  In the framework of a basic semiclassical time-dependent nonlinear two-state
problem, we study the weak coupling limit of the nonlinear Landau-Zener
transition at coherent photo- and magneto-association of an atomic
Bose-Einstein condensate. Using an exact third-order nonlinear differential
equation for the molecular state probability, we develop a variational approach
which enables us to construct an accurate analytic approximation describing
time dynamics of the coupled atom-molecular system for the case of weak
coupling. The approximation is written in terms of the solution to an auxiliary
linear Landau-Zener problem with some effective Landau-Zener parameter. The
dependence of this effective parameter on the input Landau-Zener parameter is
found to be unexpected: as the generic Landau-Zener parameter increases, the
effective Landau-Zener parameter first monotonically increases (starting from
zero), reaches its maximal value and then monotonically decreases again
reaching zero at some point. The constructed approximation quantitatively well
describes many characteristics of the time dynamics of the system, in
particular, it provides a highly accurate formula for the final transition
probability to the molecular state. The present result for the final transition
probability improves the accuracy of the previous approximation by Ishkhanyan
et al. [Phys. Rev. A 69, 043612 (2004); J. Phys. A 38, 3505 (2005)] by order of
magnitude.
