Extension of the Rosen-Zener solution to the two-level problem
R. T. Robiscoe · Physical Review A · 1978
We solve the problem of two discrete quantum levels which are coupled by a time-dependent radio-frequency pulse $W(t)=V(t) {e}^{i\ensuremath{ u}t}$, where the envelope function is of a form suggested by Rosen and Zener: $V(t)={V}_{0}sech(\frac{\ensuremath{\pi}t}{T})$. When a level damping constant $\ensuremath{\gamma}$ 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 $V(t)=\mathrm{const}$ 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 $\frac{1}{\ensuremath{\omega}}$. Our results allow a continuous and quantitatively exact comparison between the extremes of the sudden ($\ensuremath{\omega}T\ensuremath{\ll}1$) and adiabatic ($\ensuremath{\omega}T\ensuremath{\gg}1$) 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.