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:
| Q= | \left\langle(\Delta\hat{n |
| ) |
2\right\rangle-\langle\hat{n}\rangle}{\langle\hat{n}\rangle}=
| \langle\hat{n | |
| (2) |
\rangle-\langle\hat{n}\rangle2}{\langle\hat{n}\rangle}-1=\langle\hat{n}\rangle\left(g(2)(0)-1\right)
where
\hat{n}
g(2)
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.
-1\leqQ<0\Leftrightarrow0\leq\langle(\Delta\hat{n})2\rangle\leq\langle\hat{n}\rangle
The minimal value
Q=-1
\Deltan=0
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
Q=\langlen\rangle
Coherent states have a Poissonian photon-number statistics for which
Q=0