Electromagnetically induced transparency in V -, Λ -, and cascade-type schemes beyond steady-state analysis

We analyze the electromagnetically induced transparency (EIT) in $V$-, $ensuremath{Lambda}$-, and cascade-type schemes in a time-dependent way via the Schr”odinger-Maxwell formalism. We derive explicitly the analytical expressions of the space-time dependent probe field, the corresponding phase shift, absorption or amplification, group velocity, and group velocity dispersion for all the three schemes. These simple analytical expressions not only demonstrate explicitly the similarities and essential differences of the three schemes but also provide a convenient basis for investigating how the many-body effects in solids modify the magnitude, spectral shape, and space and time dependence of EIT and EIT-related quantum coherence phenomena.