QED and relativistic nuclear recoil corrections to the 413-nm tune-out wavelength for the 2 ${}^{3}S_1$ state of helium

Comparison of high-accuracy calculations with precision measurement of the 413-nm tune-out wavelength of the $\mathrm{He}\left(2{}^3{S}_1\right)$ state provides a unique test of quantum electrodynamics (QED). We perform large-scale relativistic-configuration-interaction (RCI) calculations of the tune-out wavelength that include the mass-shift operator and fully account for leading relativistic nuclear recoil terms in the Dirac-Coulomb-Breit (DCB) Hamiltonian. We obtain the QED correction to the tune-out wavelength using perturbation theory, and the effect of finite nuclear size is also evaluated. The resulting tune-out wavelengths for the $2{}^3{S}_1(M_J=0)$ and $2{}^3{S}_1(M_J=\pm 1)$ states are 413.084 26(4) nm and 413.090 15(4) nm, respectively. When we incorporate the retardation correction of 0.000 560 0236 nm obtained by Drake et al. [Hyperfine Interact 240, 31 (2019)] to compare results with the only current experimental value of $413.0938\left(9_{\text{stat}}\right)\left({20}_{\text{syst}}\right)$ nm for the $2{}^3{S}_1(M_J=\pm 1)$ state, there is $1.4\sigma$ discrepancy between theory and experiment, which stimulates further theoretical and higher precision experimental investigations on the 413-nm tune-out wavelength. In addition, we also determine the QED correction for the static dipole polarizability of the $\mathrm{He}\left(2{}^3{S}_1\right)$ state to be 22.5 ppm, which may enable a new test of QED in the future.

Figure 1

Relative contributions of various corrections to the static dipole polarizability ${\alpha }_1\left(0\right)$ and the tune-out wavelength ${\lambda }_t$ for the $2{}^3{S}_1(M_J=\pm 1)$ state of ${}^4\mathrm{He}$.

Figure 2

Comparisons of the tune-out wavelength ${\lambda }_t$ (in nm) for the $2{}^3{S}_1(M_J=\pm 1)$ state of ${}^4\mathrm{He}$.