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. It is defined as the normalized variance of the boson distribution:

[math]\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) }[/math]

where [math]\displaystyle{ \hat{n} }[/math] is the photon number operator and [math]\displaystyle{ g^{(2)} }[/math] 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.

[math]\displaystyle{ -1\leq Q \lt 0 \Leftrightarrow 0\leq \langle (\Delta \hat{n})^2 \rangle \leq \langle \hat{n} \rangle }[/math]

The minimal value [math]\displaystyle{ Q=-1 }[/math] is obtained for photon number states (Fock states), which by definition have a well-defined number of photons and for which [math]\displaystyle{ \Delta n=0 }[/math].

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 [math]\displaystyle{ Q=\langle n\rangle }[/math].

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

Further reading

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