= Solution
Four complementary classical features of a large-amplitude <coherent state> are:
* It is an eigenstate of the <annihilation operator>, $a|z\rangle=z|z\rangle$. Normally ordered field <amplitudes> therefore behave like a classical complex <amplitude>; normalized <photon subtraction> leaves the same state.
* Free harmonic evolution preserves its form, $|z\rangle\to|ze^{-i\omega t}\rangle$. The mean <electric field> follows a classical sinusoidal wave with an <amplitude> and <phase> set by $z$.
* Its <photon number> has <Poisson distribution>, with $\Delta n/\bar n=1/\sqrt{\bar n}$ and $g^{(2)}(0)=1$. Relative <intensity> fluctuations become small at large <amplitude>.
* Its <optical quadratures> $X=(a+a^\dagger)/2$ and $P=(a-a^\dagger)/(2i)$ have minimum, balanced uncertainties, $\Delta X=\Delta P=1/2$ and $\Delta X\Delta P=1/4$. 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.
Back to article page
