By measurement of the frequency of a vibrational overtone transition in the molecular hydrogen ion HD+, we demonstrate optical spectroscopy of trapped molecular ions with submegahertz accuracy. We use a diode laser, locked to a stable frequency comb, to perform resonance-enhanced multiphoton dissociation spectroscopy on sympathetically cooled HD+ ions at 50 mK. The achieved 2-ppb relative accuracy is a factor of 150 higher than previous results for HD+, and the measured transition frequency agrees well with recent high-accuracy ab initio calculations, which include highorder quantum electrodynamic effects. We also show that our method bears potential for achieving considerably higher accuracy and may, if combined with slightly improved theoretical calculations, lead to a new and improved determination of the electron-proton mass ratio.

The relative accuracy achieved in the spectroscopy here is considerably better than obtained with methods based on buffer gas-cooled trapped molecular ions and obtained with ion beams, buffer gas-cooled by a supersonic jet, where Doppler broadening due to 10 K temperatures limits the resolution achievable with single-photon spectroscopy. The main ingredients of the method, sympathetic cooling by lasercooled ions and a destructive spectroscopic scheme in combination with mass-specific detection, lend themselves for extension of this form of high-resolution spectroscopy to other ionic species such as, for example, the astronomically relevant H3+ ion and complex molecules. Therefore, it could lead to a vast improvement of the accuracy of spectroscopic constants for many molecular ions and give an impetus to developments in molecular theory.