The electric quadrupole moment of molecular hydrogen ions and their potential for a molecular ion clock

TL;DR

This paper develops an ab-initio model for the electric quadrupole interactions in molecular hydrogen ions, assessing their impact on high-precision spectroscopy and proposing methods to minimize related frequency uncertainties.

Contribution

It introduces a comprehensive theoretical framework for quadrupole shifts in hydrogen molecular ions and evaluates their effects on transition frequencies, aiding the development of molecular ion clocks.

Findings

Quadrupole shifts range from 0.2 to 10 Hz for certain transitions.

Two-photon E1 transitions have vanishing quadrupole shifts.

Estimated transition frequency uncertainties can be reduced below 1.10^(-15).

Abstract

The systematic shifts of the transition frequencies in the molecular hydrogen ions are of relevance to ultra-high-resolution radio-frequency, microwave and optical spectroscopy of these systems, performed in ion traps. We develop the ab-initio description of the interaction of the electric quadrupole moment of this class of molecules with the static electric field gradients present in ion traps. In good approximation, it is described in terms of an effective perturbation hamiltonian. An approximate treatment is then performed in the Born-Oppenheimer approximation. We give an expression of the electric quadrupole coupling parameter valid for all hydrogen molecular ion species and evaluate it for a large number of states of H2+, HD+, and D2+. The systematic shifts can be evaluated as simple expectation values of the perturbation hamiltonian. Results on radio-frequency (M1), one-photon electric dipole (E1) and two-photon E1 transitions between hyperfine states in HD+ are reported. For two-photon E1 transitions between rotationless states the shifts vanish. For a subset of rovibrational one-photon transitions the quadrupole shifts range from 0.2 to 10 Hz for an electric field gradient of 0.1 GV/m2. We point out an experimental procedure for determining the quadrupole shift which will allow reducing its contribution to the uncertainty of unperturbed rovibrational transition frequencies to the 1.10^(-15) relative level and, for selected transitions, even below it. The combined contributions of black-body radiation, Zeeman, Stark and quadrupole effects are considered for a large set of transitions and it is estimated that the transition frequency uncertainty of selected transitions can be reduced below the 1.10^(-15) level.