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.


  1. 1.

    Karshenboim, S. G. (ed.) Precision Physics of Simple Atoms and Molecules (Springer-Verlag, 2008).

  2. 2.

    Pachucki, K., Patkóš, V. & Yerokhin, V. A. Testing fundamental interactions on the helium atom. Phys. Rev. A 95, 062510 (2017).

  3. 3.

    Leach, C. A. & Moss, R. E. Spectroscopy and quantum mechanics of the hydrogen molecular cation: a test of molecular quantum mechanics. Annu. Rev. Phys. Chem. 46, 55–82 (1995).

  4. 4.

    Roth, B. et al. in Precision Physics of Simple Atoms and Molecules (ed. Karshenboim, S. G.) 205–232 (Springer-Verlag, 2008).

  5. 5.

    Wing, W. H., Ruff, G. A., Lamb, W. E. & Spezeski, J. J. Observation of the infrared spectrum of the hydrogen molecular ion HD+. Phys. Rev. Lett. 36, 1488–1491 (1976).

  6. 6.

    Arcuni, P. W., Fu, Z. W. & Lundeen, S. R. Energy difference between the (ν = 0, R = 1) and the (ν = 0, R = 3) states of \({{\rm{H}}}_{2}^{+}\), measured with interseries microwave spectroscopy of H2 Rydberg states. Phys. Rev. A 42, 6950–6953 (1990).

  7. 7.

    Carrington, A., McNab, I. R., Montgomerie-Leach, C. A. & Kennedy, R. A. Vibration-rotation spectroscopy of the HD+ ion near the dissociation limit. Mol. Phys. 72, 735–762 (1991).

  8. 8.

    Fu, Z. W., Hessels, E. A. & Lundeen, S. R. Determination of the hyperfine structure of \({{\rm{H}}}_{2}^{+}\) (ν = 0, R = 1) by microwave spectroscopy of high-L, n = 27 Rydberg states of H2. Phys. Rev. A 46, R5313–R5316 (1992).

  9. 9.

    Critchley, A. D. J., Hughes, A. N. & McNab, I. R. Direct measurement of a pure rotation transition in \({{\rm{H}}}_{2}^{+}\). Phys. Rev. Lett. 86, 1725–1728 (2001).

  10. 10.

    Osterwalder, A., Wüest, A., Merkt, F. & Jungen, C. High-resolution millimeter wave spectroscopy and multichannel quantum defect theory of the hyperfine structure in high Rydberg states of molecular hydrogen \({{\rm{H}}}_{2}^{+}\). J. Chem. Phys. 121, 11810–11838 (2004).

  11. 11.

    Koelemeij, J. C. J., Roth, B., Wicht, A., Ernsting, I. & Schiller, S. Vibrational spectroscopy of HD+ with 2-ppb accuracy. Phys. Rev. Lett. 98, 173002 (2007).

  12. 12.

    Bressel, U. et al. Manipulation of individual hyperfine states in cold trapped molecular ions and application to HD+ frequency metrology. Phys. Rev. Lett. 108, 183003 (2012).

  13. 13.

    Haase, C., Beyer, M., Jungen, C. & Merkt, F. The fundamental rotational interval of para-\({{\rm{H}}}_{2}^{+}\) by MQDT-assisted Rydberg spectroscopy of H2. J. Chem. Phys. 142, 064310 (2015).

  14. 14.

    Biesheuvel, J. et al. Probing QED and fundamental constants through laser spectroscopy of vibrational transitions in HD+. Nat. Commun. 7, 10385 (2016).

  15. 15.

    Korobov, V. I., Hilico, L. & Karr, J.-P. Fundamental transitions and ionization energies of the hydrogen molecular ions with few ppt uncertainty. Phys. Rev. Lett. 118, 233001 (2017).

  16. 16.

    Alighanbari, S., Hansen, M. G., Korobov, V. I. & Schiller, S. Rotational spectroscopy of cold and trapped molecular ions in the Lamb–Dicke regime. Nat. Phys. 14, 555–559 (2018).

