  Ultracold atomic gases can undergo phase transitions that mimic relativistic
vacuum decay, allowing us to empirically test early-Universe physics in
tabletop experiments. We investigate the physics of these analog systems, going
beyond previous analyses of the classical equations of motion to study quantum
fluctuations in the cold-atom false vacuum. We show that the fluctuation
spectrum of this vacuum state agrees with the usual relativistic result in the
regime where the classical analogy holds, providing further evidence for the
suitability of these systems for studying vacuum decay. Using a suite of
semiclassical lattice simulations, we simulate bubble nucleation from this
analog vacuum state in a 1D homonuclear potassium-41 mixture, finding
qualitative agreement with instanton predictions. We identify realistic
parameters for this system that will allow us to study vacuum decay with
current experimental capabilities, including a prescription for efficiently
scanning over decay rates, and show that this setup will probe the quantum
(rather than thermal) decay regime at temperatures $T\lesssim10\,\mathrm{nK}$.
Our results help lay the groundwork for using upcoming cold-atom experiments as
a new probe of nonperturbative early-Universe physics.
