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

Abstract

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 ${k}_{B}\phantom{\rule{4pt}{0ex}}\ifmmode\times\else\texttimes\fi{}$ (12 pK). A small (2--3 $\ensuremath{\sigma}$) 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 [L.-N. Wu and L. You, Phys. Rev. A 93, 033608 (2016)].