Rayleigh Scattering View of the Tune-out Wavelength: Application to the 1s2s $^3$S - 1s3p $^3$P Transition of Helium

The tune-out wavelength is usually viewed a zero in the frequency-dependent polarizability [1,2]. This view is appropriate for an atom in an optical lattice that is fixed in space. However, for an atom interacting with a traveling plane wave from a laser, it is more appropriate to view the tune-out wavelength as a zero in the Rayleigh scattering cross section for coherent scattering. In lowest order, the two approaches are equivalent, but not when higher-order retardation corrections are taken into account. This paper presents a development of the theory, starting from the relativistic scattering matrix of QED to obtain a formulation of the problem in the velocity gauge [3]. Gauge invariance is discussed, and an equivalent length form is obtained for the leading retardation correction for S-states. The $xp_z$ retardation correction to the tune-out wavelength of helium near 304 nm is calculated to be $0.000\,560\,0236$ nm. \newline [1] B. M. Henson et al., Phys.\ Rev.\ Lett.\ {\bf 115}, 043004 (2015).\newline [2] Y.-H. Zhang et al., Phys.\ Rev.\ A {\bf 93}, 052516 (2016).\newline [3] G. W. F. Drake et al. Hyperfine Int.\ submitted (2019).