Out of Equilibrium Dynamics and Critical Behavior in Spin-Orbit-Coupled Bose-Einstein Condensates

Many-body quantum systems exhibit emergent collective behaviors, including superfluidity, magnetism, and macroscopic entanglement. In this work, we investigate the out-of-equilibrium dynamics of a Bose-Einstein condensate (BEC) with a Raman-induced spin-orbit coupling (SOC) and/or optical lattice in a regime where collective excitations play a crucial role. We focus on two scenarios: the scattering of a condensate quenched far from equilibrium and the quench dynamics across a quantum phase transition. We observe distinct quantum effects like macroscopic self-trapping, formation of defects near phase transitions, and scattering patterns due to dynamical instability. Laboratory observations of the SOC system near criticality or undergoing a scattering process revealed pattern formation in the density profiles, which we simulate using two approaches: a mean-field model using a non-linear GP equation and the truncated Wigner approximation (TWA) to account for quantum fluctuations. Additionally, we developed an analytical model using Bogoliubov analysis to describe quantum excitations. The TWA simulations and the analytical model showed strong agreement at short time scales, providing a consistent framework for identifying quantum signatures. Our results highlight the critical role of beyond-mean-field corrections in shaping the system's dynamics. By revealing the intricate interplay between nonlinear interactions and critical behavior, this study provides new insights into the nonequilibrium physics of BECs with multiple momentum components such as spin-orbit-coupled quantum gases.