Relaxation of superfluid turbulence in highly oblate Bose-Einstein condensates

We investigate thermal relaxation of superfluid turbulence in a highly oblate Bose-Einstein condensate. We generate turbulent flow in the condensate by sweeping the center region of the condensate with a repulsive optical potential. The turbulent condensate shows a spatially disordered distribution of quantized vortices, and the vortex number of the condensate exhibits nonexponential decay behavior which we attribute to the vortex pair annihilation. The vortex-antivortex collisions in the condensate are identified with crescent-shaped, coalesced vortex cores. We observe that the nonexponential decay of the vortex number is quantitatively well described by a rate equation consisting of one-body and two-body decay terms. In our measurement, we find that the local two-body decay rate is closely proportional to $T^2/\mu$, where $T$ is the temperature and $\mu$ is the chemical potential.

Figure 1

Generation of turbulent flow in a BEC. A repulsive laser beam penetrates through the condensate and sweeps its center region by horizontally translating the trapped condensate by $37 \mu \mathrm{m}$. Images of the trapped condensate (a) before and (b) after the translation. (c) Number of the vortices in the perturbed condensate as a function of the translation speed $v$. The speed of sound is estimated to be $c\approx 4.6$ mm/s at the center of the condensate.

Figure 2

Vortex number counting for a turbulent condensate with a large number of vortices. (a) Absorption image of the condensate after expansion. (b) A binary image obtained from (a) and its blurred version (see the text). Black particles correspond to the vortex cores and red particles, having area sizes less than 15 pixels, are artifacts due to image defects. (c) Histogram of the particle area size from 24 image data. We set the vortex number transition lines (dashed lines) in the middle of the peaks. (d)–(f) display another example set of the image analysis, where the condensate contains about 30 vortices.

Figure 3

Relaxation of turbulent superflow in a highly oblate BEC. Examples of images for various relaxation times, where the atom number of the condensate $N_0\approx 1.8\times {10}^6$ and the condensate fraction $\eta \approx 0.8$.

Figure 4

Nonexponential decay of the vortex number. (a) Mean number of vortices versus relaxation time. The sample condition is the same as in Fig. 3. Each measurement point consists of at least 12 realizations of the same experiment and the error bars indicate the standard deviations of the mean vortex number. The solid line denotes a fit of the rate equation in Eq. (1) to the data. The inset shows the residue of the data from the fitting line. (b) Decay rate of the mean vortex number, $-d{N}_v/dt$, as a function of $N_v$. The dashed and dotted lines are power-law fitting lines to the data points for $N_v<10$ and $N_v>10$, giving the exponents of 1.29(17) and 1.77(20), respectively.

Figure 5

Vortex pair annihilation in the turbulent superflow. Density-depleted regions with a crescent shape are observed in the turbulent condensates. In a vortex-antivortex collision event, two vortex cores can coalesce (a), evolve into a dark soliton (b), and disappear as atoms fill up the density-depleted region (c). The bending structure results from the linear momentum of the vortex dipole.

Figure 6

The number of crescent-shaped vortex cores versus the total vortex number $N_v$ in a turbulent condensate. Each measurement point consists of at least 12 realizations of the same experiment. The inset shows an image of a turbulent condensate with two crescent-shaped vortex cores (red dashed circles).

Figure 7

Temporal evolution of the vortex number. (a) Decay curves obtained for various condensate fractions $\eta$. Solid lines are fitting curves from the rate equation in Eq. (1). The inset shows the atom number of the condensate, $N_0$, for each $\eta$. The data points for $\eta =0.80$ are identical to those in Fig. 4. Decay rates (b) ${\Gamma }_1$ and (c) ${\Gamma }_2$ as functions of the sample temperature $T$. Solid lines are power-law fits to the data with exponents of 0.87(20) and 1.88(12) for ${\Gamma }_1$ and ${\Gamma }_2$, respectively.

Figure 8

Decay rate ${\Gamma }_2$ as a function of (a) the radial trapping frequency ${\omega }_r$ ($\eta \approx 0.80$ and $N\approx 2.4\times {10}^6$) and (b) the total atom number $N$ ($\eta \approx 0.63$ and ${\omega }_r/2\pi =15$ Hz). The power-law fits (solid lines) give the exponents of (a) 1.82(9) and (b) $-0.10$(6) for ${\omega }_r$ and $N$, respectively.

Figure 9

Local two-body decay rate ${\gamma }_2=\left(\pi R^2\right){\Gamma }_2$ versus ${\left(k_{B}T\right)}^2/\mu$. The solid line denotes a linear fit to the data. Black solid circles, red open squares, and blue open diamonds correspond to the ${\Gamma }_2$ data points in Figs. 7, 8, and 8, respectively.