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

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.

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• A singular Lambert-W Schrödinger potential exactly solvable in terms of the confluent hypergeometric functions
• Generalized formula for the Landau-Zener transition in interacting Bose-Einstein condensates
• Adiabatic tracking for photo- and magneto-association of Bose-Einstein condensates with Kerr nonlinearities
• Quadratic-nonlinear Landau-Zener transition for association of an atomic Bose-Einstein condensate with inter-particle elastic interactions included
• Expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta and Gamma functions
• Landau-Zener transition in quadratic-nonlinear two-state systems
• Higher order transmission resonances in above-barrier reflection of ultra-cold atoms
• Change in the adiabatic invariant in a nonlinear Landau-Zener problem
• A Schrödinger potential involving $x^{\frac{2}{3}}$ and centrifugal-barrier terms conditionally integrable in terms of the confluent hypergeometric functions
• An ansatz for the nonlinear Demkov-Kunike problem for cold molecule formation
• Adiabatic asymmetric scattering of atoms in the field of a standing wave