The exact solution of the multistate Landau-Zener type model: the generalized bow-tie model

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.

Multiparticle Landau-Zener problem: Application to quantum dots

“…In its general form this problem is still unsolved, but a number of exact results for special choices of the matrices B and A were found [9][10][11][12][13][14][15][16][17][18][19]. In almost all available exact solutions the transition probabilities are expressed in terms of the genuine two-level LZ formula successively applied at each diabatic level intersection.…”

Multistate Landau-Zener models with all levels crossing at one point

“…We found that, indeed, scattering matrix elements in all previously solved bipartite models satisfy (25). There is difference from Eq.…”

“…However, in models with higher dimensions, scanning of a single parameter is usually insufficient to observe such points. We also found that some symmetric choices of parameters, for example LZ-chains with equal couplings and equidistant level slopes, do not generally satisfy (25) for N ≥ 4. So, our numerical investigation suggests that there are numerous solvable but still unknown bipartite models with asymmetric choices of parameters.…”

Multiple-time-scale Landau-Zener transitions in many-body systems

“…for adiabatic evolution) is non-perturbative. Extensions to multi-level problems [27][28][29], many-body situations [22,[30][31][32][33], and non-linear LZ transitions [23,34,35] have been considered. When the LZ model becomes non-linear both the exponential dependence and the smoothness of P may be lost [23].…”