#
Two-Photon Vibrational Transitions in ^{16}O_{2}^{+} as Probes of Variation of the Proton-to-Electron Mass Ratio

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Procedures

#### 2.1. State Preparation: Resonance-Enhanced Multi-Photon Ionization

^{−1}above the neutral ${}^{16}{\mathrm{O}}_{2}\phantom{\rule{0.166667em}{0ex}}X{\phantom{\rule{0.166667em}{0ex}}}^{3}{\Sigma}_{g}^{-}$ ground state [59], such that it can be excited by two 301-nm photons. We use a frequency-doubled pulsed dye laser (the dye is a mix of Rhodamine 610 and 640) to drive the transition. The transition has a linewidth of approximately 2 cm${}^{-1}$ [55,59].

^{−1}above the ${}^{16}{\mathrm{O}}_{2}\phantom{\rule{0.166667em}{0ex}}d{\phantom{\rule{0.166667em}{0ex}}}^{1}{\Pi}_{g}$ state, such that a third photon from the 301-nm laser has sufficient energy to remove an electron. Although it also has enough energy to reach the first vibrationally-excited state of ${}^{16}{\mathrm{O}}_{2}^{+}$, the near diagonal Franck–Condon factors between the Rydberg and ionic states largely suppress any vibrational excitation. Ionization into the $J={\textstyle \frac{1}{2}}$ rotational state could be enhanced with a second laser of wavelength 323 nm, which would need to be tuned to have enough energy to ionize the molecule, but not enough to reach the $|X{\phantom{\rule{0.166667em}{0ex}}}^{2}{\Pi}_{g,\frac{1}{2}},v=0,J={\textstyle \frac{3}{2}}\rangle $ state.

^{−1}, which is two 301.280-nm photons. For ${N}^{\prime}=2$, it is 66,390.2 cm

^{−1}, which is two 301.249-nm photons. In both cases, the peak is three overlapping transitions from the neutral’s triplet ground state and is broader to the low-energy side.

#### 2.2. Probe: Two-Photon Transition

^{−1}deep with approximately 55 vibrational levels [42]. In principle, two-photon transitions can be driven from $v=0$ to any ${v}_{X}$. The transition frequency (for example, in hertz) for the $v=0\to \to {v}_{X}$ vibrational overtone is:

#### 2.3. Detection: Selective Dissociation

## 3. Transition Rates and Electric-Dipole-Related Systematics

^{−1}= 1.20 PHz/c apart [41,42,43]) suppresses these systematic effects, but also increases the laser intensity required for the two-photon transition. We do not include other excited states because they have either a much larger detuning, much smaller Franck–Condon overlap (for example, the $1{\phantom{\rule{0.166667em}{0ex}}}^{2}{\Sigma}_{u}^{+}$ state), a different spin multiplicity (for example, the $a{\phantom{\rule{0.166667em}{0ex}}}^{4}{\Pi}_{u}$ state), or the wrong center-of-inversion symmetry (g/u). We also neglect continuum transitions.

#### 3.1. Calculating the Transition Dipole Moments

^{−1}for vibrational levels in the $A{\phantom{\rule{0.166667em}{0ex}}}^{2}{\Pi}_{u}$ state [41,43] such that any change in the detuning is negligible. The calculations below involve sums over states, and the sum over the rotational and fine-structure states is effectively the identity.

^{−1}for X and 3 cm

^{−1}for A; all residuals were less than 10 cm

^{−1}. We then numerically calculated the wavefunction overlap integrals and r-centroids. We calculated the electronic transition dipole moment in the r-centroid approximation ${D}_{e}\left({\overline{r}}_{{v}_{A}{v}_{X}}\right)$ using the fit to theoretical values of ${D}_{e}\left(r\right)$ found in [85].

#### 3.2. Transition Rate

#### 3.3. Stark Shifts

#### 3.3.1. Probe Laser $(\Delta {f}_{\mathrm{probe}})$

#### 3.3.2. Trapping Fields $(\Delta {f}_{\mathrm{trap}})$

#### 3.3.3. Blackbody Radiation $(\Delta {f}_{\mathrm{BBR}})$

## 4. Additional Systematic Effects

#### 4.1. Doppler Shifts

#### 4.2. Electric Quadrupole Shift

#### 4.3. Zeeman Shift

## 5. Prospects

#### 5.1. Choice of State and Techniques

#### 5.2. Reference Transitions

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**General scheme of the proposed experiments. (

**a**) The ${}^{16}{\mathrm{O}}_{2}^{+}$ ion is prepared in its ground vibrational state by photoionization through the neutral $d{\phantom{\rule{0.166667em}{0ex}}}^{1}{\Pi}_{g}$ Rydberg state with (2 + 1) photons at 301 nm. (

**b**) Two photons drive the vibrational overtone, which is the spectroscopy transition. One photon dissociates any molecules in the excited vibrational state. The overtone shown is ${v}_{X}=11$.

