Route to non-Abelian quantum turbulence in spinor Bose-Einstein condensates

We have studied computationally the collision dynamics of spin-2 Bose-Einstein condensates initially confined in a triple-well trap. Depending on the phase structure of the initial-state spinor wave function, the collision of the three condensate fragments produces one of many possible vortex-antivortex lattices, after which the system transitions to quantum turbulence. We find that the emerging vortex lattice structures can be described in terms of multiwave interference. We show that the three-fragment collisions can be used to systematically produce staggered vortex-antivortex honeycomb lattices of fractional-charge vortices, whose collision dynamics are known to be non-Abelian. Such condensate collider experiments could potentially be used as a controllable pathway to generating non-Abelian superfluid turbulence with networks of vortex rungs.

Figure 1

Schematic of our numerical experiment showing the total particle density at three different times of evolution. (a) The five hyperfine states behind the total density images of the $F=2$ Bose-Einstein condensate are initially confined in a species-independent triple-well trap, superposed with a global harmonic potential. (b) Upon removal of the triple-well potential, the three condensate fragments expand and interfere in the central portion of the harmonic trap. The multiwave interference of the condensate fragments produces transient honeycomb vortex-antivortex lattices in each of the occupied hyperfine spin states. (c) At later times the system transitions to quantum turbulence.

Figure 2

Selection of axisymmetric vortices in an $F=2$ spinor condensate represented in terms of the winding numbers $w$ of each hyperfine spin state $m_F$ of the condensate. Circles mark hyperfine spin components with nonzero condensate population. In this graphical notation each vortex corresponds to a straight line.

Figure 3

Structure of vortex lattices with black and white circles corresponding to the locations of vortices and antivortices, respectively. (a) A honeycomb lattice structure due to three-plane-wave interference in a single spin state of a condensate. (b) AB stacking of two honeycomb lattices. (c) ABC stacking of three honeycomb lattices.

Figure 4

The lattice vortices of a selection of initial states represented in terms of the winding numbers $w$ of each hyperfine spin state $m_F$ of the condensate. Circles mark hyperfine spin components with nonzero condensate population. In this graphical notation each axisymmetric vortex corresponds to a set of nodes for each populated spin state joined by a straight line, and is labeled using the naming convention listed in Table 1. (a)–(c) show the vortices that emerge in the lattice when using an initial-state vortex $\left(\frac{1}{2},\frac{1}{2}\right)$, $\left(0,1\right)$, and $\left(\frac{1}{3},\frac{1}{3}\right)$, respectively.

Figure 5

Initial state of the $\left(\frac{1}{2},\frac{1}{2}\right)$ vortex. (a) and (b) show the probability densities in the $m_F=1$ and $-1$ spin states normalized to the maximum density. (c) and (d) show the phases of the two spin states where the phase has been set to zero for regions of low density. In all figures the range of the colorbar is from the minimum to maximum value of the observable in arbitrary units. The field of view of each frame is $(23.4\times 23.4)$ $a_{\mathrm{osc}}$.

Figure 6

The lattice for the $\left(\frac{1}{2},\frac{1}{2}\right)$ vortex initial state from a Gross-Pitaevskii simulation. (a)–(d) show the probability density of the $m_F=-1$ and 1 spin states normalized to the maximum density and the corresponding phases. (e)–(g) show the total particle density, magnetization density, and the spin singlet pair amplitude density. The field of view of each frame is $(23.4\times 23.4)$ $a_{\mathrm{osc}}$ and the images are for a time $\tau =0.55$ $1/\omega$ after the triple-well trapping potential is switched off. See also movies S1–S14 in Ref. [70].

Figure 7

An expanded view, local to the trap center, of the lattice for the $\left(\frac{1}{2},\frac{1}{2}\right)$ vortex initial state. (a)–(c) The results from a Gross-Pitaevskii simulation showing the total particle density, magnetization density, and spin singlet pair amplitude density for a time $\tau =0.55$ $1/\omega$ after the triple-well trapping potential is switched off. (d)–(f) The corresponding observables calculated from the semianalytical spinor in Eq. (13) where ${\varphi }_{-1j}=0$ and ${\varphi }_{1j}=0,\frac{2\pi }{3},\frac{4\pi }{3}$ for $j=1,2,3$, with the other spin components empty. The locations of the $\left(\frac{1}{2},\frac{1}{2}\right)$, $\left(-\frac{1}{2},\frac{1}{2}\right)$, and $\left(0,-1\right)$ lattice vortices are denoted by colored dots of red, pink, and green, respectively. The field of view of each frame is $(2.3\times 2.3)$ $a_{\mathrm{osc}}$. For corresponding movies S12–S14, see Ref. [70].

Figure 8

An expanded view, local to the trap center, of the lattice for the $\left(0,1\right)$ vortex initial state. (a)–(c) The results from a Gross-Pitaevskii simulation showing the total particle density, magnetization density, and spin singlet pair amplitude density for a time $\tau =0.55$ $1/\omega$ after the triple-well trapping potential is switched off. (d)–(f) The corresponding observables calculated with the semianalytical spinor in Eq. (13) where ${\varphi }_{-2j}=0,\frac{8\pi }{3},\frac{4\pi }{3}$, ${\varphi }_{0j}=0$ and ${\varphi }_{2j}=0,\frac{4\pi }{3},\frac{8\pi }{3}$ for $j=1,2,3$, with the other spin components empty. The locations of the $\left(0,-\frac{1}{2}\right)$, $\left(-\frac{1}{2},\frac{1}{4}\right)+\left(1,0\right)$, and $\left(\frac{1}{2},\frac{1}{4}\right)+\left(-1,0\right)$ lattice vortices are denoted by colored dots of green, yellow, and purple, respectively. The field of view of each frame is $(2.3\times 2.3)$ $a_{\mathrm{osc}}$. See also movies S30–32 in Ref. [70].

Figure 9

An expanded view, local to the trap center, of the lattice for the $\left(\frac{1}{3},\frac{1}{3}\right)$ vortex initial state. (a) and (b) The results from a Gross-Pitaevskii simulation showing the total particle density and magnetization density for a time $\tau =0.55$ $1/\omega$ after the triple-well trapping potential is switched off. (c) and (d) The corresponding observables calculated with the semianalytical spinor in Eq. (13) where ${\varphi }_{-1j}=0$ and ${\varphi }_{2j}=0,\frac{2\pi }{3},\frac{4\pi }{3}$ for $j=1,2,3$, with the other spin components empty. The locations of the $\left(\frac{1}{3},\frac{1}{3}\right)$, $\left(-\frac{2}{3},\frac{1}{3}\right)$, and $\left(\frac{1}{3},-\frac{2}{3}\right)$ lattice vortices are denoted by colored dots of cyan, yellow, and brown, respectively. The field of view of each frame is $(2.7\times 2.7)$ $a_{\mathrm{osc}}$.

Figure 10

Emergence of non-Abelian two-dimensional quantum turbulence showing a selection of snapshots from the $\left(\frac{1}{3},\frac{1}{3}\right)$ initial-state simulation. (a)–(e) The total probability density of the condensate including the initial state, lattice, and turbulent regime. The field of view of each frame is $(29.2\times 29.2)$ $a_{\mathrm{osc}}$. (f)–(j) The same frames as in (a)–(e) but with a field of view of $(2.92\times 2.92)$ $a_{\mathrm{osc}}$. The colored dots of blue, red, green, and yellow denote the vortices (blue and green) and antivortices (red and yellow) in the $m_F=2$ and $-1$ spin states, respectively. See also movies S33–S44 in Ref. [70].