Inner-shell electrons naturally sense the electric field close to the nucleus, which can reach extreme values beyond 10 15 V cm −1 for the innermost electrons 1 . Especially in few-electron, highly charged ions, the interaction with the electromagnetic fields can be accurately calculated within quantum electrodynamics (QED), rendering these ions good candidates to test the validity of QED in strong fields. Consequently, their Lamb shifts were intensively studied in the past several decades 2 , 3 . Another approach is the measurement of gyromagnetic factors ( g factors) in highly charged ions 4 – 7 . However, so far, either experimental accuracy or small field strength in low- Z ions 5 , 6 limited the stringency of these QED tests. Here we report on our high-precision, high-field test of QED in hydrogen-like 118 Sn 49+ . The highly charged ions were produced with the Heidelberg electron beam ion trap (EBIT) 8 and injected into the ALPHATRAP Penning-trap setup 9 , in which the bound-electron g factor was measured with a precision of 0.5 parts per billion (ppb). For comparison, we present state-of-the-art theory calculations, which together test the underlying QED to about 0.012%, yielding a stringent test in the strong-field regime. With this measurement, we challenge the best tests by means of the Lamb shift and, with anticipated advances in the g -factor theory, surpass them by more than an order of magnitude.

Helium-3 has nowadays become one of the most important candidates for studies in fundamental physics1-3, nuclear and atomic structure4,5, magnetometry and metrology6, as well as chemistry and medicine7,8. In particular, 3He nuclear magnetic resonance (NMR) probes have been proposed as a new standard for absolute magnetometry6,9. This requires a high-accuracy value for the 3He nuclear magnetic moment, which, however, has so far been determined only indirectly and with a relative precision of 12 parts per billon10,11. Here we investigate the 3He+ ground-state hyperfine structure in a Penning trap to directly measure the nuclear g-factor of 3He+ [Formula: see text], the zero-field hyperfine splitting [Formula: see text] Hz and the bound electron g-factor [Formula: see text]. The latter is consistent with our theoretical value [Formula: see text] based on parameters and fundamental constants from ref. 12. Our measured value for the 3He+ nuclear g-factor enables determination of the g-factor of the bare nucleus [Formula: see text] via our accurate calculation of the diamagnetic shielding constant13 [Formula: see text]. This constitutes a direct calibration for 3He NMR probes and an improvement of the precision by one order of magnitude compared to previous indirect results. The measured zero-field hyperfine splitting improves the precision by two orders of magnitude compared to the previous most precise value14 and enables us to determine the Zemach radius15 to [Formula: see text] fm.

A critical set of two-loop quantum electrodynamics corrections to the g factor of hydrogenlike ions is calculated in the Furry picture. These corrections are due to the polarization of the external magnetic field by the quantum vacuum, which is dressed by the binding field. The result obtained for the self-energy-magnetic-loop diagrams is compared with the current state-of-the-art result, derived through a perturbative expansion in the binding strength parameter Z alpha, with Z the atomic number and alpha the fine-structure constant. Agreement is found in the Z -> 0 limit. However, even for very light ions, the perturbative result fails to approximate the magnitude of the corresponding correction to the g factor. The total correction to the g factor coming from all diagrams considered in this work is found to be highly relevant for upcoming experimental tests of fundamental physics with highly charged ions.

Two-loop self-energy corrections to the bound-electron g factor are investigated theoretically to all orders in the nuclear binding strength parameter Zα. The separation of divergences is performed by dimensional regularization, and the contributing diagrams are regrouped into specific categories to yield finite results. We evaluate numerically the loop-after-loop terms, and the remaining diagrams by treating the Coulomb interaction in the electron propagators up to first order. The results show that such two-loop terms are mandatory to take into account for projected near-future stringent tests of quantum electrodynamics and for the determination of fundamental constants through the g factor.

QED corrections to the g factor of Li-like and B-like ions in a wide range of nuclear charges are presented. Many-electron contributions as well as radiative effects on the one-loop level are calculated. Contributions resulting from the interelectronic interaction, the self-energy effect, and most of the terms of the vacuum-polarization effect are evaluated to all orders in the nuclear coupling strength Zα. Uncertainties resulting from nuclear size effects, numerical computations, and uncalculated effects are discussed.

Recent years have witnessed a remarkable improvement in the theoretical description of bound-electron g factors, paralleled with a quantum jump in the experimental accuracy in the investigation of these quantities. In the present article we give a brief summary of the latest developments, with emphasis on the influence of quantum electrodynamic and nuclear effects on the g factor of few-electron highly charged ions, and on the possible determination of fundamental constants.

A theoretical description of the g factor of a muon bound in a nuclear potential is presented. One-loop self-energy and multiloop vacuum polarization corrections are calculated, taking into account the interaction with the binding potential exactly. Nuclear effects on the bound-muon g factor are also evaluated. We put forward the measurement of the bound-muon g factor via the continuous Stern-Gerlach effect as an independent means to determine the muon's magnetic moment anomaly and mass. The scheme presented enables the increase of the accuracy of the mass by more than an order of magnitude.

The determination of the electron mass from Penning- trap measurements with C-12(5+) ions and from theoretical results for the bound- electron g factor is described in detail. Some recently calculated contributions slightly shift the extracted mass value. Prospects of a further improvement of the electron mass are discussed both from the experimental and from the theoretical points of view. Measurements with He-4(+) ions will enable a consistency check of the electron- mass value, and in the future an improvement of the He-4 nuclear mass and a determination of the fine-structure constant.

The muonic vacuum polarization contribution to the $g$-factor of the electron bound in a nuclear potential is investigated theoretically. The electric as well as the magnetic loop contributions are evaluated. We found these muonic effects to be observable in planned trapped-ion experiments with light and medium-heavy highly charged ions. The enhancement due to the strong Coulomb field boosts these contributions much above the corresponding terms in the free-electron $g$-factor. Due to their magnitude, muonic vacuum polarization terms are also significant in planned determinations of the fine-structure constant from the bound-electron $g$-factor.