Magic wavelengths for optical cooling and trapping of potassium

We carry out a systematic study of the static and dynamic polarizabilities of the potassium atom using a first-principles high-precision relativistic all-order method in which all single, double, and partial triple excitations of the Dirac-Fock wave functions are included to all orders of perturbation theory. Recommended values are provided for a large number of electric-dipole matrix elements. Static polarizabilities of the $4s$, $4p_j$, $5s$, $5p_j$, and $3d_j$ states are compared with other theory and experiment where available. We use the results of the polarizability calculations to identify magic wavelengths for the $4s-np$ transitions for $n = 4, 5$, {\em i.e.} those wavelengths for which the two levels have the same ac Stark shifts. These facilitate state-insensitive optical cooling and trapping. The magic wavelengths for the $4s-5p$ transitions are of particular interest for attaining a quantum gas of potassium at high phase-space density. We find 20 such wavelengths in the technically interest region of $1050-1130$~nm. Uncertainties of all recommended values are estimated.