Hyperpolarizabilities of alkaline-earth metal ions Be+, Mg+, and Ca+

Abstract

The knowledge of the hyperpolarizabilities of atoms and ions is helpful for the analysis of the high order effects of the frequency shifts in precision spectroscopy experiments. Liu et al. [Phys. Rev. Lett. 114, 223001 (2015)] proposed to establish all-optical trapped ion clocks using laser at the magic wavelength for clock transition. To evaluate the high-order frequency shifts in this new scheme of optical clocks, hyperpolarizabilities are needed, but absent. Using the finite field method based on the B-spline basis set and model potentials, we calculated the electric-field-dependent energy shifts of the ground and low-lying excited states in Be+, Mg+, and Ca+ in the field strength range of 0.0-6×10−5 a.u.. The scalar and tensor polarizabilities (α 0, α 2) and hyperpolarizabilities (γ 0, γ 2, γ 4) were deduced. The results of the hyperpolarizabilities for Be+ showed good agreement with the values in literature, implying that the present method can be applied for the effective estimation of the atomic hyperpolarizabilities, which are rarely reported but needed in experiments. The feasibility of optical trapping of Ca+ is discussed, and the contributions of hyperpolarizabilities to the transition frequency shift for Ca+ in the optical dipole trap are estimated using quasi-electrostatic approximation.