High-precision Penning-trap spectroscopy of the ground-state spin structure of HD+

TL;DR

The work targets HD$^+$ hyperfine structure and the bound-electron $g$-factor to test high-precision QED in a molecular ion and to constrain fundamental constants. Using a cryogenic 4 T Penning-trap with a double-trap ESR sequence, six ground-state transitions are measured and a global fit determines $g_{e,\mathrm{bound}}(0,0)$, $E_4(0,0)$, and $E_5(0,0)$. The extracted values are $g_{e,\mathrm{bound}}(0,0) = -2.00227854096(40)$, $E_4(0,0) = 925395.758(39)$ kHz, and $E_5(0,0) = 142287.821(20)$ kHz, in agreement with ab initio theory incorporating corrections up to order $\alpha^5$. The results represent the most precise bound-electron g-factor measurement in a molecule and establish a platform for precision molecular QED tests and future determinations of fundamental constants.

Abstract

We present high-precision spectroscopy of the ground-state hyperfine structure of HD$^+$ at 4~T. We determine the bound-electron $g$ factor, $g_{e,\mathrm{bound}} = -2.002\,278\,540\,96(40)$, to a relative uncertainty of $2\times$10$^{-10}$, the most precise determination of a bound-electron $g$ factor of a molecular ion to date. The experimental value agrees with recently developed ab initio theory that now includes quantum-electrodynamical effects up to order $α^5$ and has reduced the theoretical uncertainty by three orders of magnitude [O. Kullie \textit{et al.}, Phys. Rev. A 112 052813 (2025)]. In addition, we extract the scalar spin-spin interaction coefficients $E_4$~=~925\,395.758(41)$\,$kHz (electron-proton) and $E_5$~=~142\,287.821(22)$\,$kHz (electron-deuteron), which show a moderate tension with another state-of-the-art theoretical prediction [M. Haidar \textit{et al.}, Phys. Rev. A 106 042815 (2022)].

Figures (3)

• Figure 1: The level scheme of HD$^+$ hyperfine structure in the rovibrational ground state at 4.02$\,$T. The blue arrows represent the six measured electron-spin-flip transitions and the red and green lines indicate the energy level splittings associated with the proton and deuteron spins, respectively. The first excited rotational state is separated from the ground state by about $1.3$ THz.
• Figure 2: (a) Representative resonance data and fit, in this case for transition VI (shown in Fig$.$\ref{['fig:level_resonances']}). Individual spin flip attempts at a given detuning to the best-fit frequency value, $\Delta\nu_{\textnormal{HFS}}$, are marked with a dark green X (successful) or a dark red dot (unsuccessful). The blue curve is a Gaussian maximum-likelihood fit to the data and the blue and orange shaded areas are the confidence bands of the fitted lineshape parameters and center value, respectively. . The black binned points are provided for illustration but are not used to fit and extract values. (b) The fitted line center of all six resonances relative to the values calculated from the effective Hamiltonian in Eq$.$ 1 using the values of $E_4$, $E_5$, and $g_{e,\rm bound}$ from both ab initio theory (gray) and a global fit to the experimental data (blue). The shaded regions indicate the 1$\sigma$ confidence bands of these predictions and text labels indicate the number of successful and attempted spin flips. The differences in experimentally determined values of $E_4$ and $E_5$ from the ab initio calculations (see Tab.$\,$\ref{['tab:results']}) shift the black points right and left depending on the respective proton and deuteron orientations $m_\mathrm{I,p}$, $m_\mathrm{I,d}$ of each transition. The absence of a uni-directional shift of all transitions reflects the good agreement between experimental and theoretical values of $g_{e,\rm bound}$. (c) The same as (b) showing only the results of the global fit that determines both the best fit values for $E_4$, $E_5$, and $g_{e,\rm bound}$ and the line centers. The agreement between the predicted values and the line centers shows that the data is well-described by the effective Hamiltionian with the experimentally determined hyperfine coefficients.
• Figure 3: a) Schematic of a typical measurement cycle. In the AT the internal state of the ion is determined (purple section) by observing an electron spin flip with 50$\%$ spin-flip probability - depicted via slanted blue lines. The precision measurement is performed in the PT (orange section). b) An axial dip and modified cyclotron double dip, seen on the axial resonator, both of which are used to determine the free cyclotron frequency $\nu_c$ -- see text for details. c) The measurement of the modified cyclotron frequency is depicted with a single phase-sensitive measurement sequence with a $5.2$ s phase evolution time during which the MW is injected. Multiple phase evolution times $\tau_i$ are required for phase unwrapping and the ion is cooled for 50 s preceding each phase measurement. Adapted from Ref$.$koenig_thesis.