Dynamical precession of spin in the two-dimensional spin-orbit coupled systems

We investigate the spin dynamics in the two-dimensional spin-orbit coupled system subject to an in-plane ($x$-$y$ plane) constant electric field, which is assumed to be turned on at the moment $t=0$. The equation of spin precession in linear response to the switch-on of the electric field is derived in terms of Heisenberg's equation by the perturbation method up to the first order of the electric field. The dissipative effect, which is responsible for bringing the dynamical response to an asymptotic result, is phenomenologically implemented \`{a} la the Landau-Lifshitz-Gilbert equation by introducing damping terms upon the equation of spin dynamics. Mediated by the dissipative effect, the resulting spin dynamics asymptotes to a stationary state, where the spin and the momentum-dependent effective magnetic field are aligned again and have nonzero components in the out-of-plane ($z$) direction. In the linear response regime, the asymptotic response obtained by the dynamical treatment is in full agreement with the stationary response as calculated in the Kubo formula, which is a time-independent approach treating the applied electric field as completely time-independent. Our method provides a new perspective on the connection between the dynamical and stationary responses.