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 ωa and ωb is applied. When the difference frequency Δa b=ωa-ωb is equal to the frequency splitting Δ12 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".

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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 ωa (near resonant with the |1〉|3〉 transition) and ωb (near resonant with the |2〉|3〉 transition).

Define the atomic system.
Specify that there will be no explicit time dependence in the density matrix (steady-state condition).
Here is what the density matrix looks like.
Define two optical fields with frequencies ωa and ωb.
The Hamiltonian for the system subject to the optical fields. Each field is assumed to interact with only one transition—the other terms are set to zero.
The level diagram for the system.
Apply the rotating-wave approximation to the Hamiltonian.
IntrinsicRelaxation and TransitRelaxation supply the relaxation matrices.
OpticalRepopulation and TransitRepopulation supply the repopulation matrices.
Here are the evolution equations.
Observables supplies the absorption and phase shift experienced by the light in terms of the density-matrix elements.
Solve the system analytically. (We temporarily set the Method option of RowReduce to speed up Solve in Mathematica 9 and later.)
Here is the probe absorption spectrum as a function of probe detuning.
The increased transmission at the Raman resonance is associated with coherence induced between the |1〉 and |2〉 states. Here is the coherence as a function of probe frequency.