  Spin-orbit coupled bosons can exhibit rich equilibrium phases at low
temperature and in the presence of particle-particle interactions. In the case
with a 1D synthetic spin-orbit interaction, it has been observed that the
ground state of a Bose gas can be a normal phase, stripe phase, or magnetized
phase in different experimentally controllable parameter regimes. The
magnetized states are doubly degenerate and consist of a many-particle
two-state system. In this work, we investigate the nonequilibrium quantum
dynamics by switching on an external perturbation to induce resonant couplings
between the magnetized phases, and predict the novel quantum spin dynamics
which cannot be obtained in the single-particle systems. In particular, due to
particle-particle interactions, the transition of the Bose condensate from one
magnetized phase to the other is forbidden when the strength of external
perturbation is less than a critical value, and a full transition can occur
only when the perturbation exceeds such critical strength. This phenomenon
manifests itself a quantum dynamical phase transition, with the critical point
behavior being exactly solvable. From the numerical simulations and exact
analytic studies we show that the predicted many-body effects can be well
observed with the current experiments.
