Precise measurements on a quantum phase transition in antiferromagnetic spinor Bose-Einstein condensates

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 mHz uncertainty, equivalent to an energy resolution of kB x (12 pK). A small (2-3 sigma) shift of the transition point was observed, from q = 0 to q = + 650 mHz, whose physical mechanism is currently unknown. In this work, we demonstrate a sub-Hz precision measurement of a phase transition in quantum gases. It paves the way toward observing shifts of the transition point due to finite particle number N that scale as 1/N, and also to potential Heisenberg limited spectroscopy with antiferromagnetic spinor gases.