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:

:$Q=\frac = \frac -1 = \langle \hat \rangle \left(g^(0)-1 \right)$

where $\hat$ is the photon number operator and $g^$ 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.

:$-1\leq Q < 0 \Leftrightarrow 0\leq \langle (\Delta \hat)^2 \rangle \leq \langle \hat \rangle$

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 $Q=\langle n\rangle$.Mandel, L., and Wolf, E., '' Optical Coherence and Quantum Optics'' (Cambridge 1995)

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

References

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Further reading

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

Quantum optics