Deuteron to Proton Mass Ratio from Precision Measurement of the Cyclotron Frequencies of H$_2^+$ and D$^+$ with H$_2^+$ in a Resolved Ro-vibrational State

Determination of the deuteron-to-proton mass ratio ($m_{d}/m_{p}$) from precision measurement of the CFR (cyclotron frequency ratio) H$_2^+/$D$^+$ benefits from a reduction in systematic error due to the use of ions of similar $m/q$. However, additional uncertainty results from lack of knowledge of the H$_2^+$ ro-vibrational state. Following a previous measurement using alternating measurements of cyclotron frequency with ions in large and small cyclotron orbits [1], we are implementing a two-ion simultaneous measurement technique, originally developed at MIT for ions with $m/q \sim 30$, to increase the precision of our measurement of the H$_2^+/$D$^+$ CFR. By measuring the cyclotron frequency of the H$_2^+$ and D$^+$ ions simultaneously, statistical uncertainty due to magnetic field variation is minimized. With feedback cooling to reduce statistical noise on the cyclotron frequency (due to thermal fluctuations in the cyclotron radius combined with special relativity) this technique may enable us to identify specific ro-vibrational decays, and hence obtain a sub-$10^{-11}$ measurement of $m_d/m_p$. \break \break [1] D. J. Fink and E. G. Myers, Phys. Rev. Lett. 124, 013001 (2020).