MATHEMATICAL ANALYSIS AND NUMERICAL SIMULATION OF A NONLINEAR BLOCH-TYPE MODEL

Abstract

In this work, we study the mathematical properties of a Bloch-type model, such as energy dissipation and exponential decay, incorporating nonlinear effects including Coulomb interaction. We show that the model is globally well-posed. Furthermore, the existence of an equilibrium state is established in the case of a reduced threelevel energy system. The equilibrium populations were computed numerically and were found to satisfy the governing system of \footnotetext{ } equations with a residual of order \(10^{-12}\). The impact of the Coulomb interaction terms was quantified by measuring the differences between the computed equilibrium populations and the corresponding Gibbs states. These differences were observed to be of the order of \(10^{-3}\). All equilibrium populations belong to the interval [0, 1]. The reduced three-level model is then discretized using an exponential Euler scheme. Finally, numerical simulations are carried out to evaluate the effects of the Coulomb terms. Numerical simulations reveal that the contributions of the Coulomb terms can exceed \(10^{-1}\) for the populations, highlighting their significant influence on the system dynamics.