Moving obstacle potential in a spin-orbit-coupled Bose-Einstein condensate

We investigate the dynamics around an obstacle potential moving in the plane-wave state of a pseudospin-$1/2$ Bose-Einstein condensate with Rashba spin-orbit coupling. We numerically investigate the dynamics of the system and find that it depends not only on the velocity of the obstacle but also significantly on the direction of obstacle motion, both of which are verified by the Bogoliubov analysis. The excitation diagram with respect to the velocity and direction is obtained. The dependence of the critical velocity on the strength of the spin-orbit coupling and the size of the obstacle is also investigated.

Figure 1

(a) Snapshot of the density distribution $|{\psi }_1{|}^2$ (upper left panel) and time evolution of the drag force $\mathit{f}$ exerted on the potential (upper right panel) for the velocity of the potential $(v,{\theta }_v)=(0.5,0)$. The density distributions, $|{\psi }_1{|}^2$ and $|{\psi }_2{|}^2$, and the phase distributions, $\arg {\psi }_1$ and $\arg {\psi }_2$, in the dashed square region are magnified into the lower panels. (b) $|{\psi }_1{|}^2$ and $\mathit{f}$ for $(v,{\theta }_v)=(0.05,\pi )$. The white circles represent the obstacle potential, and the thick arrows indicate the moving direction. The curved arrows indicate the directions of the circulations of the vortices. The parameters are $R=0.5$, $\kappa =1$, and $\gamma =0.8$. See Supplemental Material for movies of the dynamics of component 1 [44].

Figure 2

Snapshots of (I–IV) the density distributions $|{\psi }_1{|}^2$ and (i–iv) the momentum-space distribution $|{\tilde{\psi }}_1{|}^2$ for $\kappa =1$, $\gamma =0.8$, and $R=0.5$, where ${\tilde{\psi }}_1$ is the Fourier transform of ${\psi }_1$. (I, i) $(v,{\theta }_v)=(0.1,\pi /3)$. (II, ii) $(v,{\theta }_v)=(0.2,\pi /3)$. (III, iii) $(v,{\theta }_v)=(0.07,5\pi /6)$. (IV, iv) $(v,{\theta }_v)=(0.4,\pi )$. The white solid circles and the red arrows in the left-hand panels are the obstacle potentials and moving directions, respectively. The dashed lines in (iii) and (iv) indicate the analytical solutions of $\omega (k_x-\kappa ,k_y)=0$ in Eq. (12). [In (iv), the dashed lines are shown only for $k_y<0$.] See Supplemental Material for movies of the dynamics of $|{\psi }_1{|}^2$ and $|{\tilde{\psi }}_1{|}^2$ [44].

Figure 3

The excitation diagram with respect to $v$ and ${\theta }_v$ for $\kappa =1$, $\gamma =0.8$, and $R=0.5$. The dynamics are classified into (I)–(IV), which correspond to those in Fig. 2. The red points indicate the critical velocity, at which the drag force increases suddenly, and the red line is a visual guide.

Figure 4

Excitation spectrum of a uniform system for $\kappa =1$ and $\gamma =0.8$ in the frame moving at velocities (a) $(v,{\theta }_v)=(0,0)$, (b) $(v,{\theta }_v)=(1,0)$, and (c) $(v,{\theta }_v)=(0.1,\pi )$. The colored regions on the $q_x\text{−}{q}_y$ plane indicate negative energy, $\omega \left(\mathit{q}\right)<0$.

Figure 5

Dependence of the critical velocity $v_c$ on the strength of the SO coupling $\kappa$ for $\gamma =0.8$ and $R=0.5$. The red circles and blue crosses indicate the critical velocities for ${\theta }_v=0$ and $\pi$, respectively. The red solid line and the blue dashed line indicate the Landau criterion for ${\theta }_v=0$ and $\pi$, respectively, in Eq. (15).

Figure 6

Dependence of critical velocities $v_c$ on the obstacle radius $R$ for $\kappa =1$ and $\gamma =0.8$. The red circles and blue crosses indicate the critical velocities for ${\theta }_v=0$ and $\pi$, respectively. The green squares indicate the critical velocity without the SOC. The black solid circles indicate the Landau criteria in Eq. (15).

Figure 7

Serial snapshots of the density distributions $|{\psi }_1{|}^2$ of the dynamics in the rest frame for $\kappa =1$ and $\gamma =0.8$. The velocity of the moving obstacle is $(v,{\theta }_v)=(0.5,0)$, and the radius is $R=2$. The time interval is 75, and the field of view is $100\times 20$. See Supplemental Material for movies of the dynamics of component 1 [44].

Figure 8

Velocity field distribution induced by an obstacle potential for $\kappa =1$ and $\gamma =0.8$. (a, b) Ground states with the obstacle potential at rest. The radii of the circular potential are $R=0.5$ and 5.0. The velocity of the potential with radius $R=0.5$ is increased as $v\left(t\right)=6.0\times {10}^{-4}t$ in (c) and (d) and $v\left(t\right)=-7.0\times {10}^{-5}t$ in (e) and (d). (c) $(v,{\theta }_v)=(0.33,0)$ at $t=550$. (d) $(v,{\theta }_v)=(0.43,0)$ at $t=717$. (e) $(v,{\theta }_v)=(0.04,\pi )$ at $t=600$. (f) $(v,{\theta }_v)=(0.06,\pi )$ at $t=835$. The arrows indicate the direction of the velocity field, and their color indicates the value of $v$. The dashed circle in each panel indicates the obstacle potential.