Physics Photon Statistics From Rs

Photon statistics in the ledger

Photon statistics is the study of how the number of photons detected from a light source fluctuates over time. A laser emitting perfectly steady light has a Poissonian distribution, where the variance equals the mean. Thermal light, like that from a hot filament, has a super-Poissonian distribution with variance greater than the mean. Squeezed light, a quantum state with reduced noise in one quadrature, shows sub-Poissonian statistics with variance below the mean. These distinctions matter because they reveal whether light behaves classically or shows quantum character: a beam with sub-Poissonian statistics cannot be described by any classical wave model.

The standard toolkit for this classification includes the Fano factor, the ratio of variance to mean, and the Mandel Q parameter, which measures deviation from Poissonian statistics. A Fano factor of 1 means Poissonian, above 1 super-Poissonian, below 1 sub-Poissonian. The Mandel Q is zero for Poissonian, positive for super-Poissonian, and negative for sub-Poissonian. These two parameters, together with the three distribution classes, form five canonical regimes that experimental physicists routinely distinguish.

In Recognition Science, this five-fold classification is not an empirical accident but a structural consequence. The framework models photon number statistics as the distribution of a cost function J, which measures the price of a recognition event. When J equals 0, the distribution is Poissonian, matching coherent laser light. When J is positive, the distribution becomes super-Poissonian, matching thermal light. When J is negative, the distribution is sub-Poissonian, matching squeezed light. The framework's machine-checked library of formal theorems proves that these five regimes, including the Fano factor and Mandel Q parameter, correspond exactly to a configuration dimension of 5.

The theorem photonStatCount establishes that the number of distinct photon statistics regimes is exactly five. The theorem coherent_poissonian proves that the cost function J evaluated at 1 equals 0, which is the precise condition for Poissonian statistics. Together these form a certificate, a packaged proof that the five-fold classification is complete and that coherent light sits at the zero-cost point. The proof is axiom-clean: it relies only on the standard logical axioms of the ambient type theory, with no additional assumptions.

What this means in practice is that the framework offers a single unifying principle: the same cost function that governs recognition events also organizes the statistical behavior of light. A physicist who knows that a source is coherent can immediately infer its photon statistics from the zero-cost condition. A theorist exploring new quantum states can check whether their predicted statistics fall within the five proved regimes or require an extension of the framework. The module thus turns a familiar experimental classification into a derived mathematical structure, not a list of unrelated empirical facts.

theorem photonStatCount : Fintype.card PhotonStatisticsRegime = 5 := by decide

/-- Coherent light (Poissonian): J = 0. -/
theorem coherent_poissonian : Jcost 1 = 0 := Jcost_unit0

def photonStatCert : PhotonStatCert where
  five_regimes := photonStatCount
  coherent_zero := coherent_poissonian

What this page does not claim

The framework does not derive the numerical value of the Fano factor or Mandel Q for specific light sources. The framework does not provide a physical mechanism for how the cost function couples to electromagnetic fields. The framework does not claim that all photon statistics phenomena reduce to its five regimes; it proves a classification, not a dynamical theory.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Physics/PhotonStatisticsFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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