Poisson counting in a coherent state ($P(n)=e^{-|\alpha {|}^2}|\alpha {|}^{2n}/n!$)

The normalized coherent state has number amplitude $e^{-|\alpha {|}^2/2}{\alpha }^n/\sqrt{n!}$, hence the displayed Poisson law with mean and variance $|\alpha {|}^2$. A product of coherent modes has factorized number probabilities.