In the present contribution, we present a general analysis of the strong coupling limit for a model nonlinear two state problem describing the photoassociation of an atomic Bose-Einstein condensate [1]. For the class of curve-crossing models, with constant external field amplitude, a general strategy for attacking the problem is developed, based on the reduction of the initial system of equations for the semi-classical atom-molecule amplitudes to a third order non-linear differential equation and further reduction of the latter equation to a limit first order nonlinear equation.
We focus our consideration on the non-linear Nikitin constant-amplitude exponential curve crossing model defined by the amplitude- and detuning-modulation functions U=const, δ=Δ[1-exp(-at)] [2]. First, we construct a uniformly convergent series solution for the case of weak coupling using a nonlinear Volterra integral equation equivalent to the basic governing equations. Further, we present the analysis of the strong nonlinearity regime for different curve crossing models. We show that in this limit, when the nonlinearity is most expressed, the governing equations are effectively replaced by a first order non-linear ordinary differential equation. This equation has a rich structure and possesses several solutions. We show that in the zeroth order approximation the time evolution of the transition probability in the strong interaction limit presents, in general, a function consisting of different pieces of the solutions of mentioned first order limit equation. We construct the limit solution for the first Nikitin exponential curve crossing model and compare this with the solution to the Landau-Zener problem [3]. We show that the limit solution allows one to linearize the problem under consideration getting a linear third order differential equation that describes well, in the first approximation, the behavior of the system everywhere. We present the approximate solution to this equation, the first correction term to the limit solution, and show that, because of a finite final detuning involved, in the limit of large field intensities applied the final transition probability tends to 1/6. Thus, the general conclusion is that the strong interaction limit, perhaps surprisingly, is not an optimal for molecule formation. We show that when the optimal intermediate regime of moderate field amplitudes is chosen, the transition probability is about 1/4, i.e., almost the half of the population of the initial atomic BEC is converted into molecular BEC. This is the main physical result of the present contribution.

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