Past exam of the laser science and quantum information processing course of the University of Oxford / 2015 / 2 / c

The photon number probabilities of a coherent state form a Poisson distribution with mean $\mu =|\alpha {|}^2$. Alternatively normal ordering gives $⟨\hat{n}(\hat{n}-1)⟩=⟨{\hat{a}}^{†2}{\hat{a}}^2⟩=|\alpha {|}^4$. ThusThe last ratio is defined only for nonzero $\alpha$. At large $|\alpha |$ the absolute fluctuation grows but its size relative to the mean becomes negligible. This is the deterministic intensity limit within quantum optics; it does not assert the disappearance of vacuum-scale quadrature noise.