Generalized Wigner Function Formulation for Quantum Systems with Open   Boundaries
  A rigorous microscopic theory for the description of quantum-transport phenomena in systems with open boundaries is proposed. We shall show that the application of the conventional Wigner-function formalism to this problem leads to unphysical results, like injection of coherent electronic states from the contacts. To overcome such basic limitation, we propose a generalization of the standard Wigner-function formulation, able to properly describe the incoherent nature of carrier injection at the device spatial boundaries as well as the interplay between phase coherence and energy relaxation/dephasing within the device active region. The proposed theoretical scheme constitutes the first rigorous derivation of the phenomenological injection model commonly employed in the simulation of open quantum devices. 
Remo Proietti Zaccaria and Fausto Rossi