Quantum-electrodynamic calculation of hyperfine-state populations in atomic sodium

A quantum-electrodynamic (QED) description of the resonant interaction of monochromatic light with the $3^2$ ${\mathit{S}}_{1/2}$(F’=2)→$3^2$ ${\mathit{P}}_{3/2}$(F=3,2,1) hyperfine transitions of the sodium $D_2$ line is formulated in terms of the Heisenberg atomic operator. Off-diagonal state coherences and all relaxation terms are included. It is found that the equations for the populations, optical coherences, and certain state coherences form a closed subset of the total system of equations. This subset is small enough to be computed numerically. The results of calculations with this model are compared with three previously developed semiclassical descriptions using the density operator. One of the semiclassical models was developed to describe the interaction of weak light with the atomic transition while another is suitable in the case of high-intensity light. For both π and σ excitation it is shown that the QED calculated values for the time-averaged, excited-level population probability converge to those of each of the semiclassical models in the appropriate limit of light intensity. For homogeneous broadening, results show that the optimum light intensity to obtain the largest fraction of atoms in the $3^2$ $P_{3/2}$ level after transversing a 1-mm-diam laser beam is 1.15 mW/${\mathrm{mm}}^2$ for π excitation and 1.43 mW/${\mathrm{mm}}^2$ for σ excitation.