Mandel Q parameter

The Mandel Q parameter measures the departure of the occupation number distribution from Poissonian statistics. It was introduced in quantum optics by L. Mandel.[1] It is a convenient way to characterize non-classical states with negative values indicating a sub-Poissonian statistics, which have no classical analog. It is defined as the normalized variance of the boson distribution:

${\displaystyle Q={\frac {\left\langle (\Delta {\hat {n}})^{2}\right\rangle -\langle {\hat {n}}\rangle }{\langle {\hat {n}}\rangle }}={\frac {\langle {\hat {n}}^{(2)}\rangle -\langle {\hat {n}}\rangle ^{2}}{\langle {\hat {n}}\rangle }}-1=\langle {\hat {n}}\rangle \left(g^{(2)}(0)-1\right)}$

where ${\displaystyle {\hat {n}}}$ is the photon number operator and ${\displaystyle g^{(2)}}$ is the normalized second-order correlation function as defined by Glauber.[2]