Extension of the Rosen-Zener solution to the two-level problem

We solve the problem of two discrete quantum levels which are coupled by a time-dependent radio-frequency pulse W(t)=V(t) e^iνt, where the envelope function is of a form suggested by Rosen and Zener: V(t)=V₀sech(πtT). When a level damping constant γ is included, in the manner of Bethe-Lamb theory, the solutions show new features which are not expected on the basis of a sudden-approximation theory, where V(t)=const over the pulse duration T. Various transient effects such as "ringing" are not present in the extended Rosen-Zener solution; these effects are related to the large impulsive forces at the step discontinuities in the sudden approximation. The final-state level amplitudes can be quite different depending on the size of the pulse rise time T as compared with the system Bohr period 1ω. Our results allow a continuous and quantitatively exact comparison between the extremes of the sudden (ωT1) and adiabatic (ωT1) approximations. A model of a "quasisudden" step function is also constructed, and remarks are made on the validity of a certain conjecture by Rosen and Zener.