Probing Kibble-Zurek Mechanism in Quenched Elongated Bose Gases

We report our numerical findings on the statistics and dynamics of spontaneous formation of defects during a gradual quench of an initially thermal atomic gas to below the critical temperature. Our study focuses on the Trento experiments [Nat.~Phys.~9,~656~(2013) and Phys.~Rev.~Lett.~113,~065302~(2014)], which showed the appearance of a few long-lived solitonic vortices [Phys.~Rev.~A~65,~043612~(2002)], as measured sometime after the system crossed the transition temperature. Our simulations access both the initial quench-driven turbulent regime where a large number of randomly-distributed defects emerge during the condensation, and the subsequent relaxation of such defects towards a few long-lived solitonic vortices, similar to those observed experimentally. We analyze our findings in the context of the Kibble-Zurek scaling law [J.~Phys.~A~9,~1387~(1976) and Nature~317,~505~(1985)], highlighting various subtle issues associated with this dynamical process, and characterize the transition through the critical region, by studying the corresponding first-order spatial correlation functions. Our simulations are based on the 3D stochastic projected Gross-Pitaevskii equation subjected to a linear temperature and chemical potential quench~[arXiv:1408.08 (Phys.~Rev.~A in press)].