Counting Photons
At low intensities, light reveals its granular nature: photodetectors register individual clicks rather than a smooth signal. The statistical distribution of these clicks carries deep information about the quantum state of the light field. Roy Glauber's 1963 theory of optical coherence showed that different light sources produce fundamentally different photon number distributions, earning him the 2005 Nobel Prize.
Three Regimes of Light
Coherent light from an ideal laser follows Poisson statistics where the variance equals the mean — this is the shot-noise limit. Thermal light from an incandescent bulb or chaotic source shows excess fluctuations (bunching) with variance σ² = ⟨n⟩ + ⟨n⟩². Remarkably, quantum light sources like single atoms or parametric downconversion can produce sub-Poissonian statistics with variance below the mean — a signature with no classical explanation.
The Fano Factor and Mandel Q
The Fano factor F = σ²/⟨n⟩ and Mandel Q = F - 1 quantify how a distribution deviates from the Poisson reference. F = 1 (Q = 0) is the classical-quantum boundary: any light source with F < 1 is certifiably nonclassical. This simple metric is used worldwide to characterize single-photon emitters, squeezed states, and other quantum light sources.
Applications in Quantum Technology
Photon statistics underpin quantum key distribution (QKD), where single-photon sources with F ≈ 0 prevent eavesdropping. In quantum metrology, sub-Poissonian light enables measurements beyond the shot-noise limit. Understanding and controlling photon statistics is essential for building quantum networks, improving gravitational wave detectors, and developing photonic quantum computers.