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Photon Statistics

Describes the probability distribution of photon energies in a quantum state, often following a Poisson or Gaussian distribution, vital for laser theory.
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The statement of the theorem

Let a^k\hat{a}_k and a^k†\hat{a}_k^{\dagger} be the annihilation and creation operators for the mode kk of the electromagnetic field, satisfying the canonical commutation relation [a^k,a^j†]=δkj[\hat{a}_k, \hat{a}_j^{\dagger}] = \delta_{kj}. Define the total photon number operator N^=∑ka^k†a^k\hat{N} = \sum_k \hat{a}_k^{\dagger} \hat{a}_k. For a quantum state ∣ψ⟩∈H|\psi\rangle \in \mathcal{H}, the probability P(n)P(n) of detecting nn photons is given by the expectation value of the number projection operator Π^n\hat{\Pi}_n: P(n)=⟨ψ∥Π^n∥ψ⟩=1n!⟨ψ∥(N^)n∥ψ⟩P(n) = \langle \psi\| \hat{\Pi}_n \| \psi\rangle = \frac{1}{n!} \langle \psi\| (\hat{N})_n \| \psi\rangle where (N^)n(\hat{N})_n is the nn-th order normally ordered moment of N^\hat{N}. Alternatively, the statistical distribution P(n)P(n) can be derived from the characteristic function χ(λ)=Tr(ρeiλN^)\chi(\lambda) = \text{Tr}(\rho e^{i \lambda \hat{N}}), where ρ=∣ψ⟩⟨ψ∣\rho = |\psi\rangle\langle \psi| is the density matrix, such that P(n) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \text{Re}\left\{ e^{-i n \lambda} \chi(\lambda) \right} d\lambda.
Source: Wikipedia