2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/133380An environment interacting with portions of a system leads to multiexponential interaction rates. Within the Keldysh formalism, we fictitiously homogenize the system-environment interaction yielding a uniform decay rate facilitating the evaluation of the propagators. Through an injection procedure we neutralize the fictitious interactions. This technique justifies a stroboscopic representation of the system-environment interaction which is useful for numerical implementation and converges to the natural continuous process. We apply this procedure to a fermionic two-level system and use the Jordan-Wigner transformation to solve a two-spin swapping gate in the presence of a spin environment.11 pages, 3 figures, title changed, some typos changedMesoscale and Nanoscale PhysicsChemical PhysicsQuantum Physicstext