  An 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.
