  Non-equilibrium dynamics of many-body systems is important in many branches
of science, such as condensed matter, quantum chemistry, and ultracold atoms.
Here we report the experimental observation of a phase transition of the
quantum coherent dynamics of a 3D many-spin system with dipolar interactions,
and determine its critical exponents. Using nuclear magnetic resonance (NMR) on
a solid-state system of spins at room-temperature, we quench the interaction
Hamiltonian to drive the evolution of the system. The resulting dynamics of the
system coherence can be localized or extended, depending on the quench
strength. Applying a finite-time scaling analysis to the observed
time-evolution of the number of correlated spins, we extract the critical
exponents v = s = 0.42 around the phase transition separating a localized from
a delocalized dynamical regime. These results show clearly that such
nuclear-spin based quantum simulations can effectively model the
non-equilibrium dynamics of complex many-body systems, such as 3D spin-networks
with dipolar interactions.
