Neutrino quantum kinetics for fast flavor conversion in a time-dependent environment

Fast flavor conversions (FFCs) of neutrinos, driven by the fast flavor instability (FFI), can reshape the neutrino flavor content in dense astrophysical environments such as core-collapse supernovae and neutron star mergers. Most studies of FFCs adopt a two-step approach, in which a flavor-unstable state containing deep electron-minus-heavy-flavor lepton number (E-XLN) angular crossings is first constructed and subsequently evolved. Because realistic crossings should instead develop gradually through neutrino transport, the validity of such setups has been called into question. We investigate this issue by solving the neutrino quantum kinetic equations with self-consistent collisional rates in a spherically symmetric supernova background whose electron fraction evolves in time through a sequence of stages, starting from a configuration free of E-XLN crossings. We find that the evolution proceeds through three characteristic episodes. In the shallow-crossing episode, FFCs develop from marginally unstable, shallow crossings, carrying small-scale structures consistent with linear stability analysis. In the near-crossing-elimination episode, the balance between collisions and FFCs keeps the system in a near-quasistationary state in which the emerging crossings are continuously eliminated, so that the strongly unstable regime is never reached. In the swapping episodes, the E-XLN reverses sign and a dynamically propagating flavor-swap E-XLN zero surface forms. We find that the evolved flavor content at the end of different time stages broadly agrees with the quasistationary solutions obtained in the corresponding two-step models adopting fixed matter backgrounds. In addition, we investigate the robustness of the effective classical transport (ECT) framework that adopts subgrid flavor redistribution using different parametrized prescriptions. Notably, except during the swapping [abridged]