# Anisotropic optical trapping as a manifestation of the complex electronic structure of ultracold lanthanide atoms: the example of holmium

## TL;DR

This paper develops an analytical formalism to calculate the anisotropic optical polarizabilities of lanthanide atoms, revealing how electronic structure and configuration interaction influence trapping efficiency and photon scattering in ultracold holmium.

## Contribution

It introduces a new analytical approach to evaluate scalar, vector, and tensor polarizabilities considering configuration interaction effects in lanthanide atoms.

## Key findings

- Polarizability components depend strongly on configuration interaction coefficients.
- Imaginary parts of polarizabilities are highly sensitive to electronic structure details.
- Holmium's electronic structure shows stronger configuration interaction than dysprosium, affecting optical trapping properties.

## Abstract

The efficiency of optical trapping is determined by the atomic dynamic dipole polarizability, whose real and imaginary parts are associated with the potential energy and photon-scattering rate respectively. In this article we develop a formalism to calculate analytically the real and imaginary parts of the scalar, vector and tensor polarizabilities of lanthanide atoms. We assume that the sum-over-state formula only comprises transitions involving electrons in the valence orbitals like $6s$, $5d$, $6p$ or $7s$, while transitions involving $4f$ core electrons are neglected. Applying this formalism to the ground level of configuration $4f^q6s^2$, we restrict the sum to transitions implying the $4f^q6s6p$ configuration, which yields polarizabilities depending on two parameters: an effective transition energy and an effective transition dipole moment. Then, by introducing configuration-interaction mixing between $4f^q6s6p$ and other configurations, we demonstrate that the imaginary part of the scalar, vector and tensor polarizabilities is very sensitive to configuration-interaction coefficients, whereas the real part is not. The magnitude and anisotropy of the photon-scattering rate is thus strongly related to the details of the atomic electronic structure. Those analytical results agree with our detailed electronic-structure calculations of energy levels, Land\'e $g$-factors, transition probabilities, polarizabilities and van der Waals $C_6$ coefficients, previously performed on erbium and dysprosium, and presently performed on holmium. Our results show that, although the density of states decreases with increasing $q$, the configuration interaction between $4f^q6s6p$, $4f^{q-1}5d6s^2$ and $4f^{q-1}5d^26s$ is surprisingly stronger in erbium ($q=12$), than in holmium ($q=11$), itself stronger than in dysprosium ($q=10$).