Stationary states of trapped spin-orbit-coupled Bose-Einstein condensates

We numerically investigate low-energy stationary states of pseudospin-1 Bose-Einstein condensates in the presence of Rashba-Dresselhaus-type spin-orbit coupling. We show that for experimentally feasible parameters and strong spin-orbit coupling, the ground state is a square vortex lattice irrespective of the nature of the spin-dependent interactions. For weak spin-orbit coupling, the lowest-energy state may host a single vortex. Furthermore, we analytically derive constraints that explain why certain stationary states do not emerge as ground states. Importantly, we show that the distinct stationary states can be observed experimentally by standard time-of-flight spin-independent absorption imaging.

Figure 1

Energies of low-energy stationary states for effectively two-dimensional condensates as functions of the SOC strength. (a) The pseudospin case $c_2=0$, (b) the antiferromagnetic case ($c_2/c_0=0.01$), and (c) the ferromagnetic case ($c_2/c_0=-0.01$). Here we take ${\tilde{c}}_0=200$. The various stationary states can be approximated by superpositions of degenerate plane-wave solutions of the single-particle Hamiltonian. The PW state can be approximated by a single plane wave, whereas the SW state corresponds to a superposition of two coherent counterpropagating plane waves. Furthermore, the SL state is formed by two pairs of counterpropagating plane waves. The dashed lines correspond to cylindrically symmetric states with different values of $\kappa$ in Eq. (5). Here, $0.5{\tilde{\alpha }}^2$ has been subtracted from the energies $\tilde{E}$ for clarity.

Figure 2

(a) The magnetization of the square-lattice state. The arrows represent the projection of the magnetization to the $xy$ plane, and the $z$ component of the magnetization is illustrated with the color map. Black corresponds to small values, and white corresponds to large values. (b) The density isosurfaces of the spinor components in the square-lattice state. Green (light gray) corresponds to ${\left|{\Psi }_{-1}\right|}^2$, black corresponds to ${\left|{\Psi }_0\right|}^2$, and red (dark gray) corresponds to ${\left|{\Psi }_1\right|}^2$. The field of view is $6.6\times 3.3\times 2.4a_r^3$, and the isosurfaces correspond to the density $0.02N/a_r^3$. The density isosurfaces are capped with a density color map, where the lightest color corresponds to the largest density. The dimensionless value of the SOC strength was taken to be $\tilde{\alpha }=5.0$.

Figure 3

Spinor components in the square-lattice state. The orientation of the axes in all the panels is the same. (a)–(c) The amplitudes and (d)–(f) the phases of ${\Psi }_1$, ${\Psi }_0$, and ${\Psi }_{-1}$, respectively. The insets in panels (a)–(c) represent the absolute value of the Fourier transform for the corresponding spinor component. The horizontal and the vertical axes on the insets correspond to $k_x$ and $k_y$, respectively. The values of the coupling constants were taken to be ${\tilde{c}}_0=200$, $c_2=0$, and $\tilde{\alpha }=5.0$.

Figure 4

Three-dimensional dynamics of the square-lattice state after the removal of the SOC and the optical trapping in the $x$ and $y$ directions. The density is integrated in the $z$ direction. The density profile at (a) $t=0,$ (b) $t=1/{\omega }_r$, and (c) $t=2/{\omega }_r$. The maxima of the color map in panels (a)–(c) are $0.03$, $0.0080,$ and $0.0035$ in units of $N/a_r^3$. The values of the coupling constants were taken to be ${\tilde{c}}_0=1000$, $c_2=0$, and $\tilde{\alpha }=5.0$.