Hydrogen molecular ions for precision physics – HYMPE

An experimental setup to produce, trap and cool H2+ ions, as well as an ultrastable mid-infrared laser source (quantum cascade laser) to probe the fundamental vibrational transition of H2+, are operational. The HYMPE project aims at developing the last missing element: a bench for absolute frequency measurements in the mid-infrared range, using optical frequency comb technology based on femtosecond lasers. The comb will be compared to the absolute frequency reference provided by the REFIMEVE+ network (European Fiber Network with European vocation). On the theoretical side, the objective is to push back the current limits of quantum electrodynamics correction calculations by adopting a new approach. This approach consists in integrating from the beginning the relativistic effects by solving the Dirac equation for the bound electron of H2+. Until now, relativistic effects have been included in the framework of a perturbative development, taking the Schrödinger equation as a starting point (the so-called non-relativistic quantum electrodynamics approach).

This project creates a natural synergy with the teams working on high resolution HD+ spectroscopy for similar objectives. In collaboration with the group of J. Koelemeij and W. Ubachs (Amsterdam VU), we achieved in 2020 a very accurate determination of the proton-electron mass ratio by HD+ spectroscopy, combining Dutch experimental results and calculations from our group (S. Patra et al., Science 369, 1238-41 (2020)).
A promising new line of research further enhancing the interest of H2+ spectroscopy was recently proposed by E. Myers (University of Tallahassee, Florida). The aim is to compare the results of spectroscopic measurements on H2+ and on its antimatter equivalent Hbar2- (made of two antiprotons and one positron) to test with an unprecedented accuracy the symmetry between matter and antimatter. A collaboration with E. Myers has started to study theoretically the feasibility of the experiment.

The very precise test of the Standard Model that we hope to achieve will have a strong impact on fundamental physics - far beyond the atomic and molecular physics community, in particular if it reveals a deviation between experimental results and theoretical predictions. The HYMPE project will also lay the groundwork for ultra-high resolution experiments using quantum logic spectroscopy. These experiments will allow to realize another fundamental test: that of a possible temporal variation of the mp/me ratio, predicted by some extensions of the standard model. Finally, the advances of quantum electrodynamics corrections calculations can be extended to other simple molecular systems on which high resolution spectroscopy projects are underway.

The finalization of the experimental setup (excluding frequency measurements) was marked by a publication demonstrating the trapping and cooling of H2+ ions produced in a selected ro-vibrational state: J. Schmidt et al., Phys. Rev. Appl. 14, 024053 (2020).
From the theoretical point of view, the calculation of transition frequencies has been revisited and updated taking into account the latest theoretical progress on QED corrections and the most recent values of fundamental constants: V.I. Korobov and J.-Ph. Karr, Phys. Rev. A 104, 032806 (2021). The description of the hyperfine structure has also been improved: J.-Ph. Karr et. al, Phys. Rev. A 102, 052827 (2020) and can be tested with high accuracy in the HYMPE project.

The hydrogen molecular ions (HMI) H2+ or HD+ are simple quantum systems that can be calculated with a very high degree of precision, and their spectra contain many narrow lines that can be probed with high resolution. The HYMPE project aims to exploit this strong metrological potential to improve the determination of fundamental physical constants, by pushing experimental and theoretical techniques to their limits. The main expected results are a determination of the electron-to-proton mass ratio me/mp at an accuracy level of 6 ppt - an improvement by nearly one order of magnitude - and an independent determination of the Rydberg constant and proton charge radius, helping to solve the current "proton-radius puzzle".
The project relies on the measurement of Doppler-free two photon transitions in the mid-infrared (MIR), between the vibrational levels v=0 and v=1 of H2+. The experiment is carried out on state-selected H2+ ions, which are trapped and sympathetically cooled by laser-cooled Be+ ions. Detection of the transitions is done by the REMPD (Resonance-Enhanced Multi Photon Dissociation) technique. The objective is to observe and measure the transition frequencies with 12 significant digits.
To this end, the last investment required to complete a complex experimental set-up is the implementation of a commercial frequency comb between 1500 and 1900 nm stabilized on the references (initially at 100 MHz, then at 1542 nm) delivered by optical fiber by the SYRTE laboratory. This comb, combined with non-linear frequency conversion techniques, will measure the frequency of the MIR laser with an accuracy of 13 to 14 digits (task 1).
Task 2 consists, first of all, in operating the entire experiment: selective creation of H2+ ions in the desired ro-vibrational state (v=0,L=0 or 2), two-photon excitation and dissociation, and measurement of the H2+ ion number, in order to observe the transitions. This will be followed by a data taking phase during which the influence of key parameters such as MIR laser power, trapping conditions and beam alignment will be studied. Only a detailed understanding of the lineshape and systematic effects (lightshifts, Zeeman effect, etc.), in conjunction with theoretical modelling, will make it possible to interpret the results and extract a new determination of me/mp at the end of the project. One of the strengths of the project is the close and daily interaction between theorists and experimentalists of our group.
Thanks to the very precise resolution of the three-body Schrödinger equation, and calculation of QED corrections up to a high order in the NRQED (Non Relativistic Quantum Electrodynamics) approach, the ro-vibrational transition frequencies of IMHs are currently predicted with a theoretical uncertainty of 7.6 ppt. This would still represent a limiting factor for the determination of me/mp. The objective of Task 3 is to bring this limit to 3 ppt by improving the calculation of the one-loop self-energy correction. An innovative approach is proposed, which consists in performing a non-perturbative calculation based on precise numerical resolution of the Dirac equation for the bound electron in a two-center potential.
A longer-term perspective of the project is to extend quantum logic spectroscopy (QLS) techniques to the H2+ molecular ion in order to reach resolutions of up to 17 digits and thus improve the current constraints on the time variations of me/mp. Task 4 aims to demonstrate a key step towards implementation of the QLS protocol; namely the cooling of the external degrees of freedom of a Be+/H2+ ion pair to the quantum ground state by the Raman sideband cooling technique.