Vibrational Spectroscopy of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>HD</mml:mi><mml:mo>+</mml:mo></mml:msup></mml:math>with 2-ppb Accuracy

By measurement of the frequency of a vibrational overtone transition in the molecular hydrogen ion ${\mathrm{HD}}^{+}$, we demonstrate the first 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 ${\mathrm{HD}}^{+}$ ions at 50 mK. The achieved 2-ppb relative accuracy is a factor of 150 higher than previous results for ${\mathrm{HD}}^{+}$, and the measured transition frequency agrees well with recent high-accuracy ab initio calculations, which include high-order 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.