STIRAP Three-Level Dark State

Stimulated Raman Adiabatic Passage (STIRAP) utilizes a counterintuitive sequence of two optical pulses (Stokes followed by Pump) to transfer population between two states via an intermediate state. The process relies on following a three-level dark eigenstate that has no amplitude in the intermediate state, thereby preventing spontaneous emission and ensuring a robust transfer.

The three-level lambda system Hamiltonian is parameterized by the Pump coupling OmegaP = Omega * sin(theta) and Stokes coupling OmegaS = Omega * cos(theta). The dark state is given by cos(theta)|1> - sin(theta)|3> and has zero energy and zero intermediate population. The bright states have energies (Delta ± sqrt(Delta² + Omega²)) / 2.

Worked example: STIRAP Dark State

Let's examine the resonant endpoints of a three-level lambda system. Consider a setup where the mixing degree is 45 degrees, total coupling is 1, and middle detuning is 0.

First, we determine the Pump and Stokes couplings: OmegaP = 1 * sin(45°) ≈ 0.7071067811865475, OmegaS = 1 * cos(45°) ≈ 0.7071067811865476.
Next, we calculate the dark state amplitudes: Dark amplitude first = cos(45°) ≈ 0.7071067811865476, Dark amplitude third = -sin(45°) ≈ -0.7071067811865475.
The middle population is 0 because the state is fully dark.
The bright state energies are determined by (Delta ± sqrt(Delta² + Omega²)) / 2. With Delta = 0 and Omega = 1, these are (0 ± 1) / 2, giving lower energy -0.5 and upper energy 0.5.
The minimum dark gap is the smallest absolute difference between the dark state energy (0) and the bright states. Both are at distance 0.5, so the gap is 0.5.

Native experiment

Open the STIRAP Three-Level Dark State interactive tool to observe the eigenstates and energy levels.

Set Mixing degrees to 30, Total coupling to 1, and Middle detuning to 0. Notice how the Pump coupling decreases while the Stokes coupling increases. The dark amplitude for state |1> becomes larger than for state |3>. The dark gap remains at 0.5.
Change the Middle detuning to 1. The bright state energies shift to -0.20710678118654757 and 1.2071067811865475. The minimum dark gap decreases to approximately 0.2071, making the adiabatic passage more sensitive.
Set the Total coupling to 0. The system reaches a degenerate uncoupled limit where the dark amplitudes are unavailable, and the gap collapses to 0.

Model limitations

This static eigenstate lesson illustrates the resonant endpoints and intermediate dark state configuration of a STIRAP process. It does not simulate the actual time evolution, nor does it guarantee that an arbitrary sweep will transfer perfectly without non-adiabatic losses. The model does not include spontaneous emission or dissipative effects, focusing strictly on the Hamiltonian eigenstates.

References

For a comprehensive review of Stimulated Raman Adiabatic Passage in atomic physics and coherent control, refer to:Rev. Mod. Phys. 89, 015006 (2017).

STIRAP Three-Level Dark State