**Figure 2.**Calculated two-photon excitation spectra from the neutral ${}^{16}{\mathrm{O}}_{2}$ $X{\phantom{\rule{0.166667em}{0ex}}}^{3}{\Sigma}_{g}^{-}$ ground state to the $d{\phantom{\rule{0.166667em}{0ex}}}^{1}{\Pi}_{g}$ Rydberg state, both in their ground vibrational state. The temperatures shown correspond to (

**a**) an effusive beam at 300 K or (

**b**) a supersonically-expanded one at 5 K. The traces are normalized such that the sum of all transitions is one; note the resulting scale change.

**Figure 3.**Calculated transition dipole moments between $X{\phantom{\rule{0.166667em}{0ex}}}^{2}{\Pi}_{g}$ and $A{\phantom{\rule{0.166667em}{0ex}}}^{2}{\Pi}_{u}$ vibrational states. Both the electronic and vibrational contributions are included. Dipole moments are given in atomic units. Numeric values are available in the Supplementary Materials.

**Table 1.**Table of two-photon-transition $(v=0\to \to {v}_{X})$ information: the wavelength ${\lambda}_{2\gamma}$ required to drive the transition, the absolute sensitivity ${f}_{\mu}$ of the transition to $\mu $ variation, estimates of photodissociation wavelength ${\lambda}_{\mathrm{PD}}$, and the calculated results for transition rates and electric-dipole-related systematics. These results are listed as the two-photon Rabi frequency ${\Omega}_{R}$ (in terms of intensity I), the probe laser AC Stark shift $\Delta {f}_{\mathrm{probe}}$ (in terms of intensity and of Rabi frequency), the AC Stark shift from the trapping field $\Delta {f}_{\mathrm{trap}}$ (in terms of the mean-squared electric field), and the blackbody radiation shift $\Delta {f}_{\mathrm{BBR}}$ at 300 K. The frequency shifts are normalized by the magnitude of the absolute sensitivity ${f}_{\mu}$, so the numbers represent relative accuracy in $\Delta \mu /\mu $, and the signs reflect the actual frequency shift.

${\mathit{v}}_{\mathit{X}}$ | ${\mathit{\lambda}}_{2\mathit{\gamma}}$ (nm) | $\mathit{f}{}_{\mathit{\mu}}$ (THz) | ${\mathit{\lambda}}_{\mathbf{PD}}$ (nm) | $\frac{{\mathbf{\Omega}}_{\mathit{R}}}{2\mathit{\pi}}/\mathit{I}$ $\left({10}^{-7}\frac{\mathbf{Hz}}{{\mathbf{W}/\mathbf{m}}^{2}}\right)$ | $\frac{\mathbf{\Delta}{\mathit{f}}_{\mathbf{probe}}}{\left|{\mathit{f}}_{\mathit{\mu}}\right|}/\mathit{I}$ $\left({10}^{-20}\phantom{\rule{0.166667em}{0ex}}{(\mathbf{W}/\mathbf{m}}^{2}{)}^{-1}\right)$ | $\frac{\mathbf{\Delta}{\mathit{f}}_{\mathbf{probe}}}{\left|{\mathit{f}}_{\mathit{\mu}}\right|}/\frac{{\mathbf{\Omega}}_{\mathit{R}}}{2\mathit{\pi}}$ $\left({10}^{-13}\phantom{\rule{0.166667em}{0ex}}{\left(\mathbf{Hz}\right)}^{-1}\right)$ | $\frac{\mathbf{\Delta}{\mathit{f}}_{\mathbf{trap}}}{\left|{\mathit{f}}_{\mathit{\mu}}\right|}/{\mathcal{E}}_{\mathbf{rms}}^{2}$ $\left({10}^{-22}\phantom{\rule{0.166667em}{0ex}}{(\mathbf{V}/\mathbf{m})}^{2}\right)$ | $\frac{\mathbf{\Delta}{\mathit{f}}_{\mathbf{BBR}}}{\left|{\mathit{f}}_{\mathit{\mu}}\right|}$ $\left({10}^{-18}\right)$ |
---|---|---|---|---|---|---|---|---|

