I am having a question regarding the theoretical classifications of a single photon source. Normally, this is accounted by finding values in the second order correlation function $g^{(2)}(0)$ as low as possible.
An issue with this classification is that there could be mixed systems in which the coincidences as measured by the $g^{(2)}(0)$ are close to zero, but higher order correlation functions are larger than 1. An example for this is given by the state
$$ \rho=\alpha|{0}\rangle\langle{0}|+\beta|{1}\rangle\langle{1}|+\gamma|{3}\rangle\langle{3}|\;\;,\alpha=\frac{297001}{300000}\,,\beta=\frac{1999}{200000}\,,\gamma=\frac{1}{600000}\\$$
Which yields $g^{(2)}(0)=\frac{\langle\hat{a}^{\dagger 2}\hat{a}^2\rangle}{\langle\hat{a}^\dagger\hat{a}\rangle^2}=\frac{6\gamma}{(\beta+3\gamma)^2}=\frac{1}{10}$ and $g^{(3)}(0)=\frac{\langle\hat{a}^{\dagger 3}\hat{a}^3\rangle}{\langle\hat{a}^\dagger\hat{a}\rangle^3}=\frac{6\gamma}{(\beta+3\gamma)^3}=10$.
I want to find a way in which these correlations functions can be grouped into a single criterion for classifying single photon sources. An example for this would be to use the n-dimensional Euclidian distance in a "correlation function space", where the origin represents an ideal SPS since it is zero for all orders. However, a problem with this is that some important thresholds, such as the value for a coherent state (which is equal to 1 for all correlation functions), would turn to be $\sqrt{n}$.
Is there any way to build a criterion that includes higher order correlation functions, where the values 0 (ideal source) and 1 (coherent states) would still serve as thresholds?
I would appreciate any ideas!
Thanks
