Magneto-optical rotation: Accurate approximated analytical solutions for single probe atomic magnetometers

We report an approximated analytical solution for a single-probe four-state atomic magnetometer where no analytical solution exists. This approximated analytical solution demonstrates excellent accuracy in broad probe power and detuning ranges when compared with the numerical solution obtained using a 4th order Runge-Kutta differential equation solver on MATLAB. The theoretical framework and results also encompass widely applied single-probe three-state atomic magnetometers for which no analytical solution, even approximated, is available to date in small detuning regions.