  17. 17.

    Jefferts, K. B. Hyperfine structure in the molecular ion \({{\rm{H}}}_{2}^{+}\). Phys. Rev. Lett. 23, 1476–1478 (1969).

  18. 18.

    Schiller, S. & Korobov, V. I. Test of time-dependence of the electron and nuclear masses with ultracold molecules. Phys. Rev. A 71, 032505 (2005).

  19. 19.

    Bakalov, D. & Schiller, S. The electric quadrupole moment of molecular hydrogen ions and their potential for a molecular ion clock. Appl. Phys. B 114, 213–230 (2014); erratum 116, 777–778 (2014).

  20. 20.

    Karr, J.-Ph. \({{\rm{H}}}_{2}^{+}\) and HD+: candidates for a molecular clock. J. Mol. Spectrosc. 300, 37–43 (2014).

  21. 21.

    Schiller, S., Bakalov, D. & Korobov, V. I. Simplest molecules as candidates for precise optical clocks. Phys. Rev. Lett. 113, 023004 (2014).

  22. 22.

    Beyer, A. et al. The Rydberg constant and proton size from atomic hydrogen. Science 358, 79–85 (2017).

  23. 23.

    Fleurbaey, H. et al. New measurement of the 1S−3S transition frequency of hydrogen: contribution to the proton charge radius puzzle. Phys. Rev. Lett. 120, 183001 (2018).

  24. 24.

    Bezginov, N. et al. A measurement of the atomic hydrogen Lamb shift and the proton charge radius. Science 365, 1007–1012 (2019).

  25. 25.

    Antognini, A. et al. Proton structure from the measurement of 2S–2P transition frequencies of muonic hydrogen. Science 339, 417–420 (2013).

  26. 26.

    Grémaud, B., Delande, D. & Billy, N. Highly accurate calculation of the energy levels of the \({{\rm{H}}}_{2}^{+}\) molecular ion. J. Phys. B 31, 383 (1998).

  27. 27.

    Moss, R. E. Energies of low-lying vibration-rotation levels of \({{\rm{H}}}_{2}^{+}\) and its isotopomers. J. Phys. B 32, L89–L91 (1999).

  28. 28.

    Taylor, J. M., Yan, Z.-C., Dalgarno, A. & Babb, J. F. Variational calculations on the hydrogen molecular ion. Mol. Phys. 97, 25–33 (1999).

  29. 29.

    Tiesinga, E., Mohr, P. J., Newell, D. B. & Taylor, B. N. Values of fundamental physical constants. NIST (2019).

  30. 30.

    Wolf, F. et al. Non-destructive state detection for quantum logic spectroscopy of molecular ions. Nature 530, 457–460 (2016).

  31. 31.

    Chou, C. et al. Preparation and coherent manipulation of pure quantum states of a single molecular ion. Nature 545, 203–207 (2017).

  32. 32.

    Schneider, T., Roth, B., Duncker, H., Ernsting, I. & Schiller, S. All-optical preparation of molecular ions in the rovibrational ground state. Nat. Phys. 6, 275–278 (2010).

  33. 33.

    Roth, B., Blythe, P., Wenz, H., Daerr, H. & Schiller, S. Ion-neutral chemical reactions between ultracold localized ions and neutral molecules with single-particle resolution. Phys. Rev. A 73, 042712 (2006).

  34. 34.

    Schiller, S., Roth, B., Lewen, F., Ricken, O. & Wiedner, M. Ultra-narrow-linewidth continuous-wave THz sources based on multiplier chains. Appl. Phys. B 95, 55–61 (2009).

  35. 35.

    Bakalov, D., Korobov, V. I. & Schiller, S. High-precision calculation of the hyperfine structure of the HD+ ion. Phys. Rev. Lett. 97, 243001 (2006).

  36. 36.

    Schiller, S. & Korobov, V. I. Canceling spin-dependent contributions and systematic shifts in precision spectroscopy of molecular hydrogen ions. Phys. Rev. A 98, 022511 (2018).

