  We study the kinetics of the first order phase separation transition in
boson-fermion cold-atom mixtures. At sufficiently low temperatures such a
transition is driven by quantum fluctuations responsible for the formation of
critical nuclei of a stable phase. Based on a microscopic description of
interacting boson-fermion mixtures we derive an effective action for the
critical droplet and obtain an asymptotic expression for the nucleation rate in
the vicinity of the phase transition and near the spinodal instability of the
mixed phase. We also discuss effects of dissipation which play a dominant role
close to the transition point, and identify the regimes where quantum
nucleation can be experimentally observed in cold-atom systems.
