Coherent state

A coherent state of the electromagnetic field is an eigenstate |α⟩ of the photon annihilation operator â for a single mode, satisfying â|α⟩ = α|α⟩ for some complex number α. Glauber introduced the formalism in 1963 and earned the 2005 Nobel Prize for it. The photon-number distribution |⟨n|α⟩|² = e^{−|α|²} |α|^{2n} / n! is Poisson with mean ⟨n⟩ = |α|² and standard deviation σ_n = |α|. The relative quantum fluctuation σ_n / ⟨n⟩ = 1 / |α| therefore decreases as 1/√⟨n⟩ as the mean photon number increases.

The coherent state is the bridge from quantum electrodynamics to classical electromagnetism. The expectation value of the electric-field operator in a coherent state ⟨α| Ê(x, t) |α⟩ is exactly the classical electric field of a wave with the corresponding amplitude and phase — α directly encodes the classical field's amplitude (|α|) and phase (arg α). In the high-occupation-number limit |α|² → ∞ the relative quantum fluctuation vanishes, the field operator Ê becomes effectively diagonal on the coherent state, and the quantum description becomes indistinguishable from the classical description. A laser at full power produces a coherent state with ⟨n⟩ ≈ 10^{15} photons per mode; the relative quantum fluctuation is ≈ 10^{−7.5}, far below any experimental sensitivity, and the laser's output is treated classically as a complex-amplitude wave to better than parts-per-million accuracy. The classical limit of QED is therefore not a special case but a high-N statistical regime — exactly analogous to the way a Bose-Einstein condensate's quantum field becomes a classical wavefunction at macroscopic occupation. Maxwell's equations are the high-N limit of the photon's quantum field, and the coherent state |α⟩ is the eigenstate that makes the limit explicit.