# Three-Level System: Electromagnetically Induced Transparency

Consider an atomic Λ-system: a three-level system with two same-parity lower states |1〉 and |2〉, and an upper state |3〉 with opposite parity. A near-resonant light field consisting of two phase-coherent components of frequency ω_{a} and ω_{b} is applied. When the difference frequency Δ_{a b}=ω_{a}-ω_{b} is equal to the frequency splitting Δ_{12} of the two lower levels (the Raman resonance condition), the system is pumped into a coherent superposition of the two lower states which no longer absorbs the bichromatic field. When one frequency component of the light field is much stronger than the other, it is referred to as a "coupling" or "drive" field, and the reduction of the absorption of the weaker probe field is called "electromagnetically induced transparency". Other names used to refer to this situation are "dark resonance" and "coherent population trapping".

We define an atomic system consisting of two even-parity lower states and an odd-parity upper state. We apply a light field with components at frequencies ω_{a} (near resonant with the |1〉|3〉 transition) and ω_{b} (near resonant with the |2〉|3〉 transition).

Observables supplies the absorption and phase shift experienced by the light in terms of the density-matrix elements. |