Dynamical spin-density waves in a spin-orbit-coupled Bose-Einstein condensate

Synthetic spin-orbit (SO) coupling, an important ingredient for quantum simulation of many exotic condensed matter physics, has recently attracted considerable attention. The static and dynamic properties of a SO-coupled Bose-Einstein condensate (BEC) have been extensively studied in both theory and experiment. Here we numerically investigate the generation and propagation of a dynamical spin-density wave (SDW) in a SO-coupled BEC using a fast moving Gaussian-shaped barrier. We find that the SDW wavelength is sensitive to the barrier's velocity while varies slightly with the barrier's peak potential or width. We qualitatively explain the generation of SDW by considering a rectangular barrier in a one-dimensional system. Our results may motivate future experimental and theoretical investigations of rich dynamics in the SO-coupled BEC induced by a moving barrier.

Figure 1

(a) Experimental setup for the observation of barrier induced dynamics in a SO-coupled BEC. The realistic barrier, created by another external laser beam, is characterized by the width $W$ (along $x$ direction), barrier peak potential $V_b$, and barrier velocity $v$. (b) Energy level diagram of the SO-coupled BEC. Two internal states are coupled by the two-photon Raman transition through a pair of counterpropagating Raman beams.

Figure 2

Excitation and propagation of the spin-dependent wave in a SO-coupled BEC by suddenly switching on a stationary barrier at the center of the trapped BEC at $t=0$ ms. (a) Spin-up component; (b) spin-down component. $V_b=10E_r,W=0.5\mu \mathrm{m}$. Because of the presence of SO coupling, the propagation of sound demonstrates an anisotropic behavior, i.e., spin-up only propagates to the left side while spin-down only propagates to the right side.

Figure 3

A moving barrier with low velocity $v=1 \mu \mathrm{m}/\mathrm{ms}$ pushes BEC to its right. $V_b=15E_r,W=0.5 \mu \mathrm{m}$. (a) Spin-up component; (b) spin-down component.

Figure 4

A fast moving barrier with a velocity of $v=50\mu \mathrm{m}/\mathrm{ms}$ penetrates the BEC and generates a dynamical SDW. $V_b=15E_r,W=0.5\mu \mathrm{m}$. (a)Spin-up component; (b) spin-down component.

Figure 5

The wavelength $\lambda$ and the contrast $C$ of the SDW generated in a fast-moving barrier as a function of (a) and (b) the barrier peak potential $V_b$ for $v=50\mu \mathrm{m}/\mathrm{ms},W=0.5\mu \mathrm{m}$; (c) and (d) the barrier velocity $v$ for $V_b=15E_r,w=0.5\mu \mathrm{m}$; (e) and (f) the width $W$ for $v=50\mu \mathrm{m}/\mathrm{ms},V_b=15E_r$.

Figure 6

Local spin polarization $\langle {\sigma }_z\rangle (x=0)$ as a function of time for Fig. 4.

Figure 7

Propagation of SDW in a SO-coupled BEC. When the barrier moves to the center of the BEC, we suddenly switch off the barrier and observe the slow propagation of the SDW to the unperturbed region. All the parameters are the same as Fig. 4.

Figure 8

SDW generation from a rectangular barrier in 1D SO-coupled BEC system for different barrier widths (a) $W=2\mu \mathrm{m}$, (b) $W=4.25\mu \mathrm{m}$, (c) $W=8.5\mu \mathrm{m}$, (d) $W=12.75\mu \mathrm{m}$. Other parameters are $V_b=4E_r,v=50\mu \mathrm{m}/\mathrm{ms}$. (e) The dynamics of the system is equivalent to the precession of a spin under the magnetic field. Without barrier, the system is in the eigenstate of a horizontal magnetic field $B_x$. A moving barrier is equivalent to suddenly quenching on a $z$-component magnetic field $B_z$. The spin then does precession about the axis determined by the total magnetic field. When the barrier passes the point, the spin could be in any possible orientations (with SDW appears) or returns to the eigenstate of $B_x$ (no SDW appears). Other parameters are $\mathrm{\Omega }=6E_r,\delta =0E_r,{\omega }_x=20$ Hz.