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 $W\left(t\right)=V\left(t\right)\mathrm{} e^{i\nu t}$, where the envelope function is of a form suggested by Rosen and Zener: $V\left(t\right)=V_{0}sech\left(\frac{\pi t}{T}\right)$. When a level damping constant $\gamma$ 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 $V\left(t\right)=\mathrm{const}$ 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 $\frac{1}{\omega }$. Our results allow a continuous and quantitatively exact comparison between the extremes of the sudden ($\omega T\ll 1$) and adiabatic ($\omega T\gg 1$) 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.