# Compensation of the trap-induced quadrupole interaction in trapped Rydberg ions

## TL;DR

This paper analyzes the quadrupole interaction in trapped Rydberg ions, showing how it can be effectively compensated and its impact on dipole-dipole interactions and spin-spin coupling.

## Contribution

It introduces an effective Hamiltonian framework to account for and mitigate trap-induced quadrupole effects in Rydberg ions.

## Key findings

- Quadrupole interaction can be replaced by rescaled laser intensities and Stark shifts.
- Compensation of quadrupole effects is achievable by increasing Rabi frequencies.
- Quadrupole coupling influences dipole-dipole and spin-spin interactions.

## Abstract

The quadrupole interaction between the Rydberg electronic states of a Rydberg ion and the radio frequency electric field of the ion trap is analyzed. Such a coupling is negligible for the lowest energy levels of a trapped ion but it is important for a trapped Rydberg ion due to its large electric dipole moment. This coupling cannot be neglected by the standard rotating-wave approximation because it is comparable to the frequency of the trapping electric field. We investigate the effect of the quadrupole coupling by performing a suitable effective representation of the Hamiltonian. For a single ion we show that in this effective picture the quadrupole interaction is replaced by rescaled laser intensities and additional Stark shifts of the Rydberg levels. Hence this detrimental quadrupole coupling can be efficiently compensated by an appropriate increase of the Rabi frequencies. Moreover, we consider the strong dipole-dipole interaction between a pair of Rydberg ions in the presence of the quadrupole coupling. In the effective representation we observe reducing of the dipole-dipole coupling as well as additional spin-spin interaction.