Analyzing the Positivity Preservation of Numerical Methods for the Liouville-von Neumann Equation

The density matrix is a widely used tool in quantum mechanics. In order to determine its evolution with respect to time, the Liouville-von Neumann equation must be solved. However, analytic solutions of this differential equation exist only for simple cases. Additionally, if the equation is coupled to Maxwell's equations to model light-matter interaction, the resulting equation set -- the Maxwell-Bloch or Maxwell-Liouville-von Neumann (MLN) equations -- becomes nonlinear. In these advanced cases, numerical methods are required. Since the density matrix has certain mathematical properties, the numerical methods applied should be designed to preserve those properties. We establish the criterion that only methods that have a completely positive trace preserving (CPTP) update map can be used in long-term simulations. Subsequently, we assess the three most widely used methods -- the matrix exponential (ME) method, the Runge-Kutta (RK) method, and the predictor-corrector (PC) approach -- whether they provide this feature, and demonstrate that only the update step of the matrix exponential method is a CPTP map.