Gauge-potential-induced rotation of spin-orbit-coupled Bose-Einstein condensates

We demonstrate that a spin-orbit-coupled Bose-Einstein condensate can be effectively rotated by adding a real magnetic field to inputting gauge angular momentum, which is distinctly different from the traditional ways of rotation by stirring or Raman laser dressing to inputting canonical angular momentum. The gauge angular momentum is accompanied by the spontaneous generation of equal and opposite canonical angular momentum in the ground states, and it leads to the nucleation of quantized vortices. We explain this by indicating that the effective rotation with the vortex nucleation results from the effective magnetic flux induced by the gauge potential, which is essentially different from the previous scheme of creating vortices by synthetic magnetic fields. In the weakly interacting regime, symmetrically placed domains separated by vortex lines as well as half-integer giant vortices are discovered. With relatively strong interatomic interaction, we predict a structure of coaxially arranged annular vortex arrays, which is in stark contrast to the familiar Abrikosov vortex lattice. The developed way of rotation may be extended to a more general gauge system.

Figure 1

Scheme of Ioffe-Pritchard magnetic-field-induced rotation in a system with Dresselhaus spin-orbit coupling. The gray lines and gray arrows depict the direction of the magnetic field in the $x-y$ plane. Arrows with other colors depict the direction of the magnetic field $\mathbf{B}$ and spin $\mathbf{S}$ (purple), gauge angular momentum ${\mathbf{J}}_{\mathrm{g}}$ (green), canonical angular momentum ${\mathbf{J}}_{\mathrm{c}}$, and effective gauge potential ${\mathbf{A}}^{*}$ (red), respectively. The integral of ${\mathbf{A}}^{*}$ around a closed loop induces a Aharonov-Bohm geometric phase factor $exp\left(i\mathrm{\Phi }/\hslash \right)$ in the wave function, where the effective magnetic flux $\mathrm{\Phi }=\oint {\mathbf{A}}^{*}·d\mathbf{l}$ is responsible for the generation of quantized vortices.

Figure 2

Ground state as a function of magnetic field gradient. (a)–(d) Density distributions of the symmetrically placed domains separated by vortex lines with (a) $B^{\prime }=0.2{\hslash }^2/{\mu }_B{g}_{F}M{a}_h^3$, (b) $B^{\prime }=1.2{\hslash }^2/{\mu }_B{g}_{F}M{a}_h^3$, (c) $B^{\prime }=2.2{\hslash }^2/{\mu }_B{g}_{F}M{a}_h^3$, and (d) $B^{\prime }=3.2{\hslash }^2/{\mu }_B{g}_{F}M{a}_h^3$. (e), (f) Density and phase distributions of the half-integer giant vortex with $B^{\prime }=4{\hslash }^2/{\mu }_B{g}_{F}M{a}_h^3$. Other parameters are fixed at $\kappa =10\hslash /M{a}_h$, $N{g}_{\uparrow \uparrow }=N{g}_{\downarrow \downarrow }=10{\hslash }^2/M$, and $N{g}_{\uparrow \downarrow }=8{\hslash }^2/M$. Here, $a_h=\sqrt{\hslash /M{\omega }_{\perp }}$ is the characteristic length of the harmonic trap.

Figure 3

Gauge angular momentum $L_z^{\mathrm{g}}$ and canonical angular momentum $L_z^{\mathrm{c}}$ per particle as well as domain number as functions of the field gradient $B^{\prime }$. Arrows indicate the jump behavior of the angular momenta in which the domain number changes; the solid line plots the proportional function of $B^{\prime }$ with slope $\kappa M{a}_h/\hslash$; and the dashed line distinguishes the regions of the domain structure and the half-integer giant vortex. The other parameters are fixed at $\kappa =10\hslash /M{a}_h$, $N{g}_{\uparrow \uparrow }=N{g}_{\downarrow \downarrow }=10{\hslash }^2/M$, and $N{g}_{\uparrow \downarrow }=8{\hslash }^2/M$.

Figure 4

Ground-state density distribution as a function of the spin-orbit coupling strength. (a)–(d) Coaxially arranged vortex arrays with (a) $\kappa =1\hslash /M{a}_h$, (b) $\kappa =2\hslash /M{a}_h$, (c) $\kappa =4\hslash /M{a}_h$, and (d) $\kappa =6\hslash /M{a}_h$. Other parameters are fixed at $B^{\prime }=6{\hslash }^2/{\mu }_B{g}_{F}M{a}_h^3$, $N{g}_{\uparrow \uparrow }=N{g}_{\downarrow \downarrow }=5000{\hslash }^2/M$, and $N{g}_{\uparrow \downarrow }=4000{\hslash }^2/M$.

Figure 5

Gauge angular momentum $L_z^{\mathrm{g}}$ and canonical angular momentum $L_z^{\mathrm{c}}$ per particle as well as layer number as functions of the spin-orbit coupling strength $\kappa$. The green solid line plots the proportional function of $\kappa$ with slope $B^{\prime }{\mu }_B{g}_{F}M{a}_h^3/{\hslash }^2$. The other parameters are fixed at $B^{\prime }=6{\hslash }^2/{\mu }_B{g}_{F}M{a}_h^3$, $N{g}_{\uparrow \uparrow }=N{g}_{\downarrow \downarrow }=5000{\hslash }^2/M$, and $N{g}_{\uparrow \downarrow }=4000{\hslash }^2/M$.