Molecular hydrogen ions (MHI), the simplest molecules, are three-body quantum systems composed of two simple nuclei and one electron. They are of high interest for fundamental physics because, in contrast to the hydrogen atom family, their internal energy depends not only on the electron-nucleus interaction but also on the nucleus-nucleus interaction and on the nuclear masses. The relative simplicity of MHIs allows predictions by ab initio theory, that as of today has reached an impressive precision approaching that of the hydrogen atom theory. The experimental study of the MHI poses its own challenges.

Over the years, we have developed a set of techniques for the precise study of molecular ions at different levels of resolution and precision. They rely firstly, on trapping MHI in a linear ion trap and sympathetic cooling by co-trapped and laser-cooled beryllium ions, allowing reaching kinetic energies of the order 10 mK [1].

A breakthrough in resolution and precision was achieved by introducing novel Doppler-free spectroscopy techniques for ensembles of molecules, rather than single molecules. For rotational spectroscopy we interrogate a prolate ion ensemble and irradiate the spectroscopy wave along the small width, achieving the Lamb-Dicke regime.  For vibrational transitions (5 µm) we work with ensembles so small that the ions arrange in a string-like fashion on the trap axis. Again, irradiation of the spectroscopy laser is performed orthogonally to the trap axis. Under these conditions we observed one-photon electric-dipole transitions with resolved carrier. Line resolution as high as 3×10¹¹ and transition frequency uncertainties as low as 3×10^-12 were achieved with HD⁺ [2,3].

The measured rotational and vibrational frequencies are in agreement with their predicted values, when CODATA fundamental constants are used as input to the predictions. Assuming the correctness of the predictions within their estimated theoretical uncertainties, we determined values of the ratio of reduced nuclear mass and electron mass. Their uncertainties are competitive with the best direct determinations using Penning trap mass spectroscopy. Further, an upper limit for the strength of a hypothetical fifth force between the two nuclei could be set that was by more than one order lower than deduced from previous experiments. 

Finally, the resolution and precision of the rotational spectroscopy is so high that we could determine the tiny quadrupole moment of the deuteron to percent precision [2]. 

Recent developments in the techniques will be presented.

This work was funded by the ERC (786 306), the DFG (Schi 431/23-1, INST-208/737-1) and FP7-2013-ITN “COMIQ” and the Prof.-Behmenburg Schenkung

[1] B. Roth, et al. Phys. Rev. A 74, 040501 (2006); 10.1103/PhysRevA.74.040501
[2] S. Alighanbari, et al., Nature 581, 152 (2020); 10.1038/s41586-020-2261-5
[3] I. Kortunov, et al., Nat. Phys. 17, 569 (2021); 10.1038/s41567-020-01150-7