The hydrogen molecular ions allow accurate quantum electrodynamics calculations of the energy levels. The experimental setups based on cold trapped HD^{+} ions and resonance-enhanced multiphoton dissociation allowed accurate measurements of optical and THz transitions, paving the way for metrology of fundamental constants, tests of the quantum electrodynamics and physics beyond the Standard Model.

Interrogations of trapped HD^{+} ions with two-photon rotational and rovibrational transitions are addressed theoretically in view of precision measurements. The two-photon transition rates are calculated with the two-photon operator formalism by taking into account the hyperfine structure (figure 1). Quantum state preparation in the ground vibrational level may be achieved by depleting selectively the population in some hyperfine states by optical or THz-wave pumping on strong transitions. Furthermore, a closed set of HD^{+} energy levels is addressed with the two-photon transitions (v,L)=(0,1)->(0,2)->(0,3) at 3.268 THz and (v,L)=(0,3)->(4,2)->(9,3) at 207.637 THz, and the photodissociation of the (v,L)=(9,3) level with 532 nm laser radiation. Blackbody radiation recycles continuously HD^{+} ions in the (v,L)=(0,0)-(0,5) levels. The calculations of the matrix elements for the two-photon operator provide the dynamic polarisabilities of the energy levels and their lightshifts. Selected hyperfine transitions between J_{z}=0āJā_{z}=0 levels and between stretched hyperfine levels have lightshifts in the range between the 10^{-12} level and the 10^{-15} level for a THz-wave intensity I_{thz}=1 mW/mm^{2}. Temporal dependences of the populations of trapped HD^{+} ions are described by a set of coupled rate equations to derive the lineshapes of the rotational and rovibrational transitions. Long natural lifetimes of HD^{+} ions energy levels allow spectral resolutions at better than the 10^{ā12} level. The rotational line may be observed with a 1 Hz level power-broadened linewidth for I_{thz}=1 mW/mm^{2} (figure 2).

The comparison of experimental frequencies of selected transitions of HD^{+} and H_{2}^{+ }with theoretical predictions, for experimental uncertainties assumed at the 10^{-12} level, may be exploited to determine the Rydberg constant, proton-to-electron and deuteron-to-proton mass ratios, proton radius, and deuteron radius, independently on previous results. Depending on possible issues of the proton radius puzzle, this approach may improve the determination of the proton-to-electron and deuteron-to-proton mass ratios beyond the 10^{-11} level.

Quantum simulation , Quantum sensors and quantum metrology , Fundamental science for quantum technologies