We describe a method of selective generation and study of polarization moments of up to the highest rank κ = 2F possible for a quantum state with total angular momentum F . The technique is based on nonlinear magneto-optical rotation with frequency-modulated light. Various polarization moments are distinguished by the periodicity of light-polarization rotation induced by the atoms during Larmor precession and exhibit distinct light-intensity and frequency dependences. We apply the method to study polarization moments of 87 Rb atoms contained in a vapor cell with antirelaxation coating. Distinct ultra-narrow (1-Hz wide) resonances, corresponding to different multipoles, appear in the magnetic-field dependence of the optical rotation. The use of the highest-multipole resonances has important applications in quantum and nonlinear optics and in magnetometry.PACS numbers: PACS 32.80. Bx,95.75.Hi High-rank polarization moments (PM) and associated high-order coherences have recently drawn attention (see [1,2,3,4,5,6,7,8] While signatures of high-order PM were detected in several experiments [3,4,5,6,8], the methods used in these investigations are not sufficiently selective and/or do not allow real-time manipulation of particular multipoles. Here we describe a method, based on nonlinear optical rotation with frequency-modulated light (FM NMOR) [16], by which one can selectively induce, control, and study any possible multipole moment. Applying the method to 87 Rb atoms in a paraffin-coated cell [17,18], we have verified the expected power and spectral dependences of the resonant signals and obtained a quantitative comparison of relaxation rates for the even-rank moments.The density matrix in the M, M ′ representation for a state with total angular momentum F can be decomposed into PM of rank κ = 0 . . . 2F , uncoupled under rotations, with components q = −κ . . . κ: with surfaces for which the distance to the origin in a given direction is proportional to the probability of finding the projection M = F along this direction. (a): "pure" quadrupole κ = 2, q = 0; (b): κ = 4, q = 0 hexadecapole; (c): same as in (b), but rotated by π/2 around the x-axis; (d): the average of (b) and (c), which has a 4-fold symmetry with respect to rotations aroundx. In all cases, the minimum necessary amount of ρ (0) 0 was added to ensure that all sublevel populations are non-negative [20]. Probability surfaces (a) and (d) rotating aroundx-directed magnetic field with the Larmor frequency correspond to the polarization states produced in this experiment.is uniquely associated with the highest PM for a given state, e.g., the quadrupole moment (κ = 2) for F = 1, or the hexadecapole (κ = 4) for F = 2. The method introduced here exploits the different axial symmetries of the PM (2-fold and 4-fold for the quadrupole and hexadecapole, respectively; Fig. 1) to selectively create and detect them (see also [3,4,5,6]).While multipole moments of rank κ ≤ 2 can be easily generated and detected with weak light (since a photon has spin one), higher-ra...