Quantum meaning of classical EM field

The classical EM field can be regarded as the expectation value of the quantum field in a special type of state.

In quantum electrodynamics, the electric and magnetic fields are non-commuting observables, roughly analogous to position and momentum in non-relativistic single-particle quantum mechanics. In both cases, we have an "uncertainty principle" that limits how sharply-defined their values can simultaneously be. In a state where their expectation values are large compared to their "uncertainties," the expectation values can be regarded as classical electric and magnetic fields.

My self-answer to this question gives more technical detail and more links to related posts, especially regarding the relationship between the classical EM field and the "wavefunction" of a photon.