Variational ansatz for the nonlinear Landau-Zener problem for cold atom association

TL;DR

This paper introduces a variational approach to accurately model the nonlinear Landau-Zener problem in ultracold atom association, significantly improving approximation accuracy across all coupling regimes.

Contribution

A new variational two-term ansatz provides a highly accurate, simple approximation for the nonlinear Landau-Zener problem in ultracold gases, covering weak, moderate, and strong coupling regimes.

Findings

The approximation achieves less than 1% error in transition probability.

Nonlinearity dominates resonance crossing in strong coupling.

Post-crossing oscillations are primarily linear in nature.

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 coupled atom-molecular system for any set of involved parameters. Ensuring an absolute error less than 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 [J. Phys. A 38, 3505 (2005)] and strong interaction [J. Phys. A 39, 14887 (2006)] limits. In addition, the constructed approximation covers the whole moderate-coupling regime, providing for this intermediate regime the same accuracy as for 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 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.