Accurate calculation of the 413 nm tune-out wavelength for 2$^{\mathrm{3}}$S$_{\mathrm{1}}$ state of helium

The tune-out wavelength is the wavelength at which the dynamic dipole polarizability vanishes. The 413 nm tune-out wavelength of the 2$^{\mathrm{3}}$S$_{\mathrm{1}}$ state of helium is proposed as a non-energy test quantum electrodynamic (QED)$^{\mathrm{[1]}}$, which sparks the great interest in high-precise measurement$^{\mathrm{[2]}}$ and high-accuracy calculations of the tune-out wavelength of helium$^{\mathrm{[3,4]}}$. So far, there exists 19 ppm discrepancy between the trapped atom dynamics measurement of 413.0938(9$_{\mathrm{stat}})$(20$_{\mathrm{syst}})$ nm$^{\mathrm{[2]}}$ and the relativistic configuration-interaction (RCI) calculation of 413.0859(4) nm$^{\mathrm{[3]}}$. In present work we performed larger-scale RCI calculation based on the Dirac-Coulomb-Breit (DCB) equation with the mass shift operators included directly in the Hamiltonian. The advantage of this developed RCI method is that the finite nuclear mass and relativistic nuclear recoil corrections on the tune-out wavelength are taken into account self-consistently in DCB framework. The QED correction on the tune-out wavelength is also estimated. Our result of tune-out wavelength is 413.090 14(5) nm with an uncertainty of 0.12 ppm, which is about 25 times more accurate than the experimental value from Ref. [2]. This work will motivate a future experimental campaign to seriously test QED at higher level of accuracy. Reference: [1] J. Mitroy and L.Y. Tang, Phys. Rev. A 88, 052515 (2013) [2] B. M. Henson, R. I. Khakimov, R. G. Dall, K. G. H. Baldwin, L.Y. Tang, and A. G. Truscott, Phys. Rev. Lett. 115, 043004 (2015) [3] Y. H. Zhang, L. Y. Tang, X. Z. Zhang, and T. Y. Shi, Phys. Rev. A 93, 052516 (2016) [4] G. W. F. Drake, private communication (2017).