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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. 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:

Contents

Q = ⟨ ( Δ n ^ ) 2 ⟩ − ⟨ n ^ ⟩ ⟨ n ^ ⟩ = ⟨ n ^ ( 2 ) ⟩ − ⟨ n ^ ⟩ 2 ⟨ n ^ ⟩ − 1 = ⟨ n ^ ⟩ ( g ( 2 ) ( 0 ) − 1 )

where 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 ≤ Q < 0 ⇔ 0 ≤ ⟨ ( Δ n ^ ) 2 ⟩ ≤ ⟨ n ^ ⟩

The minimal value Q = − 1 is obtained for photon number states, which by definition have a well-defined number of photon and for which Δ 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 = ⟨ n ⟩ .

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

References

Mandel Q parameter Wikipedia
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