  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.
