  We investigate the nonlinear dynamics of cold atom systems that can in
principle serve as quantum simulators of false vacuum decay. The analog false
vacuum manifests as a metastable vacuum state for the relative phase in a
two-species Bose-Einstein condensate (BEC), induced by a driven periodic
coupling between the two species. In the appropriate low energy limit, the
evolution of the relative phase is approximately governed by a relativistic
wave equation exhibiting true and false vacuum configurations. In previous
work, a linear stability analysis identified exponentially growing
short-wavelength modes driven by the time-dependent coupling. These modes
threaten to destabilize the analog false vacuum. Here, we employ numerical
simulations of the coupled Gross-Pitaevski equations (GPEs) to determine the
non-linear evolution of these linearly unstable modes. We find that unless a
physical mechanism modifies the GPE on short length scales, the analog false
vacuum is indeed destabilized. We briefly discuss various physically expected
corrections to the GPEs that may act to remove the exponentially unstable
modes. To investigate the resulting dynamics in cases where such a removal
mechanism exists, we implement a hard UV cutoff that excludes the unstable
modes as a simple model for these corrections. We use this to study the range
of phenomena arising from such a system. In particular, we show that by
modulating the strength of the time-dependent coupling, it is possible to
observe the crossover between a second and first order phase transition out of
the false vacuum.
