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

摘要

In this work, the use of the Boltzmann collision operator for dissipativequantum transport is analyzed. Its mathematical role on the description of thetime-evolution of the density matrix during a collision can be understood asprocesses of adding and subtracting states. We show that unphysical results canbe present in quantum simulations when the old states (that built the densitymatrix associated to an open system before the collision) are different fromthe additional states generated by the Boltzmann collision operator. As aconsequence of the Fermi Golden rule, the new generated sates are usuallyeigenstates of the momentum or kinetic energy. Then, the differenttime-evolutions of old and new states involved in a collision process canoriginate negative values of the charge density, even longer after thecollision. This unphysical feature disappears when the Boltzmann collisionoperator generates states that were already present in the density matrix ofthe quantum system before the collision. Following these ideas, in this paper,we introduce an algorithm that models phonon-electron interactions through theBoltzmann collision operator without negative values of the charge density. Themodel only requires the exact knowledge, at all times, of the states that buildthe density matrix of the open system.