Decoherence under many-body system-environment interactions: a stroboscopic representation based on a fictitiously homogenized interaction rate

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.