Coherent population trapping linewidths for open transitions: Cases of different transverse laser intensity distribution

Abstract

We calculated the coherent population trapping (CPT) for the open $D1$ line transition ${F}_{g}=2\ensuremath{\rightarrow}{F}_{g}=1$ in $^{87}\mathrm{Rb}$ by solving the time-dependent optical Bloch equations, integrating the results for the total excited state populations over atomic trajectories, and averaging over velocities (components perpendicular and parallel to the laser beam) and over atom incident angles to the laser beam. We obtained a square root dependence of the CPT linewidths as a function of the laser intensity for both steplike and Gaussian transverse beam profiles. The results obtained with the Gaussian transverse laser beam profile are in good agreement with recent experimental results. Results for both steplike and Gaussian profiles of the laser beam show asymptotic $1∕\sqrt{d}$ dependence ($d$ is the laser beam diameter) of the CPT linewidths, for a large range of the laser intensity.