arXiv:2404.19293 · Physical Review Research

Two atoms in a harmonic trap with spin-orbital-angular-momentum coupling

Xiao-Long Chen, Aixi Chen, Shi-Guo Peng

Published Aug 21, 2024• 1 citation

Abstract

We study the problem of two harmonically trapped atoms in the presence of spin-orbital-angular-momentum (SOAM) coupling. The two-body energy spectrum is numerically calculated by utilizing the exact diagonalization method. We analyze how the degeneracy of energy levels is lifted under the interplay between the interatomic interaction and SOAM coupling. The exact numerical results show excellent agreement with that of perturbation theory in the weak-interaction limit as well as that in the absence of SOAM coupling. The properties of correlations between the two atoms are also discussed with respect to the interaction strength. The findings in this paper may provide valuable insights into few-body physics subjected to SOAM coupling and the possible experimental detection of spectrum functions in many-body systems, such as radiofrequency spectroscopy. Published by the American Physical Society 2024

DOI: 10.1103/physrevresearch.6.033200

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arXiv:2404.19293 · Physical Review Research

Two atoms in a harmonic trap with spin-orbital-angular-momentum coupling

Xiao-Long Chen, Aixi Chen, Shi-Guo Peng

Published Aug 21, 2024• 1 citation

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevresearch.6.033200

Related papers

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Chuanzhou Zhu, Lin Dong, Han Pu

Abstract We study harmonically trapped one-dimensional atoms subjected to an equal combination of Rashba and Dresselhaus spin–orbit coupling induced by Raman transition. We first examine the wave function and the degeneracy of the single-particle ground state, followed by a study of two weakly interacting bosons or fermions. For the two-particle ground state, we focus on the effects of the interaction on the degeneracy, the spin density profiles, and the density–density correlation functions. Finally we show how these studies help us to understand the many-body properties of the system.Abstract We study harmonically trapped one-dimensional atoms subjected to an equal combination of Rashba and Dresselhaus spin–orbit coupling induced by Raman transition. We first examine the wave function and the degeneracy of the single-particle ground state, followed by a study of two weakly interacting bosons or fermions. For the two-particle ground state, we focus on the effects of the interaction on the degeneracy, the spin density profiles, and the density–density correlation functions. Finally we show how these studies help us to understand the many-body properties of the system.

arXiv:1601.01335Journal of Physics B Atomic Molecular and Optical PhysicsJun 23, 201624 citations

{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}

{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.

{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.

s

-wave interaction exists between two spin species that are SOAM coupled and an atom with SOAM coupled internal spins interacts state selectively with another atom. For both cases, we identify the condition for the emergence of molecular states with finite orbital angular momenta. These molecular states with quantized orbital angular momenta correspond to the SOAM-coupling-induced vortices in the corresponding Fermi superfluid. We propose to detect the molecules through Raman spectroscopy with Laguerre-Gaussian lasers. As the molecular states can form above the superfluid temperature, they offer an experimentally more accessible route toward the study of the underlying pairing mechanism under SOAM coupling.

We numerically investigate the following two-body stationary Schrödinger equation (SE):

\left\{ { - \frac{{{{\rlap{--} h}^2}}}{{2m}}\frac{\partial }{{x_1^2}} - \frac{{{{\rlap{--} h}^2}}}{{2m}}\frac{\partial }{{x_2^2}} + V\left( {{x_2}} \right) + g\partial \left( {{x_1} - {x_2}} \right)} \right\}x\Psi \left( {{x_1},{x_2}} \right) = E\Psi \left( {{x_1},{x_2}} \right),

where

{x_1},{x_2} \in \mathbb{R},V\left( {{x_1}} \right) = {V_0}{\sin ^2}\left( {{k_x}{x_i}} \right)

is the potential describing the interaction of atoms with a trap and gδ(x 1–x 2) is the interatomic potential. Previously, a similar problem has been solved analytically for the harmonic interaction

{V_h}\left( {{x_i}} \right) = \frac{1}{2}m\omega x_i^2

with the trap, which leads to the separation of coordinates

y = \frac{{{x_1} + {x_2}}}{{\sqrt 2 }},{\kern 1pt} x = \frac{{{x_1} - {x_2}}}{{\sqrt 2 }}

for the center-of-mass and relative motion. The anharmonicity of the trap couples these motions and, there-fore, the problem becomes significantly more complicated. In previous works, the anharmonicity V a = V–V h of the trap has been taken into account in the framework of perturbation theory. In this work, the energy level shifts of a two-body atomic system are calculated beyond the perturbation theory for different magnitudes of parameter g of the interatomic interaction. The results are compared to those computed using the perturbation theory.