1 | 10,614 | −28 | 113 | 4.94 | −0.59 | −0.12 | −0.16 | −3.59 |

2 | 5386 | −54 | 123 | 1.24 | −0.40 | −0.32 | −0.11 | −2.43 |

3 | 3617 | −80 | 134 | 1.56 | −0.47 | −0.30 | −0.12 | −2.86 |

4 | 2738 | −104 | 144 | 0.91 | −0.45 | −0.49 | −0.12 | −2.69 |

5 | 2211 | −128 | 155 | 0.99 | −0.51 | −0.52 | −0.13 | −3.07 |

6 | 1859 | −151 | 166 | 0.91 | −0.53 | −0.59 | −0.14 | −3.14 |

7 | 1609 | −172 | 179 | 0.68 | −0.62 | −0.90 | −0.15 | −3.56 |

8 | 1421 | −193 | 192 | 0.91 | −0.67 | −0.73 | −0.16 | −3.79 |

9 | 1275 | −213 | 206 | 0.51 | −0.78 | −1.53 | −0.18 | −4.26 |

10 | 1158 | −231 | 221 | 0.92 | −0.88 | −0.95 | −0.20 | −4.59 |

11 | 1063 | −249 | 238 | 0.40 | −1.03 | −2.56 | −0.22 | −5.14 |

12 | 984 | −266 | 256 | 0.93 | −1.22 | −1.31 | −0.24 | −5.65 |

13 | 917 | −281 | 276 | 0.33 | −1.53 | −4.57 | −0.28 | −6.39 |

14 | 860 | −296 | 299 | 0.94 | −1.97 | −2.09 | −0.31 | −7.14 |

15 | 810 | −310 | 323 | 0.30 | −2.79 | −9.20 | −0.35 | −8.14 |

16 | 767 | −323 | 351 | 0.96 | −4.72 | −4.91 | −0.40 | −9.20 |

17 | 730 | −334 | − | 0.31 | −1.48 | −4.77 | −0.46 | −10.58 |

18 | 696 | −345 | − | 0.98 | 9.10 | 9.29 | −0.52 | −12.06 |

19 | 667 | −355 | − | 0.35 | 17.52 | 49.40 | −0.60 | −13.95 |

20 | 640 | −364 | − | 1.01 | 1.79 | 1.77 | −0.69 | −16.06 |

21 | 616 | −371 | − | 0.43 | −6.69 | −15.59 | −0.81 | −18.81 |

22 | 594 | −378 | − | 1.01 | 5.88 | 5.79 | −0.95 | −21.97 |

23 | 575 | −384 | − | 0.56 | 3.55 | 6.30 | −1.12 | −26.10 |

24 | 557 | −389 | − | 1.01 | 3.34 | 3.29 | −1.32 | −30.82 |

25 | 541 | −393 | − | 0.75 | 2.66 | 3.55 | −1.58 | −36.94 |

26 | 526 | −395 | − | 0.99 | 2.18 | 2.21 | −1.88 | −43.99 |

27 | 513 | −397 | − | 0.99 | 2.04 | 2.06 | −2.27 | −53.31 |

28 | 500 | −398 | − | 0.86 | 1.85 | 2.14 | −2.74 | −64.36 |

29 | 489 | −398 | − | 1.28 | 1.80 | 1.40 | −3.35 | −79.02 |

30 | 478 | −397 | − | 0.59 | 1.71 | 2.88 | −4.09 | −96.75 |

31 | 468 | −395 | − | 1.58 | 1.69 | 1.07 | −5.06 | −120.31 |

32 | 459 | −392 | − | 0.22 | 1.66 | 7.62 | −6.29 | −150.64 |

33 | 451 | −388 | − | 1.84 | 1.68 | 0.91 | −7.92 | −191.28 |

34 | 444 | −383 | − | 0.38 | 1.72 | 4.47 | −9.98 | −242.92 |

35 | 437 | −377 | − | 1.91 | 1.73 | 0.91 | −12.32 | −303.11 |

36 | 430 | −369 | − | 1.21 | 1.78 | 1.47 | −15.20 | −379.12 |

37 | 424 | −361 | − | 1.69 | 1.77 | 1.05 | −18.99 | −483.29 |

38 | 418 | −352 | − | 2.28 | 1.86 | 0.82 | −24.72 | −646.53 |

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**MDPI and ACS Style**

Carollo, R.; Frenett, A.; Hanneke, D. Two-Photon Vibrational Transitions in ^{16}O_{2}^{+} as Probes of Variation of the Proton-to-Electron Mass Ratio. *Atoms* **2019**, *7*, 1.
https://doi.org/10.3390/atoms7010001

**AMA Style**

Carollo R, Frenett A, Hanneke D. Two-Photon Vibrational Transitions in ^{16}O_{2}^{+} as Probes of Variation of the Proton-to-Electron Mass Ratio. *Atoms*. 2019; 7(1):1.
https://doi.org/10.3390/atoms7010001

**Chicago/Turabian Style**

Carollo, Ryan, Alexander Frenett, and David Hanneke. 2019. "Two-Photon Vibrational Transitions in ^{16}O_{2}^{+} as Probes of Variation of the Proton-to-Electron Mass Ratio" *Atoms* 7, no. 1: 1.
https://doi.org/10.3390/atoms7010001