Magic frequencies in atom-light interaction for precision probing of the density matrix.

We analyze theoretically and experimentally the existence of a magic frequency for which the absorption of a linearly polarized light beam by a vapor of alkali-metal atoms is independent of the population distribution among the Zeeman sublevels and the angle between the beam and a magnetic field. The phenomenon originates from a peculiar cancellation of the contributions of higher moments of the atomic density matrix, and is described using the Wigner-Eckart theorem and inherent properties of Clebsch-Gordan coefficients. One important application is the robust measurement of the hyperfine population.