Derivation of coherent-population-trapping resonance signals from density-matrix equations with all relevant sublevels of Cs atoms and confirmation of experimental results

Coherent-population-trapping resonance in vapor cell is a quantum interference effect that appears in the two-photon transitions between the ground-state hyperfine levels of alkali-metal atoms and is often utilized in miniature and centimeter-scale clock devices. To quantitatively understand and predict the performance of this phenomenon, it is necessary to consider the transitions and relaxations between all hyperfine Zeeman sublevels involved in the different excitation processes of the atom. In this paper, we constructed a computational multilevel atomic model of the Liouville density-matrix equation for 32 Zeeman sublevels involved in the ${D}_{1}$ line of $^{133}\mathrm{Cs}$ irradiated by two frequencies with circularly polarized components and then simulated the amplitude and shape of the resonance spectrum of the transmitted light through centimeter-scale Cs-vapor cells. We show that the numerical solutions of the equation and analytical investigations adequately explain a variety of the characteristics observed in the experiment.