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

We analyze theoretically and experimentally the existence of a {\it magic frequency} for which the absorption of a linearly polarized light beam by vapor alkali atoms is independent of the population distribution among the Zeeman sub-levels and the angle between the beam and a magnetic field. The phenomenon originates from a peculiar cancelation 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.