Emergence and Decay of Turbulence in Stirred Atomic Bose-Einstein Condensates

We show that a “weak” elliptical deformation of an atomic Bose-Einstein condensate rotating at close to the quadrupole instability frequency leads to turbulence with a Kolmogorov energy spectrum. The turbulent state is produced by energy transfer to condensate fragments that are ejected by the quadrupole instability. This energy transfer is driven by breaking the twofold rotational symmetry of the condensate. Subsequently, vortex-sound interactions damp the turbulent state leading to the crystallization of a vortex lattice.

Figure 1

The condensate energy (black line) and energy of the outer cloud (dashed line), defined as the region with $n<0.2n_0$, as a function of time, with a trap rotation frequency $\Omega =0.77{\omega }_r$. The four distinct stages of the evolution (see text) are indicated.

Figure 2

Fragmentation: (a) Snapshots of the condensate density in the range $(0–0.05n_0)$ at times $t\approx$ (i) 400, (ii) 800, (iii) 1200, and (iv) 6000 $(\xi /c)$. Each plot represents a region $[-30,30]\xi \times [-30,30]\xi$ (while the numerical box is much larger). Dark represents high density. (b) Total condensate energy (solid line), energy of inner cloud (dotted line), and energy of outer cloud (dashed line).

Figure 3

Symmetry breaking: Evolution of (a) total condensate energy and (b) condensate asymmetry parameter for no trap jitter (solid line), and trap jitters of $\gamma =0.0001$ (dashed line) and $\gamma =0.1$ (dotted line). (c)–(d) Snapshots of the total condensate density and phase, respectively, during the symmetry-breaking stage at $t=9600(\xi /c)$ in the absence of trap jitter.

Figure 4

Turbulence: Snapshots of the turbulent (a) condensate density and (b) phase at $t=11000(\xi /c)$. The vortices are characterized by a node in the density and an azimuthal phase change of $2\pi$. (c) Energy spectrum in $k$ space during (i) the turbulent stage at $t=11000(\xi /c)$ (bold line); (ii) fragmentation at $t=8000(\xi /c)$ (intermediate gray line); and crystallization at $t=20000(\xi /c)$ (light gray line). The turbulent Kolmorogov behavior $E_k\propto k^{-5/3}$ is indicated (dot-dashed line).

Figure 5

Crystallization: (a) Total condensate energy (solid line), sound energy (dotted line), and vortex energy (dashed line). Snapshots of the (a) total condensate density $(0–n_0)$ and (b) low density $(0–0.05n_0)$ at times (i) 20 000 and (ii) $100000(\xi /c)$.