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. 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:


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

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

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.


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

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

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 $$ Q=\langle n\rangle $$.

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