Two quantum levels interact and avoid crossing. Sweeping the detuning slowly causes adiabatic following; fast sweeps induce diabatic transitions. The LZ formula gives the exact transition probability.
Energy Levels
State Populations |c₁|², |c₂|²
0.50
0.50
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P_LZ (formula)
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P_sim (computed)
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Γ = πΔ²/2v
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Regime
Landau-Zener Formula (1932)
Hamiltonian: H = ½[vt·σz + Δ·σx]
Transition prob: P_LZ = exp(−2πΓ) where Γ = πΔ²/(2ℏv)
Adiabatic (slow): Γ≫1 → P_LZ → 0 (stays on adiabat)
Diabatic (fast): Γ≪1 → P_LZ → 1 (crosses)
Gap at crossing: ΔE_min = Δ (avoided level repulsion)