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 Leonard 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]

Non-classical value

Negative values of Q corresponds to state which variance of photon number is less than the mean (equivalent to sub-Poissonian statistics). In this case, the phase space distribution cannot be interpreted as a classical probability distribution.

${\displaystyle -1\leq Q<0\Leftrightarrow 0\leq \langle (\Delta {\hat {n}})^{2}\rangle \leq \langle {\hat {n}}\rangle }$

The minimal value ${\displaystyle Q=-1}$ is obtained for photon number states (Fock states), which by definition have a well-defined number of photons and for which ${\displaystyle \Delta n=0}$.

Examples

For black-body radiation, the phase-space functional is Gaussian. The resulting occupation distribution of the number state is characterized by a Bose–Einstein statistics for which ${\displaystyle Q=\langle n\rangle }$.^[3]

Coherent states have a Poissonian photon-number statistics for which ${\displaystyle Q=0}$.