LANDAU-ZENER TRANSITION IN NONLINEAR QUANTUM SYSTEMS

摘要

A comprehensive theory of the generalized Landau-Zener problem for quadratic-and cubic-nonlinear quantum two-state models is developed. Combining analytical and numerical methods, a simple analytic formula involving elementary functions only is constructed for the final transition probability for the quadratic-nonlinear Landau-Zener problem. The formula provides a highly accurate approximation for the whole rage of the variation of the Landau-Zener parameter. Further, a rigorous analysis of the cubic-nonlinear Landau-Zener problem applicable, e.g., to the interacting Bose-Einstein condensates is presented. It is shown that for any set of involved parameters the time evolution of the system is accurately described by a two-term variational ansatz. Applying an exact third order nonlinear differential equation a fifth order polynomial equation is constructed for the final transition probability. A root of this equation can be viewed as a generalized Landau-Zener formula for cubic-nonlinear quantum systems.