# Study of systematic effects in the HfF$^+$ ion experiment to search for the electron electric dipole moment

## TL;DR

This paper investigates systematic effects in the HfF$^+$ ion experiment aimed at detecting the electron electric dipole moment by calculating hyperfine energy splittings under external fields using two theoretical approaches.

## Contribution

It provides a detailed theoretical analysis of hyperfine level shifts in HfF$^+$ ions, aiding in understanding systematic errors in eEDM experiments.

## Key findings

- Calculated hyperfine energy splittings as functions of electric and magnetic fields.
- Compared two approaches: rotating frame transformation and quantization of rotating fields.
- Results help identify potential systematic errors in eEDM measurements.

## Abstract

The energy splitings for $J = 1$, $F=3/2$, $|M_F|=3/2$ hyperfine levels of the $^3\Delta_1$ electronic state of $^{180}$Hf$^{19}$F$^+$ ion are calculated as functions of the external variable electric and magnetic fields within two approaches. In the first one transition to the rotating frame is performed, whereas in the second approach the quantization of rotating electromagnetic field is performed. Calculations are required for understanding possible systematic errors in the experiment to search for electron electric dipole moment (eEDM) on $^{180}$Hf$^{19}$F$^+$ ion.