Atomic clocks highly sensitive to the variation of the fine-structure constant based on Hf ii, Hf iv, and W vi ions

Abstract

We demonstrate that several metastable excited states in Hf ii, Hf iv, and W vi ions may be good clock states since they are sufficiently long-living and are not sensitive to the perturbations. The cooling electric dipole ($E1$) transition is available for Hf ii, while sympathetic cooling is possible for Hf iv and W vi using ${\mathrm{Ca}}^{+}$ or ${\mathrm{Sr}}^{+}$ ions. Energy levels; Land\'e $g$ factors; transition amplitudes for electric dipole ($E1$), electric quadrupole ($E2$), and magnetic dipole ($M1$) transitions; lifetimes; and electric quadrupole moments for Hf ii, Hf iv, and W vi ions are investigated using a combination of several methods of relativistic many-body calculations including the configuration interaction (CI), linearized coupled-cluster single-doubles (SD), and many-body perturbation theory ($\mathrm{CI}+\mathrm{SD}$), and also the configuration interaction with perturbation theory (CIPT). Scalar polarizabilities of the ground states and the clock states have been calculated to determine the blackbody radiation (BBR) shifts. We have found that the relative BBR shifts for these transitions range between ${10}^{\ensuremath{-}16}$ and ${10}^{\ensuremath{-}18}$. A linear combination of two clock transition frequencies allows one to further suppress BBR. Several $5d\text{\ensuremath{-}}6s$ single-electron clock transitions ensure high sensitivity of the transition frequencies to the variation of the fine-structure constant $\ensuremath{\alpha}$ and may be used to search for dark matter producing this variation of $\ensuremath{\alpha}$. The enhancement coefficient for the $\ensuremath{\alpha}$ variation reaches $K=8.3$. Six stable isotopes of Hf and five stable isotopes in W allow one to make King plots and study its nonlinearities in order to put limits on the new interactions mediated by scalar particles or other mechanisms.