Propagation Dynamics and Transient Amplification in Warm and Cold Atomic EIT Systems

We study the limitations on observing transient amplification in atomic systems exhibiting electromagnetically induced transparency (EIT) and evaluate the limits of optical Bloch equation (OBE) models. Using propagation-based Maxwell-Bloch simulations, we show that single-atom, spatially uniform OBE treatments overestimate gain by neglecting propagation dynamics. In two-level systems, this yields incorrect transmission, while in three-level systems, it predicts unrealistically large amplification. Furthermore, we show that Doppler averaging in warm vapor suppresses oscillatory ringing and the maximum achievable gain. Our results explain discrepancies between OBE predictions and experimental observations, and establish practical limits on transient gain in cold and thermally broadened EIT media.