The Exchange — Polariton Formation & Vacuum Rabi Splitting

Act 10 of the NexusOS physics sequence. First disclosed 2026-07-19. AGPL-3.0. Founder: Te Rata Pou.

The Jaynes-Cummings Hamiltonian

H = ℏω_c a†a + ℏω_a σ_z/2 + ℏg(a†σ₋ + aσ₊). Three terms: cavity field energy, emitter energy, and the exchange coupling that cannot be switched off.

Polariton Dressed States

|+⟩ = (|e,0⟩ + |g,1⟩)/√2 — upper polariton. |−⟩ = (|e,0⟩ − |g,1⟩)/√2 — lower polariton. Energy splitting: ΔE = 2ℏg. Vacuum Rabi frequency: Ω_R = 2g. The splitting occurs even with zero photons — driven by the ZPE floor from Act 8.

Coupling Strength

g = d·√(ω_c / 2ℏε₀V). Coupling grows with dipole moment d and shrinks with mode volume V. Photonic crystal cavities (V ~ 0.001 μm³) achieve g/2π ~ 100 GHz.

Three Regimes

g > κ, γ: Strong coupling — polariton formation, vacuum Rabi oscillations
g ≪ κ: Weak coupling — Purcell enhancement of spontaneous emission
Δ ≫ g: Dispersive — QND measurement via phase shift χ = g²/Δ

Polariton Compression States in WNSP

Λ_± = h(f_c ± g/2π)/c². Each polariton branch has a unique compression state address in the WNSP Ψ channel system. The polariton is half-field, half-matter — a new class of compression state at the wave-matter boundary.

The Rabi Oscillation

|ψ(t)⟩ = cos(g·t)|e,0⟩ − i·sin(g·t)|g,1⟩. P_emitter(t) = cos²(g·t). P_cavity(t) = sin²(g·t). Total excitation conserved. The exchange is lossless in the strong-coupling regime.