Propagation of coherent states through avoided energy level crossing

We study the propagation of a coherent state for a one-dimensional system of two coupled Schrödinger equations in the semi-classical limit.Couplings are induced by a cubic nonlinearity and a matrix-valued potential, whose eigenvalues present an ”avoided crossing” : at one given point, the gap between them reduces as the semi-classical parameter becomes smaller.We show that when an initial coherent state polarized along an eigenvector of the potential propagates through the avoided crossing point, there are transitions between the modes at leading order.In the regime we consider, we observe a nonlinear propagation far from the crossing region while the transition probability can be computed with the linear Landau-Zener formula.