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Experiments with vortex beams have shown a surge of interest in controllingcold atoms. Most of the controlling protocols are dominated by circularlypolarized light due to its ability to induce vector polarization at atoms,which is impossible for paraxial linearly polarized light. Here we develop atheory for frequency dependent polarizability of an atomic state interactingwith a focused linearly polarized vortex beam. The naturally induced spin-orbitcoupling in this type of linearly polarized beam produces vector component ofthe valence polarizability to an atomic state, obeying the total angularmomentum conservation of the beam. The theory is employed on $^{87}$Sr$^{+}$ion to accurately calculate the magic wavelengths for the clock transitions andtune-out wavelengths for the clock states using relativistic coupled-clustermethod. The induced vector component in the dynamic polarizability due to thelinearly polarized focused vortex beam promotes fictitious magnetic field tothe atomic state. We demonstrate that this fictitious magnetic field, dependingon the focusing angle and OAM of the beam, improves the flexibility of thecoherent heteronuclear spin oscillations in a spin-1 mixture of $^{87}$Rb and$^{23}$Na atoms.

physical review. a/physical review, a

Vector polarizability of atomic state induced by a linearly polarized  vortex beam: External control of magic, tune-out wavelengths, and  heteronuclear spin oscillations