  We study the quantum dynamics of a two-level system driven by a pulse that
starts near-resonant for small amplitudes, yielding nonadiabatic evolution, and
induces an adiabatic evolution for larger amplitudes. This problem is analyzed
in terms of lifting of degeneracy for rising amplitudes. It is solved exactly
for the case of linear and exponential rising. Approximate solutions are given
in the case of power law rising. This allows us to determine approximative
formulas for the lineshape of resonant excitation by various forms of pulses
such as truncated trig-pulses. We also analyze and explain the various
superpositions of states that can be obtained by the Half Stark Chirped Rapid
Adiabatic Passage (Half-SCRAP) process.
