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Among the best known quantities obtainable from photon correlationmeasurements are the $g^{(m)}$~correlation functions. Here, we introduce a newprocedure to evaluate these correlation functions based on higher-orderfactorial cumulants $C_{\text{F},m}$ which integrate over the time dependenceof the correlation functions, i.e., summarize the available information atdifferent time spans. In a systematic manner, the information content ofhigher-order correlation functions as well as the distribution of photonwaiting times is taken into account. Our procedure greatly enhances thesensitivity for probing correlations and, moreover, is robust against a limitedcounting efficiency and time resolution in experiment. It can be applied evenin case $g^{(m)}$ is not accessible at short time spans. We use the newevaluation scheme to analyze the photon emission of a plasmonic cavity coupledto a single quantum dot. We derive criteria which must hold if the system canbe described by a generic Jaynes-Cummings model. A violation of the criteriacan be explained by the presence of an additional excited quantum dot state.

Higher-Order Photon Statistics as a New Tool to Reveal Hidden Excited States in a Plasmonic Cavity