Two-photon vibrational transitions in ${\rm O}_2^+$ 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 ${\rm O}_2^+$, including state-preparation through photoionization, a two-photon probe, and detection. We calculate transition dipole moments between all $X\,^2\Pi_g$ vibrational levels and those of the $A\,^2\Pi_u$ 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 ${\rm O}_2^+$ provide multiple routes to improved searches for $\mu$ variation.