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The evolution of additional (hidden) quantum variables in the interference of Bose-Einstein condensates

Additional variables (also often called ``hidden variables'') are sometimes added to standard quantum mechanics in order to remove its indeterminism or ``incompletness,'' and to make the measurement process look more classical. Here we discuss a case in which an additional variable arises almost spontaneously from the quantum formalism: the emergence of relative phase between two highly populated Fock state Bose-Einstein condensates. The model simulated here involves the interference of two Bose condensates, one with all up spins, and the other with down spins, along a z-axis. With the clouds overlapping, we consider the results of measuring spins in a transverse plane (the general direction is studied in an appendix). The determination of the previously ``hidden'' phase becomes progressively more definite as additional measurements are made. We also provide an analysis of a recent and closely related experiment.