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Precise test of quantum electrodynamics and determination of fundamental constants with HD+ ions


Bound three-body quantum systems are important for fundamental physics1,2 because they enable tests of quantum electrodynamics theory and provide access to the fundamental constants of atomic physics and to nuclear properties. Molecular hydrogen ions, the simplest molecules, are representative of this class3. The metastability of the vibration–rotation levels in their ground electronic states offers the potential for extremely high spectroscopic resolution. Consequently, these systems provide independent access to the Rydberg constant (R), the ratios of the electron mass to the proton mass (me/mp) and of the electron mass to the deuteron mass (me/md), the proton and deuteron nuclear radii, and high-level tests of quantum electrodynamics4. Conventional spectroscopy techniques for molecular ions5,6,7,8,9,10,11,12,13,14 have long been unable to provide precision competitive with that of ab initio theory, which has greatly improved in recent years15. Here we improve our rotational spectroscopy technique for a sympathetically cooled cluster of molecular ions stored in a linear radiofrequency trap16 by nearly two orders in accuracy. We measured a set of hyperfine components of the fundamental rotational transition. An evaluation resulted in the most accurate test of a quantum-three-body prediction so far, at the level of 5 × 10−11, limited by the current uncertainties of the fundamental constants. We determined the value of the fundamental constants combinations \({R}_{\infty }{m}_{{\rm{e}}}({m}_{{\rm{p}}}^{-1}+{m}_{{\rm{d}}}^{-1})\) and mp/me with a fractional uncertainty of 2 × 10−11, in agreement with, but more precise than, current Committee on Data for Science and Technology values. These results also provide strong evidence of the correctness of previous key high-precision measurements and a more than 20-fold stronger bound for a hypothetical fifth force between a proton and a deuteron.

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Fig. 1: Energy diagram of the spin structures and favoured transitions.
Fig. 2: Hyperfine components of the fundamental rotational transition of HD+ at 1.3 THz.
Fig. 3: Exclusion plot (95% confidence limit) for a Yukawa-type interaction between a proton and a deuteron, deduced from spectroscopy of MHIs.
Fig. 4: Comparison of results of this work with literature values.

Data availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.


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We thank M. G. Hansen for assistance with optimization of the apparatus and J.-Ph. Karr for checking theoretical expressions. This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement number 786306, ‘PREMOL’), and from the Deutsche Forschungsgemeinschaft in project Schi 431/23-1. S.A. acknowledges a fellowship of the Prof.-W.-Behmenburg-Schenkung. V.I.K. acknowledges support from the Russian Science Foundation under grant number 18-12-00128.

Author information




S.A. and G.S.G. performed the measurements and analysed data, F.L.C. contributed to the measurements. S.A. developed and maintained the apparatus. V.I.K. performed the ab initio calculations, S.S. performed data and theoretical analyses, prepared the manuscript and supervised the work. All authors contributed to discussion and manuscript editing.

Corresponding author

Correspondence to S. Schiller.

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The authors declare no competing interests.

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Peer review information Nature thanks Brian Odom, Richard Thompson and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Fig. 1 Conceptual view of the arrangement used for high-resolution spectroscopy of HD+ using TICTES.

The spectroscopy wave (1.3 THz) crosses the ion cluster perpendicular to its long axis, enabling spectroscopy in the Lamb–Dicke regime. The ion cluster comprises atomic Be+ ions (blue dots) and HD+ molecular ions (red dots). The indicated laser beams implement the Doppler cooling of Be+ ions (313 nm), rotational cooling of HD+ (2.7 µm and 5.48 μm) and detection by REMPD (266 nm and 1.4 μm). The magnetic field B lifts the degeneracy of Zeeman sublevels during terahertz spectroscopy. The polarizer and the half-wave plate enable adjustment of the polarization and intensity of the terahertz radiation.

Extended Data Fig. 2 Beryllium ion fluorescence during one preparation–spectroscopy cycle.

Spectroscopy (terahertz wave on) occurs during the interval marked ‘REMPD’. Beryllium laser cooling is on all the time. SE, secular excitation. B, a magnetic flux strength B is applied during REMPD. B0, a strength B0 is applied for rotational laser cooling. CPS, counts per second. The signal obtained from the spectroscopy cycle is indicated in cyan.

Extended Data Fig. 3 Systematic shifts of the Zeeman component 19+ of the rotational hyperfine transition line 19.

a, The trap’s amplitude is decreased by 2.5 V from \({V}_{{\rm{RF}}}^{(1)}\) to \({V}_{{\rm{RF}}}^{(2)}\). The FWHM linewidth is 4 Hz, corresponding to 3 × 10−12 fractional FWHM. b, The light shift induced by the 266 nm and 1.4 μm dissociation lasers, determined by comparing two spectroscopy modes. ‘Continuous’ indicates that the lasers are on when the terahertz radiation is applied. ‘Interleaved’ indicates that the lasers and terahertz radiation are on alternatingly.

Extended Data Table 1 Spin Hamiltonian coefficients, spin-structure frequencies and spin-frequency derivatives

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Alighanbari, S., Giri, G.S., Constantin, F.L. et al. Precise test of quantum electrodynamics and determination of fundamental constants with HD+ ions. Nature 581, 152–158 (2020).

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