Rotational spectroscopy of cold and trapped molecular ions in the Lamb-Dicke regime

Sympathetic cooling of trapped ions has been established as a powerful technique for the manipulation of non-laser-coolable ions^1-4. For molecular ions, it promises vastly enhanced spectroscopic resolution and accuracy. However, this potential remains untapped so far, with the best resolution achieved being not better than 5 × 10^-8 fractionally, due to residual Doppler broadening being present in ion clusters even at the lowest achievable translational temperatures⁵. Here we introduce a general and accessible approach that enables Doppler-free rotational spectroscopy. It makes use of the strong radial spatial confinement of molecular ions when trapped and crystallized in a linear quadrupole trap, providing the Lamb-Dicke regime for rotational transitions. We achieve a linewidth of 1 × 10^-9 fractionally and 1.3 kHz absolute, an improvement of ≃50-fold over the previous highest resolution in rotational spectroscopy. As an application, we demonstrate the most precise test of ab initio molecular theory and the most accurate (1.3 × 10^-9) determination of the proton mass using molecular spectroscopy. The results represent the long overdue extension of Doppler-free microwave spectroscopy of laser-cooled atomic ion clusters⁶ to higher spectroscopy frequencies and to molecules. This approach enables a wide range of high-accuracy measurements on molecules, both on rotational and, as we project, vibrational transitions.