  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 1/6. 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 1/4
(i.e., the half of the initial atomic population).
