Solution

Four complementary classical features of a large-amplitude coherent state are:

It is an eigenstate of the annihilation operator, $a ∣ z ⟩ = z ∣ z ⟩$ . Normally ordered field amplitudes therefore behave like a classical complex amplitude; normalized photon subtraction leaves the same state.
Free harmonic evolution preserves its form, $∣ z ⟩ \to ∣ z e^{- iω t} ⟩$ . The mean electric field follows a classical sinusoidal wave with an amplitude and phase set by $z$ .
Its photon number has Poisson distribution, with $Δ n / \overset{n}{ˉ} = 1/ \overset{n}{ˉ}$ and $g^{(2)} (0) = 1$ . Relative intensity fluctuations become small at large amplitude.
Its optical quadratures $X = (a + a^{†}) /2$ and $P = (a - a^{†}) / (2 i)$ have minimum, balanced uncertainties, $Δ X = Δ P = 1/2$ and $Δ X Δ P = 1/4$ . The fixed vacuum-scale fluctuation becomes small relative to the large phase-space displacement.

These properties identify a particular classical oscillation, rather than a state of definite photon number. Absolute quantum fluctuations remain even when their relative size is small.

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