Simple rate-equation model for polarization-dependent probe-absorption in Doppler-broadened ⁸⁷Rb vapor

In a multi-level atomic vapor such as Rb, Zeeman pumping (within the magnetic sub-levels of the "bright" hyperfine ground state) and hyperfine pumping (to another "dark" hyperfine ground state) compete to determine the absorption, even for a weak probe (intensity<< I_sat0 , where I_sat0 is the saturation intensity predicted by the conventional 2-level model) [1-3]. Zeeman pumping depends on the probe polarization, whereas hyperfine pumping is independent of it. For the Doppler broadened ⁸⁵Rb D₂ line, two analytical approaches derived from the full multi-level density-matrix model [1] and rate equation model [2] have been used to evaluate these effects. However, these models have limitations, deviating considerably from the experimental data and/or valid only for linearly or circularly polarized probes. Here, we present an ab-initio but simple, reduced 7-level rate equation model (∼100Χ faster/less memory intensive than full multi-level models) that includes the dependence of probe absorption on its intensity, polarization, and beam diameter (via the transit-relaxation) to consider the effect of hyperfine pumping and Zeeman pumping. Our model predictions are experimentally verified for the ⁸⁷Rb D₂ line, without any fitting parameters (residual error<5%), for various input probe polarizations (linear, elliptical, and circular) at probe intensities ranging from ∼0.01I_sat0 to I_sat0. The interplay of hyperfine and Zeeman pumping is decided by whether the atomic transitions being probed are primarily closed or open.

[1] Shin S R and Noh H R 2011 Optics Communications 284 1243–1246.

[2] Shin S R and Noh H R 2009 Journal of the Physical Society of Japan 78 084302–084302

[3] Bala R et al. 2022 Journal of Physics B: Atomic, Molecular and Optical Physics 55-165003