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 $Q=0$.

Further reading

• L. Mandel, E. Wolf Optical Coherence and Quantum Optics (Cambridge 1995)
• R. Loudon The Quantum Theory of Light (Oxford 2010)