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Laser spectroscopy of a rovibrational transition in the molecular hydrogen ion \({\mathbf{H}}_{\mathbf{2}}^{\mathbf{+}}\)


Comparison of precise predictions of the energy levels of the molecular hydrogen ion \({\rm{H}}_{2}^{+}\)—the simplest molecule—with measured vibrational transition frequencies would allow a direct determination of the proton-to-electron mass ratio and of the proton’s charge radius. Here we report vibrational laser spectroscopy of trapped and sympathetically laser-cooled \({{{{\rm{H}}}}}_{2}^{+}\), which represents a step towards this goal. We studied a first-overtone electric-quadrupole transition and measured its two hyperfine components. The determined spin-averaged vibrational transition frequency has a fractional uncertainty of 1.2 × 10−8 and is in agreement with the theoretically predicted value. We measured an analogous electric-quadrupole transition in HD+ to estimate systematic uncertainties. Here, we observed a vastly improved line quality factor compared to previous electric-quadrupole spectroscopy of molecular ions. Our work demonstrates that first-overtone electric-quadrupole transitions are suitable for precision spectroscopy of molecular ions, including \({{{{\rm{H}}}}}_{2}^{+}\), and that determining the proton-to-electron mass ratio with laser spectroscopy could become competitive with mass spectrometry using Penning traps. Furthermore, achieving precision spectroscopy of \({{{{\rm{H}}}}}_{2}^{+}\) is an essential prerequisite for a future test of combined charge, parity and time reversal symmetry based on a comparison with its antimatter counterpart.

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Fig. 1: Energy levels of \({{{{\rm{H}}}}}_{2}^{+}\) and transitions relevant to this work.
Fig. 2: The two electron-spin-rotation components of the E2 rovibrational transition (v = 1, N = 0) → (v′ = 3, N′ = 2) in H +2 .
Fig. 3: Two Zeeman components measured under different operating conditions.

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Data availability

Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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No custom code or software was used for analysing or presenting the data associated with this paper.


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We are indebted to V. I. Korobov for putting at our disposal his codes for the computation of the \({{{{\rm{H}}}}}_{2}^{+}\) properties and for many important communications on the properties of MHI. We are grateful to S. Schlemmer (Universität zu Köln) for generously providing para-H2 gas. G. S. Giri contributed to early tests of the spectroscopy. The assistance of U. Rosowski in the maintenance of the H maser and data analysis has been important. C. Wellers and V. Vogt contributed in obtaining and characterizing the OPO. M. G. Hansen and I. V. Kortunov are acknowledged for their help on setting up the locking and metrology scheme of the OPO. We also thank E. Wiens for support in the optical frequency measurements. We thank S. Sturm, F. Heiße, C. König and S. Ulmer for discussions about the CPT tests. This work has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 786306, PREMOL (S.S.)) and from both the German Research Foundation and the state of North-Rhine-Westphalia (Grant Nos. INST-208/774-1 FUGG (S.S.) and INST-208/796-1 FUGG (S.S.)).

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M.R.S. and S.A. performed the experiments and analysed the data. M.R.S. developed and characterized the spectroscopy laser system. S.A. maintained the apparatus. S.S. conceived the experiment, supervised the project and performed analyses. All authors contributed to writing the manuscript and reviewing the data.

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Correspondence to S. Schiller.

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Nature Physics thanks Shui-Ming Hu, Krzysztof Pachucki and Xin Tong for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 The \({{{{\rm{H}}}}}_{2}^{+}\) spectroscopy cycle.

The black trace is the Be+ fluorescence detected by a photomultiplier tube in mega counts per second (Mcps). During the depletion of the excited state population (see Methods, section 1.a), the fluorescence is multiplied by 0.5 for illustration purposes. The red ovals indicate the radial secular excitation resonance of the trapped \({{{{\rm{H}}}}}_{2}^{+}\) and \({{{{\rm{H}}}}}_{3}^{+}\) ions. The actual spectroscopy signal, that is the reduction of the number of trapped \({{{{\rm{H}}}}}_{2}^{+}\) ions by the REMPD process, is highlighted in the top (black) signal by the red dashed lines and red arrows. The interleaved shuttering during the REMPD is shown simplified with only 10 repetitions. In actuality, there are 310 repetitions of duration  62 ms each. Each consists of two 30 ms and two 1 ms intervals. During the 30 ms intervals, either the 2.4 μm wave or the 405 nm wave is sent to the ions. The 1 ms intervals act as a buffers in-between. The cycle for measuring the background is analogous, except that the 2.4 μm wave is off during the whole cycle. During the exposure to the REMPD lasers, the magnetic field was B≤0.2 μT, while at all other times a magnetic field Bcooling for the purpose of Doppler-cooling was applied. The values and status of each parameter are displayed on the right-hand vertical axis.

Extended Data Table 1 Hamiltonians for rovibrational levels (v, N = 0, 2) in H +2
Extended Data Table 2 Approximate expressions for the energies of the states of H +2 in rotational levels N = 2
Extended Data Table 3 Theoretical values of the spin and Zeeman contributions of some Zeeman components of the studied transition (v = 1, N = 0) → (v′ = 3, N′ = 2), in weak magnetic field B in units of μT

Source data

Source Data for Fig. 2

Source data for the data points shown in Fig. 2, including errors.

Source Data for Fig. 3

Source data for the data points shown in Fig. 3, including errors.

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Schenkel, M.R., Alighanbari, S. & Schiller, S. Laser spectroscopy of a rovibrational transition in the molecular hydrogen ion \({\mathbf{H}}_{\mathbf{2}}^{\mathbf{+}}\). Nat. Phys. 20, 383–388 (2024).

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