This studies the non-renewal statistics (NRS) of photon streams from a variety of configurations, though with an emphasis on (three-level) cascades:
Renewal statistics is when the probability density function only depends on the arrival time of the previous photon, so with no memory effect conditionned on previous events. The Authors claim that "nonrenewal photon statistics is rather the norm, and renewal the exception".
This paper introduces a new measure of NRS based on the correlation between consecutive interphoton times.
Interesting overview of various measures of photon statistics:
The cascade is found to have NRS, although in regime of rate invariance. This should be investigated further in relation to our circular quantum cascades.