  Analog condensed matter systems present an exciting opportunity to simulate
early Universe models in table-top experiments. We consider a recent proposal
for an analog condensed matter experiment to simulate the relativistic quantum
decay of the false vacuum. In the proposed experiment, two ultra-cold
condensates are coupled via a time-varying radio-frequency field. The relative
phase of the two condensates in this system is approximately described by a
relativistic scalar field with a potential possessing a series of false and
true vacuum local minima. If the system is set up in a false vacuum, it would
then decay to a true vacuum via quantum mechanical tunnelling. Should such an
experiment be realized, it would be possible to answer a number of open
questions regarding non-perturbative phenomena in quantum field theory and
early Universe cosmology. In this paper, we illustrate a possible obstruction:
the time-varying coupling that is invoked to create a false vacuum for the
long-wavelength modes of the condensate leads to a destabilization of shorter
wavelength modes within the system via parametric resonance. We focus on an
idealized setup in which the two condensates have identical properties and
identical background densities. Describing the system by the coupled
Gross-Pitaevskii equations (GPE), we use the machinery of Floquet theory to
perform a linear stability analysis, calculating the wavenumber associated with
the first instability band for a variety of experimental parameters. However,
we demonstrate that, by tuning the frequency of the time-varying coupling, it
may be possible to push the first instability band outside the validity of the
GPE, where dissipative effects are expected to damp any instabilities. This
provides a viable range of experimental parameters to perform analog
experiments of false vacuum decay.
