  We study the dynamics of a two-mode Bose-Einstein condensate in the vicinity
of a mean-field dynamical instability. Convergence to mean-field theory (MFT),
with increasing total number of particles $N$, is shown to be logarithmically
slow. Using a density matrix formalism rather than the conventional
wavefunction methods, we derive an improved set of equations of motion for the
mean-field plus the fluctuations, which goes beyond MFT and provides accurate
predictions for the leading quantum corrections and the quantum break time. We
show that the leading quantum corrections appear as decoherence of the reduced
single-particle quantum state; we also compare this phenomenon to the effects
of thermal noise. Using the rapid dephasing near an instability, we propose a
method for the direct measurement of scattering lengths.
