Optical Bloch Equations describe the evolution of a two-level quantum system driven by a classical electromagnetic field, including spontaneous emission. The Bloch vector (u,v,w) represents the quantum state on the unit sphere.
du/dt = Δv − γ₂u | dv/dt = −Δu + Ωw − γ₂v | dw/dt = −Ωv − γ(w+1)
Without decay (γ=γ₂=0), the Bloch vector precesses about the generalized Rabi vector Ω_eff = (Ω, 0, Δ) at frequency Ω_R = √(Ω²+Δ²). At resonance (Δ=0), this gives perfect Rabi oscillations between ground |g⟩ and excited |e⟩ states. Detuning reduces and oscillates the excitation probability. Decay drives the system to steady state.