Two-photon vibrational transitions in ${\mathrm{O}}_2^{+}$ as probes of variation of the proton-to-electron mass ratio

Vibrational overtones in deeply bound molecules are sensitive probes for variation of the proton-to-electron mass ratio $\mu$. In nonpolar molecules, these overtones may be driven as two-photon transitions. Here, we present procedures for experiments with ${\mathrm{O}}_2^{+}$, including state-preparation through photoionization, a two-photon probe, and detection. We calculate transition dipole moments between all $X{}^2{\mathrm{\Pi }}_g$ vibrational levels and those of the $A{}^2{\mathrm{\Pi }}_u$ excited electronic state. Using these dipole moments, we calculate two-photon transition rates and AC-Stark-shift systematics for the overtones. We estimate other systematic effects and statistical precision. Two-photon vibrational transitions in ${\mathrm{O}}_2^{+}$ provide multiple routes to improved searches for $\mu$ variation.