  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.
