Nonadiabatic Transitions for a Decaying Two-Level-System: Geometrical and Dynamical Contributions

We study the Landau-Zener Problem for a decaying two-level-system described by a non-hermitean Hamiltonian, depending analytically on time. Use of a super-adiabatic basis allows to calculate the non-adiabatic transition probability P in the slow-sweep limit, without specifying the Hamiltonian explicitly. It is found that P consists of a ``dynamical'' and a ``geometrical'' factors. The former is determined by the complex adiabatic eigenvalues E_(t), only, whereas the latter solely requires the knowledge of \alpha_(+-)(t), the ratio of the components of each of the adiabatic eigenstates. Both factors can be split into a universal one, depending only on the complex level crossing points, and a nonuniversal one, involving the full time dependence of E_(+-)(t). This general result is applied to the Akulin-Schleich model where the initial upper level is damped with damping constant $\gamma$. For analytic power-law sweeps we find that Stueckelberg oscillations of P exist for gamma smaller than a critical value gamma_c and disappear for gamma > gamma_c. A physical interpretation of this behavior will be presented by use of a damped harmonic oscillator.