Double-quantum spin vortices in SU(3) spin-orbit-coupled Bose gases

We show that double-quantum spin vortices, which are characterized by doubly quantized circulating spin currents and unmagnetized filled cores, can exist in the ground states of SU(3) spin-orbit-coupled Bose gases. It is found that the SU(3) spin-orbit coupling and spin-exchange interaction play important roles in determining the ground-state phase diagram. In the case of effective ferromagnetic spin interaction, the SU(3) spin-orbit coupling induces a threefold degeneracy to the magnetized ground state, while in the antiferromagnetic spin interaction case, the SU(3) spin-orbit coupling breaks the ordinary phase rule of spinor Bose gases and allows the spontaneous emergence of double-quantum spin vortices. This exotic topological defect is in stark contrast to the singly quantized spin vortices observed in existing experiments and can be readily observed by the current magnetization-sensitive phase-contrast imaging technique.

Figure 1

Scheme for creating SU(3) spin-orbit coupling in spinor BECs. (a) Laser geometry. Three laser beams with different frequencies and polarizations, intersecting at an angle of $2\pi /3$, illuminate the cloud of atoms. (b) Level diagram. Each of the three Raman lasers dresses one hyperfine Zeeman level from $|F=1,m_F=1\rangle , |F=1,m_F=0\rangle$, and $|F=1,m_F=-1\rangle$ of the ${}^{\text{87}}\mathrm{Rb}5S_{1/2}, F=1$ ground state. ${\delta }_1, {\delta }_2$, and ${\delta }_3$ correspond to the detuning in the Raman transitions. (c) Triple-well dispersion relation. The SU(3) spin-orbit coupling induces three discrete minima of the single-particle energy band on the vertices of an equilateral triangle in the $k_x-k_y$ plane. (d) Projection of the first energy band on a 2D plane. Units with $\hslash =m=1$ are used for simplicity.

Figure 2

Two distinct phases present in SU(3) spin-orbit-coupled BECs. (a)–(d) The topologically nontrivial lattice phase for antiferromagnetic spin interaction ($c_2>0$) with (a) the density and phase of ${\mathrm{\Psi }}_1$ represented by heights and colors, (b) the phase within one unit cell showing the positions of vortices (white circles) and antivortices (black circles), (c) the corresponding momentum distributions, and (d) the structural schematic drawing of the phase separation. (e)–(f) The threefold-degenerate magnetized phase for ferromagnetic spin interaction ($c_2<0$) with (e) the density and phase distributions of ${\mathrm{\Psi }}_1$ and (f) the corresponding momentum distributions.

Figure 3

(a) Energy comparison between the lattice and stripe phases. The energy difference $\mathrm{\Delta }E$ between the numerical simulation and the variational calculation are shown by solid (lattice state) and dashed (stripe state) lines. (b)–(d) The ground-state density, phase, and momentum distributions of the stripe phase with the parameters $c_2=20{\kappa }^2$ and $c_0=10c_2$.

Figure 4

Vortex configurations in antiferromagnetic spinor BECs with SU(3) spin-orbit coupling. (a) Vortex arrangement among the three components of the condensates. One can identify three types of vortices, including a polar-core vortex with winding combination $\langle -1,0,1\rangle$ (blue line) and two ferromagnetic-core vortices with winding combinations $\langle 1,-1,0\rangle$ (green line) and $\langle 0,1,-1\rangle$ (red line). (b)–(d) Spherical-harmonic representation of the three types of vortices. The surface plots of ${\left|\mathrm{\Phi }(\theta ,\varphi )\right|}^2$ for (b) the polar-core vortex $\langle -1,0,1\rangle$, (c) the ferromagnetic-core vortex $\langle 1,-1,0\rangle$, and (d) the ferromagnetic-core vortex $\langle 0,1,-1\rangle$ are shown with the colors representing the phase of $\mathrm{\Phi }(\theta ,\varphi )$. Here, $\mathrm{\Phi }(\theta ,\varphi )={\sum }_{m=-1}^1{Y}_{1m}(\theta ,\varphi ){\mathrm{\Psi }}_m$, and $Y_{1m}$ is the rank-1 spherical-harmonic function.

Figure 5

Double-quantum spin vortex in antiferromagnetic spinor BECs with SU(3) spin-orbit coupling. (a) Spatial maps of the transverse magnetization with colors indicating the magnetization orientation. (b) Longitudinal magnetization. (c) Amplitude of the total magnetization $\left|\mathbf{F}\right|$. Two kinds of topological defects, double-quantum spin vortex (DSV) and half-Skyrmion (HS) are marked by big and small circles, respectively. The transverse magnetization orientation $\arg F_{+}$ along a closed path (indicated by big circles) surrounding the unmagnetized core shows a net winding of $4\pi$, revealing the presence of a double-quantum spin vortex.

Figure 6

(a)–(c) Phases of the polar-core vortex described by the wave function in Eq. (22), displaying the winding combination $\langle -1,0,1\rangle$. (d) Direction of the transverse magnetization, indicating the emergence of spin current with two quanta of circulation.

Figure 7

Stable double-quantum spin vortex under a random fluctuation. The images are taken at $t=20\text{ms}$ in the real-time evolution, with thermal fluctuation in the energy scale of $k_{B}T$ with $T\sim 300$ nK. (a) Spatial maps of the transverse magnetization with colors indicating the magnetization orientation. (b) Longitudinal magnetization. (c) Amplitude of the total magnetization $\left|\mathbf{F}\right|$. It is shown that the double-quantum spin vortices are topologically stable under external fluctuations with a fairly long lifetime of tens of milliseconds.