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Hydrogen-like light muonic ions, in which one negative muon replaces all theelectrons, are extremely sensitive probes of nuclear structure, because thelarge muon mass increases tremendously the wave function overlap with thenucleus. Using pulsed laser spectroscopy we have measured three 2S-2Ptransitions in the muonic helium-3 ion ($\mu^3$He$^+$), an ion formed by anegative muon and bare helium-3 nucleus. This allowed us to extract the Lambshift $E(2P_{1/2}-2S_{1/2})= 1258.598(48)^{\rm exp}(3)^{\rm theo}$ meV, the 2Pfine structure splitting $E_{\rm FS}^{\rm exp} = 144.958(114)$ meV, and the2S-hyperfine splitting (HFS) $E_{\rm HFS}^{\rm exp} = -166.495(104)^{\rmexp}(3)^{\rm theo}$ meV in $\mu^3$He$^+$. Comparing these measurements totheory we determine the rms charge radius of the helion ($^3$He nucleus) to be$r_h$ = 1.97007(94) fm. This radius represents a benchmark for few nucleontheories and opens the way for precision tests in $^3$He atoms and $^3$He-ions.This radius is in good agreement with the value from elastic electronscattering, but a factor 15 more accurate. Combining our Lamb shift measurementwith our earlier one in $\mu^4$He$^+$ we obtain $r_h^2-r_\alpha^2 =1.0636(6)^{\rm exp}(30)^{\rm theo}$ fm$^2$ to be compared to results from theisotope shift measurements in regular He atoms, which are however affected bylong-standing tensions. By comparing $E_{\rm HFS}^{\rm exp}$ with theory wealso obtain the two-photon-exchange contribution (including higher orders)which is another important benchmark for ab-initio few-nucleon theories aimingat understanding the magnetic and current structure of light nuclei.

The helion charge radius from laser spectroscopy of muonic helium-3 ions