= Photon statistics
{title2=$p_n=\Pr(N=n)$}
<Photon statistics> is the <probability distribution> of the <number operator> in specified field modes, or the distribution of registered counts in a specified gate. Those distributions agree only for an appropriate ideal measurement. Mode selection, losses, and <detector dead time> matter. For a one-mode <density matrix>, $p_n=\langle n|\rho|n\rangle$ and
$$
\mu=\sum_{n=0}^{\infty}np_n,\qquad
\langle N(N-1)\rangle=\sum_{n=0}^{\infty}n(n-1)p_n.
$$
These two sums already distinguish preparations with the same <expected value>.
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