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 L. Mandel.[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:

where is the photon number operator and is the normalized second-order correlation function as defined by Glauber.[2]

Non-classical value[edit]

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.

The minimal value is obtained for photon number states (Fock states), which by definition have a well-defined number of photons and for which .


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 .[3]

Coherent states have a Poissonian photon-number statistics for which .


  1. ^ Mandel, L. (1979). "Sub-Poissonian photon statistics in resonance fluorescence". Optics Letters. 4 (7): 205–7. doi:10.1364/OL.4.000205. ISSN 0146-9592. PMID 19687850.
  2. ^ Glauber, Roy J. (1963). "The Quantum Theory of Optical Coherence". Physical Review. 130 (6): 2529–2539. doi:10.1103/PhysRev.130.2529. ISSN 0031-899X.
  3. ^ Mandel, L., and Wolf, E., Optical Coherence and Quantum Optics (Cambridge 1995)

Further reading[edit]

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