  Understanding the non-equilibrium behavior of quantum systems is a major goal
of contemporary physics. Much research is currently focused on the dynamics of
many-body systems in low-dimensional lattices following a quench, i.e., a
sudden change of parameters. Already such a simple setting poses substantial
theoretical challenges for the investigation of the real-time post-quench
quantum dynamics. In classical many-body systems the
Kolmogorov-Mehl-Johnson-Avrami model describes the phase transformation
kinetics of a system that is quenched across a first order phase transition.
Here we show that a similar approach can be applied for shedding light on the
quench dynamics of an interacting gas of Rydberg atoms, which has become an
important experimental platform for the investigation of quantum
non-equilibrium effects. We are able to gain an analytic understanding of the
time-evolution following a sudden quench from an initial state devoid of
Rydberg atoms and identify strikingly different behaviors of the excitation
growth in the classical and quantum regimes. Our approach allows us to describe
quenches near a non-equilibrium phase transition and provides an approximate
analytic solution deep in the quantum domain.
