Final answer:
For modeling atom excitation by lasers, an Optical Bloch Equation solver is required instead of a rate equation solver, due to the coherent interactions that must be accounted for to achieve accurate populations of the low-lying states.
Step-by-step explanation:
You are correct in suspecting that a regular rate equation solver might not suffice for modeling the populations of low-lying, long-lived "dark" electronic states in an atom excited by a laser. This is due to the coherent nature of the interaction between the light and the atom, which cannot be captured by rate equations that assume incoherent processes and independent populations. Instead, an Optical Bloch Equation (OBE) solver is needed since the OBE takes into account the coherent interaction between the atomic states and the driving laser field, as well as the decay processes.
To model the populations of these states accurately, we should use the OBE approach especially when we deal with phenomena such as stimulated emission, population inversion, and the sensitive balance between absorption and emission processes which are fundamental to laser operation. Einstein's realization that stimulated emission and absorption are equally probable is an integral part of this. Therefore, for a realistic and accurate portrayal of the dynamics of excited states in laser-irradiated atoms, a more sophisticated approach like the OBE solver is essential.