Tune-out wavelengths and landscape modulated polarizabilities of alkali Rydberg atoms in infrared optical lattices

Intensity modulated optical lattice potentials can change sign for an alkali metal Rydberg atom, and the atoms are not always attracted to intensity minima in optical lattices with wavelengths near the CO$_2$ laser band. Here we demonstrate that such IR lattices can be tuned so that the trapping potential seen by the Rydberg atom can be made to vanish for atoms in \targeted" Rydberg states. Such state selective trapping of Rydberg atoms can be useful in controlled cold Rydberg collisions, cooling Rydberg states, and species-selective trapping and transport of Rydberg atoms in optical lattices. We tabulate wavelengths at which the trapping potential vanishes for the $n$s, $n$p, and $n$d Rydberg states of Na and Rb atoms, and discuss advantages of using such optical lattices for state selective trapping of Rydberg atoms. We also develop exact analytic expressions for the lattice induced polarizability for the $m_z = 0$ Rydberg states, and derive an accurate formula predicting tune-out wavelengths at which the optical trapping potential becomes invisible to Rydberg atoms in targeted $l = 0$ states.