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Landau-Zener Transition for Association of an Atomic Bose-Einstein Condensate With Inter-Particle Elastic Interactions Included

N. Sahakyan, H. H. Azizbekyan, Ruzan Sokhoyan, Claude Leroy, Y. Pashayan-Leroy, Artur Mikhailovich Ishkhanyan

Published Oct 14, 2008• 0 citations
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Abstract

We study strong coupling limit of the non-linear Landau-Zener transition at coherent photo- and magneto-association of cold atoms, focusing on the role of the elastic interspecies scatterings. Using an exact third-order nonlinear differential equation for the molecular state probability, we develop a nontrivial version of the strained parameter method which enables us to construct a highly accurate and simple analytic approximation describing time dynamics of the coupled atom-molecular system.

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From atomic to molecular Bose-Einstein condensates: a physically realizable term-crossing model for cold atom association

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Using an exact third-order NL di®erential equation for the molecular state probability, we develop a variational approach which enables us to construct highly accurate analytic approximations describing time dynamics of the coupled atom-molecular system in each of the interaction regimes. We show that the approximation describing time evolution of the molecular state probability both in the weak interaction limit and in the large detuning regime of the strong interaction limit can be written as a sum of two distinct terms; the ¯rst one, being a solution to a limit ¯rst-order NL equation, e®ectively describes the process of the molecule formation while the second one, being a scaled solution to the linear DK problemb (but now with modi¯ed parameters), models the remaining oscillations which come up when the process of molecule formation is over. The graphs of the approximate and numerical solutions are practically indistinguishable for almost all the time range.

•Sep 7, 2010•0 citations
We study strong coupling limit of the non-linear Landau-Zener transition at coherent photo- and magneto-association of cold atoms, focusing on the role of the elastic interspecies scatterings. Using an exact third-order nonlinear differential equation for the molecular state probability, we develop a nontrivial version of the strained parameter method which enables us to construct a highly accurate and simple analytic approximation describing time dynamics of the coupled atom-molecular system.
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.
We discuss the strong interaction regime of the nonlinear Landau-Zener problem coming up at coherent photo- and magneto-association of ultracold atoms. We apply a variational approach to an exact third-order nonlinear differential equation for the molecular state probability and construct an accurate approximation describing the whole time dynamics of the coupled atom-molecular system. The resultant solution improves the accuracy of the previous approximation by A. Ishkhanyan et al. [J. Phys. A 39, 14887 (2006)]. The obtained results reveal a remarkable observation that in the strong coupling limit the resonance crossing is mostly governed by the nonlinearity while the coherent atom-molecular oscillations coming up soon after the resonance has been crossed are principally of linear nature. This observation is supposed to be general for all the nonlinear quantum systems having the same generic quadratic nonlinearity, due to the basic attributes of the resonance crossing processes in such systems. The constructed approximation turns out to have a larger applicability range (than it was initially expected) covering the whole moderate coupling regime for which the proposed solution accurately describes all the main characteristics of the system's evolution except the amplitude of the coherent atom-molecule oscillation, which is rather overestimated.
We discuss the strong-coupling regime of the nonlinear Landau-Zener problem occurring at coherent photo- and magneto-association of ultracold atoms. We apply a variational approach to an exact third-order nonlinear differential equation for the molecular state probability and construct an accurate approximation describing the time dynamics of the coupled atom-molecule system. The resultant solution improves the accuracy of the previous approximation [22]. The obtained results reveal a remarkable observation that in the strong-coupling limit, the resonance crossing is mostly governed by the nonlinearity, while the coherent atom-molecule oscillations occurring soon after crossing the resonance are principally of a linear nature. This observation is supposedly general for all nonlinear quantum systems having the same generic quadratic nonlinearity, due to the basic attributes of the resonance crossing processes in such systems. The constructed approximation turns out to have a larger applicability range than it was initially expected, covering the whole moderate-coupling regime for which the proposed solution accurately describes ail the main characteristics of the system evolution except the amplitude of the coherent atom-molecule oscillation, which is rather overestimated.
The nonlinear Landau–Zener problem for two-mode photoassociation of cold atoms is studied. Based on an exact nonlinear Volterra integral equation for molecular state probability, a limit nonlinear first-order differential equation is applied to construct the first-order approximation to the solution of the problem at a strong coupling limit. An accurate approximate expression for the final transition probability to the molecular state is derived. The non-transition probability turns out to be inversely proportional to the Landau–Zener parameter in contrast to the linear two-state case when the dependence is exponential.
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.
Using an exact third-order NL di®erential equation for the molecular state probability, we develop a variational approach which enables us to construct highly accurate analytic approximations describing time dynamics of the coupled atom-molecular system in each of the interaction regimes. We show that the approximation describing time evolution of the molecular state probability both in the weak interaction limit and in the large detuning regime of the strong interaction limit can be written as a sum of two distinct terms; the ¯rst one, being a solution to a limit ¯rst-order NL equation, e®ectively describes the process of the molecule formation while the second one, being a scaled solution to the linear DK problemb (but now with modi¯ed parameters), models the remaining oscillations which come up when the process of molecule formation is over. The graphs of the approximate and numerical solutions are practically indistinguishable for almost all the time range.