  There have been many discussions of two-mode models for Bose condensates in a
double well potential, but few cases in which parameters for these models have
been calculated for realistic situations. Recent experiments lead us to use the
Gross-Pitaevskii equation to obtain optimum two-mode parameters. We find that
by using the lowest symmetric and antisymmetric wavefunctions, it is possible
to derive equations for a more exact two-mode model that provides for a
variable tunneling rate depending on the instantaneous values of the number of
atoms and phase differences. Especially for larger values of the nonlinear
interaction term and larger barrier heights, results from this model produce
better agreement with numerical solutions of the time-dependent
Gross-Pitaevskii equation in 1D and 3D, as compared with previous models with
constant tunneling, and better agreement with experimental results for the
tunneling oscillation frequency [Albiez et al., cond-mat/0411757]. We also show
how this approach can be used to obtain modified equations for a second
quantized version of the Bose double well problem.
