Time development of a driven three-level lambda system: A case study

How does a driven system with many energy levels approach its steady state? Insights are gained by studying a system with three energy levels when the ground state is excited by a laser. The time-dependent occupation probabilities of the three energy levels show students how the system develops in time. The occupation probabilities come from the numerical solution of the Liouville-von Neumann equations for the density operator matrix elements when relaxation is included. A combination of the Interaction Picture, the Rotating Wave Approximation, and the assumption of resonance permit the eigenvalues of the Liouville-von Neumann equations to be found numerically and in closed-form in certain limits. The two methods are complementary and help students understand time-dependent systems. In addition, the eigenvalues allow the short-time and the long-time occupation probabilities to be connected to the relaxation parameters and the magnitude of the laser's electric field. Thus, this model three-level system illuminates how a driven system behaves over time and provides guidance for students studying time-dependent systems.