  37. 37.

    Bakalov, D., Korobov, V. I. & Schiller, S. Magnetic field effects in the transitions of the HD+ molecular ion and precision spectroscopy. J. Phys. B 44, 025003 (2011); corrigendum 45, 049501 (2012).

  38. 38.

    Korobov, V. I., Koelemeij, J. C. J., Hilico, L. & Karr, J.-P. Theoretical hyperfine structure of the molecular hydrogen ion at the 1 ppm level. Phys. Rev. Lett. 116, 053003 (2016).

  39. 39.

    Menasian, S. C. & Dehmelt, H. G. High-resolution study of (1,1/2,1/2)−(1,1/2,3/2) HFS transition in \({{\rm{H}}}_{2}^{+}\). Bull. Am. Phys. Soc. 18, 408 (1973).

  40. 40.

    Heiße, F. et al. High-precision mass spectrometer for light ions. Phys. Rev. A 100, 022518 (2019).

  41. 41.

    Fink, D. J. & Myers, E. G. Deuteron-to-proton mass ratio from the cyclotron frequency ratio of \({{\rm{H}}}_{2}^{+}\) to D+ with \({{\rm{H}}}_{2}^{+}\) in a resolved vibrational state. Phys. Rev. Lett. 124, 013001 (2020).

  42. 42.

    Sturm, S. et al. High-precision measurement of the atomic mass of the electron. Nature 506, 467–470 (2014).

  43. 43.

    Pastor, P. C. et al. Absolute frequency measurements of the 23S 1 → 23P 0,1,2 atomic helium transitions around 1083 nm. Phys. Rev. Lett. 92, 023001 (2004).

  44. 44.

    Hori, M. et al. Buffer-gas cooling of antiprotonic helium to 1.5 to 1.7 K, and antiproton-to-electron mass ratio. Science 354, 610–614 (2016).

  45. 45.

    Rengelink, R. J. et al. Precision spectroscopy of helium in a magic wavelength optical dipole trap. Nat. Phys. 14, 1132–1137 (2018).

  46. 46.

    Hori, M. et al. Two-photon laser spectroscopy of antiprotonic helium and the antiproton-to-electron mass ratio. Nature 475, 484–488 (2011).

  47. 47.

    Udem, T. Quantum electrodynamics and the proton size. Nat. Phys. 14, 632–632 (2018); correction 14, 767 (2018).

  48. 48.

    Schiller, S., Bakalov, D., Bekbaev, A. K. & Korobov, V. I. Static and dynamic polarizability and the Stark and blackbody-radiation frequency shifts of the molecular hydrogen ions \({{\rm{H}}}_{2}^{+}\), HD+, and \({{\rm{D}}}_{2}^{+}\). Phys. Rev. A 89, 052521 (2014).

  49. 49.

    Berkeland, D. J., Miller, J. D., Bergquist, J. C., Itano, W. M. & Wineland, D. J. Minimization of ion micromotion in Paul trap. J. Appl. Phys. 83, 5025–5033 (1998).

  50. 50.

    Shen, J., Borodin, A. & Schiller, S. A simple method for characterization of the magnetic field in an ion trap using Be+ ions. Eur. Phys. J. D 68, 359 (2014).

  51. 51.

    Bakalov, D. & Schiller, S. The electric quadrupole moment of molecular hydrogen ions and their potential for a molecular ion clock. Appl. Phys. B 114, 213–230 (2014); corrigendum 116, 777–778 (2014).

  52. 52.

    Salumbides, E. J., Ubachs, W. & Korobov, V. I. Bounds on fifth forces at the sub-Å length scale. J. Mol. Spectrosc. 300, 65–69 (2014).

  53. 53.

    Pavanello, M., Tung, W.-C. & Adamowicz, L. Determination of deuteron quadrupole moment from calculations of the electric field gradient in D2 and HD. Phys. Rev. A 81, 042526 (2010).

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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.

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Correspondence to S. Schiller.

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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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