Deuteron-to-Proton Mass Ratio from Simultaneous Measurement of the Cyclotron Frequencies of ${\mathrm{H}}_2^{+}$ and ${\mathrm{D}}^{+}$

By simultaneously measuring the cyclotron frequencies of an ${\mathrm{H}}_2^{+}$ ion and a deuteron in a coupled magnetron orbit we have made an extended series of measurements of their cyclotron frequency ratio. From the observed changes in ${\mathrm{H}}_2^{+}$ mass energy we have followed the decay of three ${\mathrm{H}}_2^{+}$ ions to the vibrational ground state. We are able to assign some of our measured ratios to specific rovibrational levels, hence reducing uncertainty due to ${\mathrm{H}}_2^{+}$ rotational energy. Assuming the most probable assignment, we obtain a deuteron-to-proton mass ratio, $m_d/m_p=1.999007501272(9)$. Combined with the atomic mass of the deuteron [S. Rau et al., Nature (London) 585, 43 (2020).] we also obtain a new value for the atomic mass of the proton, $m_p=1.007276466574(10)\mathrm{u}$.

Figure 1

${\mathrm{H}}_2^{+}/{\mathrm{D}}^{+}$ cyclotron frequency ratio measurements using three different ${\mathrm{H}}_2^{+}$ ions. The data are for 5 ms cyclotron drive time, except for ${\mathrm{H}}_2^{+}(1)$, $v=1$, where data at 6 ms have been corrected to 5 ms. The offset is 0.999 231 659.

Figure 2

Minimum chi-squared with respect to $R_{00}(5\mathrm{ms})$, with a step size of ${10}^{-12}$, for assignments of rovibrational levels to our averaged CFRs. The markers indicate whether the $N$ values for the three ${\mathrm{H}}_2^{+}$ ions are odd or even as follows: solid round $ooo$, solid diamond $eoo$, solid square $eee$, solid triangle $oee$; open round $oeo$, open diamond $eeo$, open square $eoe$, open triangle $ooe$.