Rabi Oscillations vs Perturbation Theory

Exact Rabi formula (RED) versus first-order perturbation theory (SLATE dashed). At resonance (δ/Ω = 0), PT diverges — it predicts probability > 100%.

Transition Probability P vs Dimensionless Time x = Ωt
0.00
Exact Rabi: P = (1/(1+(δ/Ω)²)) · sin²(√(1+(δ/Ω)²) · x/2)
Perturbation Theory: PPT (first order)
P at x = π vs Detuning (0 to 5 δ/Ω)
Exact Rabi P(x=π)
PT P(x=π)

Current State

δ/Ω (detuning ratio) 0.000
Generalized Rabi freq √(1+(δ/Ω)²) 1.000 Ω
P_exact max over [0, 4π] 1.000
P_PT max over [0, 4π] 9.870
P_PT at x = π 2.467
P_exact at x = π 1.000
P_exact at x = 2π 0.000
P_PT at x = 2π (expect π²≈9.87 at δ=0) 9.870
⚠ PT PREDICTS P > 1 — UNPHYSICAL
Perturbation theory has broken down

Physics Notes

At resonance (δ=0): Exact Rabi cycles between 0 and 1. PT gives P = (Ωt)²/4 — grows without bound. At x = 2π (one full Rabi cycle), exact P = 0, but PT predicts π² ≈ 9.87 (987%!).

Off-resonance (large δ/Ω): Both methods agree when δ/Ω ≫ 1 (weak-coupling / short-time limit). At δ/Ω = 3, curves converge.

PT validity: First-order PT requires Ωt ≪ 1 at resonance. The "perturbation is small" assumption fails when the coupling drives significant population transfer.

Formulas

Ω_gen = √(Ω² + δ²)
P_exact = (Ω/Ω_gen)² sin²(Ω_gen t/2)
δ=0: P_PT = (Ωt)²/4 = x²/4
δ≠0: P_PT = sin²(δt/2)/(δt/2)² · (Ωt)²/4
x = Ωt (dimensionless)