Exact description of non-Markovian effect in open quantum system: the   discretized environment method
  An approach, called discretized environment method, is introduced to treat exactly non-Markovian effects in open quantum systems. In this approach, a complex environment described by a spectral function is mapped into a finite set of discretized states with an appropriate coupling to the system of interest. The finite set of system plus environment degrees of freedom are then explicitly followed in time leading to a quasi-exact description. The present approach is anticipated to be particularly accurate in the low temperature and strongly non-Markovian regime. The discretized environment method is validated on a two-level system (qubit) coupled to a bosonic or fermionic heat bath. A perfect agreement with the quantum Langevin approach is found. Further illustrations are made on a three-level system (qutrit) coupled to a bosonic heat-bath. Emerging processes due to strong memory effects are discussed. 
Denis Lacroix, V.V. Sargsyan, G.G. Adamian and N.V. Antonenko
quant-ph cond-mat.other math.DS nucl-th