Physics: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. Denoting by $\mathrm{\Delta }\hat{n}:=\hat{n}-⟨\hat{n}⟩$ the (operator-valued) photon-number fluctuation, the Mandel Q parameter is defined as the normalized variance of the boson distribution:

$Q=\frac{⟨(\mathrm{\Delta }\hat{n}{)}^2⟩-⟨\hat{n}⟩}{⟨\hat{n}⟩}=\frac{⟨{\hat{n}}^2⟩-⟨\hat{n}{⟩}^2}{⟨\hat{n}⟩}-1=⟨\hat{n}⟩(g^{(2)}(0)-1)$

where $\hat{n}$ is the photon number operator and $g^{(2)}$ is the normalized second-order correlation function as defined by Glauber. Note that $\mathrm{\Delta }\hat{n}$ denotes the fluctuation operator itself, not the (scalar) standard deviation of the photon number, which is instead given by $\sqrt{⟨(\mathrm{\Delta }\hat{n}{)}^2⟩}$.

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.

$-1\le Q<0\iff 0\le ⟨(\mathrm{\Delta }\hat{n}{)}^2⟩\le ⟨\hat{n}⟩$

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 $⟨(\mathrm{\Delta }\hat{n}{)}^2⟩=0$.

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 $Q=0$.

Further reading

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