| A semi-classical treatment of the two-state atom subjected to a time-dependent applied force leads to a set of two coupled, complex, first-order ordinary differential equations governing the time evolution of the system's state vector that are to date, not solvable in closed form. Contained in this paper is a demonstration of how the system is parameterized by a single variable theta, which in turn reduces the problem to one real, nonlinear, second-order ordinary differential equation. Utilizing a non-standard perturbation expansion in the variable 'A' (the Field Strength Parameter) on this reduction subsequently allows for both a first-order Weak Field Approximation and a first-order Strong Field Approximation. In addition, a technique is outlined for obtaining the full power series solution in the Weak Field Limit (|A|<< l). However, a detailed discussion of the power series solution as well as its consequences is deferred due to the fact that it is presently a collaborative work in progress between Dr. Robert L. Anderson and myself. When applied to the specific case of both resonant and near-resonant linearly polarized light incident on an atom, both the Weak Field Approximation and the Strong Field Approximation are shown to be in good agreement with numerically generated solutions for the probability amplitudes of the state vector. Furthermore, this new Weak Field Approximation reveals the defect in the ansatz of discarding the 'rapidly oscillating' term in the traditional Rotating Wave Approximation. Finally, the resonance case of the first-order Weak Field Approximation is found to contain large-time behavior. This large-time behavior is extracted and the new approximation is referred to as the Long-Time Weak Field Approximation. The resonance power series solution is demonstrated to contain large-time behavior, which is found to reduce to the first-order Long-Time Weak Field Approximation, but again a detailed analysis of the power series is deferred. |
