No abstract

The generalized bow-tie model (Yu N Demkov and V N Ostrovsky 2000 Phys. Rev. A 61 32705) is a particular generalization of the famous two-state Landau-Zener model widely used in atomic physics and beyond. It comprises an arbitrary number of states; the diabatic-potential curves are linear functions of time whereas the coupling matrix elements are constant. We derive a rigorous solution of the model by the contour integral method. The complete set of transition amplitudes is obtained by considering solution asymptotes for t → ±∞. It agrees with the transition probabilities evaluated earlier by heuristically appealing but non-rigorous reduction of the model to the sequence of two-state transitions. An unusual quasi-factorization property of the transition amplitude matrix is established. The entire matrix is expressed via a single complexvalued vector.

Various problems in atomic physics can be formulated in terms of two bands of potential curves which cross each other. Each band consists of parallel (non-interacting) diabatic potential curves equally spaced on the energy axis. The bands with infinite number of states are considered under the assumption of 'translational' symmetry along the energy axis. The adiabatic potential curves are constructed explicitly. The system of avoided crossings appears not only in the weak coupling case (which is obvious), but also in the strong coupling limit. This feature also holds in the generalized model where the bands are finite and non-equidistant. In the case of an infinite number of states in each band the dynamic description (i.e. evolution in time) is reduced to the two-state problem which contains an additional continuous parameter (analogue of the quasimomentum). Some peculiarities of the time propagation are discussed. The present model generalizes the famous Landau-Zener two-state case and Demkov-Osherov model (one level interacting with a band of levels).