Phase-space Mixing in Dynamically Unstable, Integrable Few-mode Quantum Systems

Quenches in isolated quantum systems are currently a subject of intense study. Here, we consider quantum few-mode systems that are integrable in their classical mean-field limit and become dynamically unstable after a quench of a system parameter. Specifically, we study a Bose-Einstein condensate (BEC) in a double-well potential and an antiferromagnetic spinor BEC constrained to a single spatial mode. We study the time dynamics after the quench within the truncated Wigner approximation (TWA), focus on the role of motion near separatrices, and find that system relaxes to a steady state due to phase-space mixing. Using the action-angle formalism and a pendulum as an illustration, we derive general analytical expressions for the time evolution of expectation values of observables and their long-time limits. We find that the deviation of the long-time expectation value from its classical value scales as (1/ln ), where is the number of atoms in the condensate. Furthermore, the relaxation of an observable to its steady-state value is a damped oscillation. The damping is Gaussian in time with a time scale of [(ln )]. We also give the quantitative dependence of the steady-state value and the damping time on the system parameters. Our results are confirmed with numerical TWA simulations.