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arXiv:1002.4512 · Journal of Contemporary Physics (Armenian Academy of Sciences)

An ansatz for the nonlinear Demkov-Kunike problem for cold molecule formation

H. H. Azizbekyan

Published Apr 29, 2010• 0 citations
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Abstract

We study nonlinear mean-field dynamics of ultracold molecule formation in the case when the external field configuration is defined by the level-crossing Demkov-Kunike model, characterized by a bell-shaped coupling and finite variation of the detuning. Analyzing the fast sweep rate regime of the strong interaction limit, which models a situation when the peak value of the coupling is large enough and the resonance crossing is sufficiently fast, we construct a highly accurate ansatz to describe the temporal dynamics of the molecule formation in the mentioned interaction regime. The absolute error of the constructed approximation is less than 3 × 10 −6 for the final transition probability while at certain time points it might increase up to 10 −3 . Examining the role of the different terms in the constructed approximation, we prove that in the fast sweep rate regime of the strong interaction limit the temporal dynamics of the atom-molecule conversion effectively consists of the process of resonance crossing, which is governed by a nonlinear equation, followed by atom-molecular coherent oscillations which are basically described by a solution of the linear problem, associated with the considered nonlinear one.

DOI
10.3103/s1068337210030023

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We study nonlinear mean-field dynamics of ultracold molecule formation in the case when the external field configuration is defined by the level-crossing Demkov-Kunike model, characterized by a bell-shaped coupling and finite variation of the detuning. Analyzing the fast sweep rate regime of the strong interaction limit, which models a situation when the peak value of the coupling is large enough and the resonance crossing is sufficiently fast, we construct a highly accurate ansatz to describe the temporal dynamics of the molecule formation in the mentioned interaction regime. The absolute error of the constructed approximation is less than 3 × 10 −6 for the final transition probability while at certain time points it might increase up to 10 −3 . Examining the role of the different terms in the constructed approximation, we prove that in the fast sweep rate regime of the strong interaction limit the temporal dynamics of the atom-molecule conversion effectively consists of the process of resonance crossing, which is governed by a nonlinear equation, followed by atom-molecular coherent oscillations which are basically described by a solution of the linear problem, associated with the considered nonlinear one.
We study the nonlinear mean-field dynamics of molecule formation at coherent photo- and magneto-association of an atomic Bose-Einstein condensate for the case when the external field configuration is defined by the quasi-linear level crossing Demkov-Kunike model, characterized by a bell-shaped pulse and finite variation of the detuning. We present a general approach to construct an approximation describing the temporal dynamics of the molecule formation in the weak interaction regime and apply the developed method to the nonlinear Demkov-Kunike problem. The presented approximation, written as a scaled solution to the linear problem associated to the nonlinear one we treat, contains fitting parameters which are determined through a variational procedure. Assuming that the parameters involved in the solution of the linear problem are not modified, we suggest an analytical expression for the scaling parameter.
We study a quadratic-nonlinear version of the level-crossing Demkov-Kunike problem relevant to coherent molecule formation via photo- or Feshbach-association of ultracold atoms. Using an exact third-order nonlinear differential equation for the molecular state probability, we construct an approximate solution to the problem in the fast sweep regime of the strong interaction limit. The constructed approximation, defined as a solution of a first-order nonlinear equation, contains a fitting parameter which we determine through a variational procedure. The suggested approach can be interpreted as a modification of the adiabatic approximation and can further be used for the treatment of analogous nonlinear two-state problems.
We study the dynamics of cold molecule formation via photo- or magneto-association of quantum degenerate atomic gases for the case when the field configuration is defined by the quasi-linear level crossing Demkov-Kunike model, which is characterized by a bell-shaped pulse and finite variation of the frequency detuning. We generalize the approach developed for the Landau-Zener model and propose a cubic polynomial equation for the asymptotic transition probability at t → + ∞ applicable in the large detuning regime of the interaction process. The proposed approximation applies to all values of the Rabi frequency; hence, it presents a unified description of previously noticed weak, moderate, and strong interaction limits of the fast sweep regime. This cubic equation for the transition probability provides improvement of previous results by an order of the magnitude.
We examine the nonlinear dynamics of molecule formation at coherent photo- and magneto-association of an atomic Bose–Einstein condensate when the external field configuration is defined by the quasi-linear level crossing Demkov–Kunike model, characterized by a bell-shaped pulse and finite variation of the detuning. We present an approach to construct an approximation describing the temporal dynamics of the molecule formation in the weak interaction regime and apply the developed method to the nonlinear Demkov–Kunike problem. The presented approximation, written as a scaled solution to the linear problem associated to the nonlinear one we treat, contains adjustable parameters which are determined through a variational procedure. Assuming that the parameters include in the solution of the linear problem are not modified, we suggest an analytical relation for the scaling parameter.
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.
The strong-coupling limit of molecule formation in an atomic Bose-Einstein condensate via two-mode one-color photoassociation or sweep across a Feshbach resonance is examined using a basic nonlinear time-dependent two-state model. For the general class of term-crossing models with constant coupling, a common strategy for attacking the problem is developed based on the reduction of the initial system of semiclassical equations for atom-molecule amplitudes to a third-order nonlinear differential equation for the molecular state probability. This equation provides deriving exact solution for a class of periodic level-crossing models. These models reveal much in common with the Rabi problem. Discussing the strong-coupling limit for the general case of variable detuning, the equation is further truncated to a limit first-order nonlinear equation. Using this equation, the strong nonlinearity regime for the first Nikitin exponential-crossing model is analyzed and accurate asymptotic expressions for the nonlinear transition probability to the molecular state are derived. It is shown that, because of a finite final detuning involved, this model displays essential deviations from the Landau-Zener behavior. In particular, it is shown that in the limit of strong coupling the final conversion probability tends to 1∕61∕61∕6. Thus, in this case the strong interaction limit is not optimal for molecule formation. We have found that if optimal field intensity is applied the molecular probability is increased up to 1∕41∕41∕4 (i.e., the half of the initial atomic population).