  We have experimentally investigated the quench dynamics of antiferromagnetic
spinor Bose-Einstein condensates in the vicinity of a zero temperature quantum
phase transition at zero quadratic Zeeman shift $q$. The rate of instability
shows good agreement with predictions based upon solutions to the Bogoliubov
de-Gennes equations. A key feature of this work was removal of magnetic field
inhomogeneities, resulting in a steep change in behavior near the transition
point. The quadratic Zeeman shift at the transition point was resolved to 250
milliHertz uncertainty, equivalent to an energy resolution of $k_B \times 12$
picoKelvin. To our knowledge, this is the first demonstration of sub-Hz
precision measurement of a phase transition in quantum gases. Our results point
to the use of dynamics, rather than equilibrium studies for high precision
measurements of phase transitions in quantum gases.
