Spectroscopy of the Hyperfine Structure of HD$^+$ in Rotationally Excited States with 10-ppb Uncertainty

Abstract

We report the first measurement of the total hyperfine interval in the manifold of the ($v=0,L=3$) rovibrational state of HD$^+$ using microwave spectroscopy of HD$^+$ ions in a linear Paul trap. To overcome the low (0.2%) occupancy of the quantum states involved, we employ a novel technique that combines intermittent Majorana depolarization of the HD$^+$ ensemble with redistribution of rotational-state population by blackbody radiation to effectively amplify the occupancy by a factor of 25. This enables the observation of a single field-insensitive magnetic subcomponent of the hyperfine transition, leading to a measured total hyperfine interval of ${1\,050\,804.511}\pm 0.011\,$kHz with an unprecedented relative uncertainty of 10 parts per billion. This differs from the theoretically predicted value ${1\,050\,802.78}\pm 0.89\,$kHz by 1.9$σ$. We show that our experimental value is consistent with the recently measured hyperfine structure of the ($v=0,L=0$) rovibrational state of HD$^+$ [C.M. König et al., High-precision Penning trap spectroscopy of the ground state spin structure of HD$^+$, Phys. Rev. Lett. $\mathbf{136}$,143002 (2026)], for which a similar deviation from theory was found. Our work may help resolve puzzling discrepancies between the theoretical and experimental hyperfine structure observed previously in optical rovibrational spectra of HD$^+$.