Dynamics and Relative Phase of Fragmenting BECs*

Recent BEC experiments\footnote{Y. Shin et al. PRL 93, 050405 (2004)}$^,$\footnote{M. Albeiz et al. PRL 95, 010402 (2005)} have shown that an initially coherent condensate, when spatially fragmented, can exhibit phase-driven phenomena such as the Josephson effect and controllable interference patterns analogous to those of a Mach-Zehnder interferometer. A theoretical treatment of the dynamics of such systems is difficult as two models are appropriate in different regimes: Mean-field theory for the initial unfragmented state and a Hubbard model for the particle number dynamics as localized lobes of the cloud form. We present a method which correctly describes dynamics of both spatial and Fock-space variables for a fragmenting condensate. This method shows the effect of non-adiabaticity on interference patterns, stability of Josephson oscillations and the degree of number squeezing in specific trap geometries. Further, this technique yields the order parameter (nor rigorously defined in mean-field theory) and gives a context in which to discuss phase uncertainty - the ultimate limit on the quality of interferometry schemes.