Hyperfine effects on potassium tune-out wavelengths and polarizabilities

The influence of hyperfine interactions on the tune-out wavelengths of the $^{39,40,41}$K isotopes of the potassium atom was investigated. The hyperfine interaction of the $4s_{1/2}$ state result in a shift and splitting of the primary tune-out wavelength near 769 nm. The $4s_{1/2}$ ground state hyperfine splittings of the primary tune-out were almost equal to the hyperfine splittings of the ground states. The splittings in the wavelengths were 0.0008, 0.0027 and 0.0005 nm for $^{39}$K, $^{40}$K and $^{41}$K respectively. The hyperfine splitting of the $np_{J}$ levels leads to the creation of additional tune-out wavelengths. The additional tune-out wavelengths could be difficult to detect due to very small differences from the transition wavelengths to the $4p_{J,F}$ states. The hyperfine Stark shift for the ground states of all three isotopes were also computed and the value for $^{30}$K was found to be compatible with the previous experiments and the most recent calculation using relativistic many body perturbation theory.