General connection between time-local and time-nonlocal perturbation expansions

The time-evolution equation of an open quantum system has long been known to come in two equivalent, yet fundamentally different flavors: a time-nonlocal one with a memory kernel and a time-local one with a generator, whose connection has been discovered only recently [phys_rev_x_11_021041_2021]. Here, the authors show that this implies a recursive relation between their perturbative series: the complicated time-local series can be computed from the simpler time-nonlocal one by disentangling the time ordering of microscopic interactions with the environment.