Quantum Spin Dynamics in a Normal Bose Gas with Spin-Orbit Coupling

In this Letter, we investigate spin dynamics of a two-component Bose gas with spin-orbit coupling realized in cold atom experiments. We derive coupled hydrodynamic equations for number and spin densities as well as their associated currents. Specializing to the quasi-one-dimensional situation, we obtain analytic solutions of the spin helix structure and its dynamics in both adiabatic and diabatic regimes. In the adiabatic regime, the transverse spin decays parabolically in the short-time limit and exponentially in the long-time limit, depending on initial polarization. In contrast, in the diabatic regime, transverse spin density and current oscillate in a way similar to the charge-current oscillation in an undamped LC circuit. The effects of Rabi coupling on the short-time spin dynamics is also discussed. Finally, using realistic experimental parameters for ${}^{87}\mathrm{Rb}$, we show that the timescales for spin dynamics is of the order of milliseconds to a few seconds and can be observed experimentally.

Figure 1

Short- and long-time behaviors of the transverse spin in the adiabatic regime. Short-time decay for initial spin density polarized close to the $xy$ plane. The decay is parabolic (left panel). Long-time decay for initial spin density polarized close to the $z$ axis. The decay is exponential (right panel). Numerical simulations of the full set of Eqs. (2), (3), (5), (6) (black solid) agree very well with the analytical results (red dashed) given by Eq. (11). Blue dashed lines show the asymptotic results, Eqs. (15), (16). The deviation at the tail of the left panel between the simulation and analytic equation (11) indicates the failure of adiabatic approximation.

Figure 2

Left panel shows time dependence of the $y$ component of spin density for spin density polarized along the $\widehat{x}$ direction at $t=0$. Numerical result (black dashed line) agrees very well with the analytical result (red solid line), Eq. (17). Right panel is the trajectory of transverse spin component in the $xy$ plane. Also indicated in the graph are the fast oscillations of the magnitude of transverse spin ($\mathrm{\Gamma }$) and its slow rotations with rate $\lambda s_{z,0}/2$.

Figure 3

The approximate demarcation of adiabatic from diabatic regions based on the condition $|{\partial }_t\overrightarrow{s}/\overrightarrow{s}|\sim \gamma$ (red shaded lines). The region of adiabaticity becomes smaller when the wave number of spin helix deviates away from $2q$.