Induced transition probabilities and energies for the strongly coupled two-level system

An exact method of solution to the time-dependent wave equation for a system interacting with a sinusoidal field, which formally treats time and the phase of the field on an equal footing, is used to discuss (1) the importance of averaging the properties of a system over the phase of the applied field and (2) the nature and behavior of the characteristic exponents derived from the Floquet solution (with particular emphasis on the frequency-sweep experiment). The results for the Zeeman tuning experiment are used to resolve recent discrepancies in the literature which relate to the validity of Shirley's important result for the two-level average induced transition probability involving the derivative of the characteristic exponents with respect to the Zeeman splitting parameter ${\omega }_0$. The explicit calculations included in this work are for the single-photon two-level problem. Important implications can be inferred from them with respect to both the importance of phase averaging and the usefulness of the characteristic exponents as a quantitative means of obtaining resonance frequency shifts and half-widths for multiphoton and multilevel problems.