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.
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