Gombojav O Ariunbold, Vaidyanathan Sivaraman
Cooperative emission from three-level ensembles is conventionally diagnosed through the radiated intensity alone, which cannot say which channels carry the photons at each instant. We promote every transition of the cooperative-decay graph into a time-dependent edge flux and show that node-wise conservation of this flux is an exact restatement of the Pauli master equation. Suppose the graph carries a depth function that decreases by one along every edge. Then the total cooperative rate is the descent speed of the mean graph depth, and its energy-weighted form gives the radiated intensity as $I(t)=-\dd\langle E\rangle/\dd t$ for arbitrary level spacings. The same ordering triangularizes the generator. This delivers the relaxation spectrum from the state exit rates, together with a generalized Bateman closed form for every population on an arbitrary finite directed acyclic graph, in which all exit-rate degeneracies are absorbed into a polynomial-times-exponential recursion.