Unphysical features in the application of the Boltzmann collision operator in the time dependent modelling of quantum transport

Abstract

In this work, the use of the Boltzmann collision operator for dissipative quantum transport is analyzed. Its mathematical role on the description of the time-evolution of the density matrix during a collision can be understood as processes of adding and subtracting states. We show that unphysical results can be present in quantum simulations when the old states (that built the density matrix associated to an open system before the collision) are different from the additional states generated by the Boltzmann collision operator. As a consequence of the Fermi Golden rule, the new generated sates are usually eigenstates of the momentum or kinetic energy. Then, the different time-evolutions of old and new states involved in a collision process can originate negative values of the charge density, even longer after the collision. This unphysical feature disappears when the Boltzmann collision operator generates states that were already present in the density matrix of the quantum system before the collision. Following these ideas, in this paper, we introduce an algorithm that models phonon-electron interactions through the Boltzmann collision operator without negative values of the charge density. The model only requires the exact knowledge, at all times, of the states that build the density matrix of the open system.