Abstract. We present a rigorous analysis of the Landau–Zener linear-in-time term crossing problem for quadratic-nonlinear systems relevant to the coherent association of ultracold atoms in degenerate quantum gases. Our treatment is based on an exact third-order nonlinear differential equation for the molecular state probability. Applying a variational two-term ansatz, we construct a simple approximation that accurately describes the whole-time dynamics of the coupled atom–molecular system for any set of involved parameters. Ensuring an absolute error of less than 10⁻⁵ for the final transition probability, the resultant solution improves by several orders of magnitude the accuracy of the previous approximations by A Ishkhanyan et al developed separately for the weak coupling (2005 J. Phys. A: Math. Gen. 38 3505) and strong interaction (2006 J. Phys. A: Math. Gen. 39 14887) limits. In addition, the constructed approximation covers the whole moderate-coupling regime, providing this intermediate regime with the same accuracy as the two mentioned limits. The obtained results reveal the remarkable observation, that for the strong-coupling limit the resonance crossing is mostly governed by the nonlinearity, while the coherent atom–molecular oscillations arising soon after the resonance has been crossed are basically of a linear nature. This observation is supposed to be of a general character, due to the basic attributes of the resonance-crossing processes in the nonlinear quantum systems of the discussed type of involved quadratic nonlinearity.

Print publication: Issue 22 (28 November 2009)
Received 30 August 2009
Published 30 October 2009