Defect Formation Across Non-equilibrium Phase Transitions in a Spin–Orbit-Coupled BEC in an Optical Lattice

We study dynamics near non-equilibrium critical points in a spin-orbit-coupled Bose-Einstein condensate (BEC) subjected to a tunable optical lattice, which enables an effective two-mode Josephson junction description in momentum space. Due to interaction-induced self-trapping, the system exhibits both a ground-state phase transition and a non-equilibrium phase transition. The transition between these phases is continuous (second order). By quenching across the critical point, we investigate the universal non-equilibrium dynamics associated with this transition. Using truncated Wigner approximation simulations and many-body quantum evolution, we extract the scaling behavior of the delay time for the onset of symmetry breaking and show that it follows a power law in the quench time, consistent with Kibble-Zurek predictions for continuous phase transitions. We also identify two dynamical regimes: one dominated by spin (momentum) dynamics with negligible spatial excitations, and another in which spatial and momentum degrees of freedom become strongly coupled and domain structures form. Our results shed light on universal critical dynamics in momentum-space condensates and their connection to Kibble-Zurek physics.