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

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 H + 2 , HD + , and D + 2 . 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 large subset of rovibrational one-photon transitions the absolute values of the quadrupole shifts range from 0.3 to 10 Hz for an electric field gradient of 10 8 V/m 2 . 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 total transition frequency uncertainty of selected transitions can be reduced below the 1 × 10 −15 level.

High-precision spectroscopy of the HD+ molecule at the 1-p.p.b. level

“…A conservative upper limit of 100 Hz to the electric-quadrupole shift for the (v, L):(0, 2)→(8, 3) transition is obtained from [40]. The second-order Doppler effect is dominated by average micromotion velocity of the HD + ions and is found to be less than 5 Hz.…”

A simple method for characterization of the magnetic field in an ion trap using Be+ ions

“…Precise measurements of transition frequencies of HD + require information about the magnetic field experienced by the ions in the trap. Therefore, it is natural to consider the coolant 9 Be + ions as an in-situ magnetic field sensor.…”

“…The J z = 0 → J ′ z = 0 component of this particular transition has a quadratic Zeeman shift that is approx. 2.9 kHz in a field of 1 G. Other metrological properties have been discussed in [9,10]. Thus, assuming that one can set a magnetic field with the small value B < 0.05 G in the trap, and that the ions are in the Lamb-Dicke regime, a shift and transition linewidth (due to a possible magnetic field gradient) of less than 10 Hz should be achievable.…”

“…1). 9 Be + ions are loaded and other species are removed from the trap [14]. During this step, the 313 nm carrier wave is red-detuned by a few hundred MHz from the cooling transition line, and the sideband power is set to a few ten percent of the carrier power.…”

“…As an example, the hyperfine transition frequency f 0 in the ground 2 S1 /2 level of 9 Be + is 1.25 GHz in zero field. Using RF spectroscopy between hyperfine states, the frequencies can be measured with kHz resolution or better.…”