Photon Statistics — Thermal, Coherent & Fock States

Mandel Q = (⟨n²⟩−⟨n⟩²)/⟨n⟩ − 1: Q<0 sub-Poissonian, Q=0 coherent, Q>0 super-Poissonian

Thermal

Coherent

Fock |n⟩

Squeezed

Mean photon ⟨n⟩ 5.0

Fock state |n⟩ 3

Squeezing r 0.5

Sampling trials 2000

Delay τ (HBT) 0

Phase offset φ 0

Photon number distributions: Thermal: P(n) = ⟨n⟩^n/(⟨n⟩+1)^(n+1) — Bose-Einstein, super-Poissonian (Q=⟨n⟩). Coherent: P(n) = e^(−⟨n⟩)⟨n⟩^n/n! — Poisson, Q=0 (shot noise limited). Fock |n⟩: definite photon number, sub-Poissonian (Q=−1). Hanbury Brown-Twiss g²(τ): thermal g²(0)=2 (bunching), coherent g²(0)=1, Fock g²(0)=(n−1)/n<1 (anti-bunching). Wigner function: quasi-probability distribution in phase space; negative regions indicate non-classical states.