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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-ref-mandel1979-1-0[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 = ⟨ ( Δ Δ n ^ ^ ) 2 ⟩ − − ⟨ ⟨ n ^ ^ ⟩ ⟩ ⟨ ⟨ n ^ ^ ⟩ ⟩ = ⟨ ⟨ n ^ ^ 2 ⟩ ⟩ − − ⟨ ⟨ n ^ ^ ⟩ ⟩ 2 ⟨ ⟨ n ^ ^ ⟩ ⟩ − − 1 = ⟨ ⟨ n ^ ^ ⟩ ⟩ ( g ( 2 ) ( 0 ) − − 1 ) {\displaystyle 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 n ^ ^ {\displaystyle {\hat {n}}} is the photon number operator and g ( 2 ) {\displaystyle g^{(2)}} is the normalized second-order correlation function as defined by Glauber.cite-ref-glauber1963-2-0[2]

• Non-classical value

• Examples

• References

• Further reading

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 ≤ ≤ Q < 0 ⇔ ⇔ 0 ≤ ≤ ⟨ ⟨ ( Δ Δ n ^ ^ ) 2 ⟩ ⟩ ≤ ≤ ⟨ ⟨ n ^ ^ ⟩ ⟩ {\displaystyle -1\leq Q<0\Leftrightarrow 0\leq \langle (\Delta {\hat {n}})^{2}\rangle \leq \langle {\hat {n}}\rangle }

The minimal value Q = − − 1 {\displaystyle Q=-1} is obtained for photon number states (Fock states), which by definition have a well-defined number of photons and for which Δ Δ n = 0 {\displaystyle \Delta n=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 = ⟨ ⟨ n ⟩ ⟩ {\displaystyle Q=\langle n\rangle } .cite-ref-3[3]

Coherent states have a Poissonian photon-number statistics for which Q = 0 {\displaystyle Q=0} .

cite-note-mandel1979-11. ↑ citerefmandel1979Mandel, L. (1979). "Sub-Poissonian photon statistics in resonance fluorescence". Optics Letters. 4 (7): 205–7. Bibcode:1979OptL....4..205M. doi:10.1364/OL.4.000205. ISSN 0146-9592. PMID 19687850.

cite-note-glauber1963-22. ↑ citerefglauber1963Glauber, Roy J. (1963). "The Quantum Theory of Optical Coherence". Physical Review. 130 (6): 2529–2539. Bibcode:1963PhRv..130.2529G. doi:10.1103/PhysRev.130.2529. ISSN 0031-899X.

cite-note-33. ↑ Mandel, L., and Wolf, E., Optical Coherence and Quantum Optics (Cambridge 1995)

• L. Mandel, E. Wolf Optical Coherence and Quantum Optics (Cambridge 1995)

• R. Loudon The Quantum Theory of Light (Oxford 2010)