the repulsively interacting two-boson system are characterized. With the presence of a sufficiently strong interaction and spin-orbit coupling strength, there is a direct energy-level crossing in the ground state of the system between states of different $J_z$, the third component of the total angular momentum, that changes its structure. This is reflected in a discontinuity in the different energy terms and it is signaled in the spatial density of the system.