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A method is proposed to describe Fermi or Bose systems coupled to one orseveral heat baths composed of fermions and/or bosons. The method, calledCoupled Equations of Motion method, properly includes non-Markovian effects.The approach is exact in the Full-Coupling approximation when only bosonicparticles are present in the system and baths. The approach provides anapproximate treatment when fermions are present either in the system and/or inone or several environments. The new approach has the advantage to properlyrespect the Pauli exclusion principle for fermions during the evolution. Weillustrate the approach for the single Fermi or Bose two-level system coupledto one or two heat-baths assuming different types of quantum statistics(Fermion or Bosons) for them. The cases of Fermi system coupled to fermion orboson heat baths or a mixture of both are analyzed in details. With the futuregoal to treat Fermi systems formed of increasing number of two-level systems(Qubits), we discuss possible simplifications that could be made in theequations of motion and their limits of validity in terms of the system--bathscoupling or of the initial heat baths temperatures.

Non-Markovian modeling of Fermi-Bose systems coupled to one or several Fermi-Bose thermal baths