Tests of time independence of the electron and nuclear masses with ultracold molecules

We propose to use laser spectroscopy of ultracold molecules to establish improved limits on the time independence of electron-to-nuclear and nuclear-nuclear mass ratios by comparing, via an optical frequency comb, the frequencies of suitable sets of transitions in the ground electronic state. Hydrogen molecular ions trapped in a radiofrequency trap, sympathetically cooled by atomic ions, are identified as an accessible system. We show that the dipole-allowed rovibrational transition frequencies of $\mathrm{H}{\mathrm{D}}^{+}$ are suitable probes for a time dependence of $m_e∕m_p$ or $m_p∕m_d$. Separate bounds on the time independence of these constants can be obtained from a comparison of $\mathrm{H}{\mathrm{D}}^{+}$ and $\mathrm{H}_2{}^{+}$ transitions frequencies. Spectroscopy of single molecular ions via a quantum jump method is proposed as an approach toward ultrahigh precision.