Precision Test of Many-Body QED in the Be${}^{+}$ $2p$ Fine Structure Doublet Using Short-Lived Isotopes

Absolute transition frequencies of the $2s; ^2{\rm S}{1/2} \rightarrow 2p;^2$\mathrm{P}${1/2,3/2}$transitionsinBe$^+$ were measured for the isotopes ${}^{7,9-12}$Be. The fine structure splitting of the $2p$ state and its isotope dependence are extracted and compared to results of $\mathit{\text{ab}}\mathit{\text{initio}}$ calculations using explicitly correlated basis functions, including relativistic and quantum electrodynamics effects at the order of $m{\alpha }^6$ and $m{\alpha }^7\ln \alpha$. Accuracy has been improved in both the theory and experiment by 2 orders of magnitude, and good agreement is observed. This represents one of the most accurate tests of quantum electrodynamics for many-electron systems, being insensitive to nuclear uncertainties.