Mandel Q parameter

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The Mandel Q parameter measures the departure of the occupation number distribution from Poissonian statistics. It was introduced in quantum optics by Leonard Mandel.{{cite journal|last1=Mandel|first1=L.|title=Sub-Poissonian photon statistics in resonance fluorescence|journal=Optics Letters|volume=4|issue=7|year=1979|pages=205–7|issn=0146-9592|doi=10.1364/OL.4.000205|pmid=19687850|bibcode=1979OptL....4..205M}} 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.{{cite journal|last1=Glauber|first1=Roy J.|title=The Quantum Theory of Optical Coherence|journal=Physical Review|volume=130|issue=6|year=1963|pages=2529–2539|issn=0031-899X|doi=10.1103/PhysRev.130.2529|bibcode=1963PhRv..130.2529G|doi-access=free}}

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

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

Mandel Q parameter

Non-classical value

Examples

References

Further reading