atoms Article Two-Photon Vibrational Transitions in 16 O + 2 as Probes of Variation of the Proton-to-Electron Mass Ratio Ryan Carollo , Alexander Frenett and David Hanneke * Physics & Astronomy Department, Amherst College, Amherst, MA 01002, USA; rcarollo@amherst.edu (R.C.); afrenett18@amherst.edu (A.F.) * Correspondence: dhanneke@amherst.edu; Tel.: +1-413-542-5525 Received: 31 October 2018; Accepted: 18 December 2018; Published: 20 December 2018   Abstract: Vibrational overtones in deeply-bound molecules are sensitive probes for variation of the proton-to-electron mass ratio µ. In nonpolar molecules, these overtones may be driven as two-photon transitions. Here, we present procedures for experiments with 16 O + 2 , including state-preparation through photoionization, a two-photon probe, and detection. We calculate transition dipole moments between all X 2 Π g vibrational levels and those of the A 2 Π 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 16 O + 2 provide multiple routes to improved searches for µ variation. Keywords: precision measurements; fundamental constants; variation of constants; proton-to-electron mass ratio; molecular ions; forbidden transition; two-photon transition; vibrational overtone 1. Introduction Even simple molecules contain a rich set of internal degrees of freedom. When these internal states are controlled at the quantum level, they have many applications in fundamental physics [1,2] such as searches for new forces [3], the investigation of parity [4,5] and time-reversal [6,7] symmetries, or searches for time-variation of fundamental constants [8–10]. Experiments with molecular ions [6,11–14] are already at the forefront of these scientific questions, taking advantage of the long interrogation times allowed in trapped systems. Because some degrees of freedom involve the motion of the nuclei themselves, molecules possess the potential to probe for changes in the proton mass relative to the electron mass. This mass ratio, µ = m p /m e , is predicted to change over time by several extensions to the standard model. Some models of quantum gravity include extra spatial dimensions or new scalar fields and suggest a drift in µ on cosmological timescales and continuing to the present day [15,16]. Ultralight dark matter could cause µ to oscillate at a frequency set by the mass of the dark matter particle [17,18]; topological dark matter could cause transient changes in µ [19]. Models typically predict that variation in µ should be approximately 40-times larger than corresponding changes in the fine structure constant α [15]. Current limits on present-day variation in µ come from atomic clock experiments and find that ˙ µ/µ =(5.3 ± 6.5) × 10 −17 year −1 [20]. The sensitivity to µ in these experiments is through the hyperfine structure of cesium. Linking the hyperfine frequency to the nuclear mass requires a model of the cesium nuclear magnetic moment [21]. The reliance on cesium clocks also means that atomic techniques are nearing their feasible limits, as the cesium microwave clock has been surpassed in stability by optical atomic clocks [22]. These optical clocks are based on electronic—not hyperfine—transitions, so they have little sensitivity to µ variation. The vibration and rotation of molecules provide a model-independent means to search for variation in µ [8–10,23–25]. The current limit for a molecular experiment is ˙ µ/µ =(−3.8 ± 5.6) × 10 −14 year −1 , which Atoms 2018, 7, 1; doi:10.3390/atoms7010001 www.mdpi.com/journal/atoms