Vinen turbulence via the decay of multicharged vortices in trapped atomic Bose-Einstein condensates

We investigate a procedure to generate turbulence in a trapped Bose-Einstein condensate which takes advantage of the decay of multicharged vortices to reduce surface oscillations. We show that the resulting singly charged vortices twist around each other, intertwined in the shape of helical Kelvin waves, which collide and undergo vortex reconnections, creating a disordered vortex state. By examining the velocity statistics, the energy spectrum, the correlation functions, and the temporal decay and comparing these properties with the properties of classical turbulence and observations in superfluid helium, we conclude that this disordered vortex state can be identified with the Vinen regime of turbulence which has been discovered in the context of superfluid helium.

Figure 1

Three-dimensional isodensity plots show the time (in units of ${\tau }_{\mathrm{HO}}$) evolution of an initial $j=4$ multicharged vortex (a); note the twisted unwinding of the vortex (b), which finally decays into four singly charged ($j=1$) vortices (c) in the disordered, anisotropic state. The density isosurfaces representing the condensate and the vortices are gray and blue, respectively. The 3D domain is displayed within the plot limits $-7{\ell }_{\text{HO}}\le x\le 7{\ell }_{\text{HO}}, -10{\ell }_{\text{HO}}\le y\le 10{\ell }_{\text{HO}}$, and $-30{\ell }_{\text{HO}}\le z\le 30{\ell }_{\text{HO}}$.

Figure 2

Decay of a quadruply charged vortex (anisotropic state). Time evolution of average velocity components $v_i$ and $|v_i|$ (in units of ${\ell }_{ho}/{\tau }_{\text{HO}}$, where $i=x,y,z$) vs time (in units of ${\tau }_{\text{HO}}$). The symbol $\langle \cdots \rangle$ denotes the spatial average over the condensate.

Figure 3

Decay of quadruply charged vortex (anisotropic state). Time (in units of ${\tau }_{\text{HO}}$) evolution of the condensate's width $w_i$ (in units of ${\ell }_{\text{HO}}$).

Figure 4

Isodensity plots showing the evolution (with time $t$ in units of ${\tau }_{\text{HO}}$) of two initial doubly charged ($j=2$) antiparallel vortices at $t=0{\tau }_{\text{HO}}$ (a) into the turbulent quasi-isotropic state at $t=13.6{\tau }_{\text{HO}}$ (b), which has finally decayed at $t=50{\tau }_{\text{HO}}$ (c). The 3D domain is displayed within the plot limits $-7{\ell }_{\text{HO}}\le x\le 7{\ell }_{\text{HO}}, -10{\ell }_{\text{HO}}\le y\le 10{\ell }_{\text{HO}}$, and $-30{\ell }_{\text{HO}}\le z\le 30{\ell }_{\text{HO}}$.

Figure 5

Decay of antiparallel doubly charged vortices (quasi-isotropic state). Time evolution of velocity components as in Fig. 2.

Figure 6

Decay of antiparallel doubly charged vortices (quasi-isotropic state). Time evolution of the condensate's width.

Figure 7

Decay of antiparallel doubly charged vortices (quasi-isotropic state). PDFs of velocity components $v_i$ ($i=x, i=y$, and $i=z$ corresponding to blue, red, and green symbols, respectively) plotted vs $v_i/{\sigma }_i$ (where ${\sigma }_i$ are the corresponding standard deviations) at $t=12.6{\tau }_{\text{HO}}$. For reference, solid curves are Gaussian fits (gPDFs) with standard deviations ${\sigma }_x=1.8, {\sigma }_y=1.7$, and ${\sigma }_z=1.4$, and mean values (plotted as vertical lines near the origin) ${\mu }_x=-0.3, {\mu }_y=-0.1$, and ${\mu }_z=0.0$.

Figure 8

Decay of antiparallel doubly charged vortices (quasi-isotropic state). Longitudinal correlation functions $f_i\left(r\right)$ ($i=x,y,z$) vs $r$ (in units of ${\ell }_{\text{HO}}$) at $t=13{\tau }_{\text{HO}}$. Dashed vertical lines indicate the length ${\ell }_c$ over which the velocity field is highly correlated, as defined in Eq. (4).

Figure 9

Decay of two antiparallel doubly charged vortices (quasi-isotropic state). Incompressible kinetic energy spectrum $E_i\left(k\right)$ (arbitrary units) vs wave number $k$ (in units of ${\ell }_{\text{HO}}^{-1}$) at $t=12.8{\tau }_{\text{HO}}$. Vertical lines mark the wave numbers corresponding to the healing length $\xi =0.24{\ell }_{\text{HO}}$, the vortex core size $a=0.96{\ell }_{\text{HO}}$, the average distance between vortex lines $\ell =6.10{\ell }_{\text{HO}}$, and the radial and axial Thomas-Fermi radii $d_{\text{TF}}=4.21{\ell }_{\text{HO}}$ and $D_{\text{TF}}=32.64{\ell }_{\text{HO}}$. Red, gray, and green lines represent the power laws $k^{-3}, k^{-5/3}$, and $k^{-1}$, respectively.

Figure 10

Time evolution of the vortex line density ($t$ in units of ${\tau }_{\text{HO}}$). The shaded area corresponds to the time elapsed for the decay of the two antiparallel multiply charged vortices, $t\approx 8{\tau }_{\text{HO}}$, after which we only see singly quantized vortices in the system in the quasi-isotropic state.

Figure 11

Transverse cross section of the condensate in the quasi-isotropic case. The green curve defines the condensate edge $C$ compared to the magenta curve, representing the radial Thomas-Fermi circumference for (a) the initial state and (b) at a turbulent instant. The circularity of both edges is given by $Q$ as defined in Eq. (A1).

Figure 12

Time evolution of the isoperimetric quotient ($t$ in units of ${\tau }_{\text{HO}}$).