Numerical Analysis of the Transient Behavior of the Non-Equilibrium Quantum Liouville Equation

The numerical analysis of transient quantum effects in heterostructure devices with conventional numerical methods tends to pose problems. To overcome these limitations, a novel numerical scheme for the transient non-equilibrium solution of the quantum Liouville equation utilizing a finite volume discretization technique is proposed. Additionally, the solution with regard to the stationary regime, which can serve as a reference solution, is inherently included within the discretization scheme for the transient regime. Resulting in a highly oscillating interference pattern of the statistical density matrix as well in the stationary as in the transient regime, the reflecting nature of the conventional boundary conditions can be an additional source of error. Avoiding these non-physical reflections, the concept of a complex absorbing potential used for the Schrödinger equation is utilized to redefine the drift operator in order to render open boundary conditions for quantum transport equations. Furthermore, the method allows the application of the commonly used concept of inflow boundary conditions.