Experimental studies of equilibrium vortex properties in a Bose-condensed gas

We characterize several equilibrium vortex effects in a rotating Bose-Einstein condensate. Specifically we attempt precision measurements of the vortex-lattice spacing and vortex-core size over a range of condensate densities and rotation rates. These measurements are supplemented by numerical simulations, and both experimental and numerical data are compared to theory. Finally, we study the effect of the centrifugal weakening of the trapping spring constants on the critical temperature for quantum degeneracy and the effects of finite temperature on vortex contrast.

Figure 1

Examples of the condensates used in the experiment viewed after expansion. Image (a) is a slowly rotating condensate. Images (b) and (c) are of rapidly rotating condensates with similar in-trap conditions. They differ only in that (c) was allowed to expand axially during the antitrapped expansion. The effect on the vortex core size is visible by eye. Images (a) and (b) are of the regularity required for the nearest-neighbor lattice spacing measurements.

Figure 2

Measured and binned lattice spacings as a function of radial position $\rho$. The solid curve is the theory result [Eq. (3)] of Sheehy and Radzihovsky [17]. The rigid-body-rotation rate lattice spacing is also plotted for comparison (dashed line). Plots (a)–(c) are experimental data with increasing rotation. Plots (a) and (c) are data taken from the condensate in Figs. 1a, 1b, respectively. Plot (d) is the same effect observed in the numerical data. One can see that theory and experiment show a similar dependence on radial position and that the fractional amplitude of the density inhomogeneity effect is suppressed at higher rotation. Plot (e) is the data in (c) plotted without suppressing the zero. The vortex-lattice spacing changes less than 2% over a region in which the atom density varies by 35%.

Figure 3

Comparison of measured core radii with the Thomas-Fermi prediction [Eq. (4)] represented by the solid line. Black squares are the core size in expansion scaled by the radial expansion of the condensate so that they correspond to the in-trap values. Data show reasonable agreement with theory. The fact that the measured core size is consistently larger is likely due to the fact that nearly all our imaging systematics lead to an overestimation of the vortex-core size.

Figure 4

Fractional condensate area occupied by vortex cores $\left(\mathcal{A}\right)$ as a function of ${\Gamma }_{LLL}^{-1}=2\hslash \Omega ∕\mu$, the inverse lowest-Landau-level parameter. Plot (a) shows a smooth transition in the numerical data from the Thomas-Fermi limit where $\mathcal{A}$ is linear in ${\Gamma }_{LLL}^{-1}$ to the LLL limit where $\mathcal{A}$ saturates. Here the Thomas-Fermi theory is represented by the dashed line and the LLL limit by the dotted line. Plot (b) is a comparison of the numerical data to experimental data. Plot (c) demonstrates the effect on the experimental measurement of allowing the condensate to expand axially during the expansion process.

Figure 5

Numerically generated vortex-core-density profiles approaching the lowest-Landau-level regime. Solid lines represent the numerical result for (a) ${\Gamma }_{LLL}=117$, (b) ${\Gamma }_{LLL}=3.6$, and (c) ${\Gamma }_{LLL}=0.72$. The dashed line is the expected profile for a LLL wave function [25] given the condensate rotation. The dotted line is the expected vortex form in the Thomas-Fermi limit [20] given the condensate density. The vertical lines in (b) and (c) designate the edge of the Wigner-Seitz unit cell. As ${\Gamma }_{LLL}$ decreases, one can see a clear transition from the interaction-dominated Thomas-Fermi regime to a LLL function where kinetic energy concerns and the vortex-core spacing dictate the shape and size of the vortex.

Figure 6

(Color online) Condensate fraction versus temperature for various sample rotation rates. Condensate number $N_{BEC}$ and total number $N$ are obtained from fits to cloud images. The temperature $T$ is extracted from the vertical width of the thermal cloud while the centrifugal distortion of the thermal cloud’s aspect ratio yields its rotation ${\Omega }_N$. The temperature is scaled by the static critical temperature $T_c^{\left(0\right)}$ for an ideal gas. The data have been grouped according to three different ranges of ${\Omega }_N$. The solid line is the theoretical dependence expected for a static ideal gas.

Figure 7

(Color online) Inferred critical temperature $T_c$, scaled by the nonrotating expectation $T_c^{\left(0\right)}$ for an ideal gas, as a function of thermal gas rotation ${\Omega }_N$. For each data point, $T_c$ is inferred from the measured condensate fraction and temperature of a sample. The total number of atoms is used to obtain $T_c^{\left(0\right)}$. The data have been grouped according to three different condensate fractions as shown. A set of static points, where the thermal cloud is not stirred before evaporation, is deliberately plotted at zero rotation. Otherwise, the rotation rate ${\Omega }_N$ is obtained from the thermal clouds aspect ratio. The rotating data have been cropped at a minimum threshold $0.2<{\Omega }_N∕{\omega }_{\rho }$ to avoid imaginary rotation values arising from noise in near-static aspect ratios. The dotted line is the theoretical expectation according to Eq. (5) with $T_c^{\left(0\right)}$ as for an ideal gas. The solid line is a fit of the data to Eq. (5) with an arbitrary overall scaling of the vertical axis. The fit result is equivalent to assuming an effective $T_c^{\left(0\right)}$ that is $87\%\pm 1\%$ of the ideal gas value. This 13% shift is consistent with what one would expect from atom-atom interactions [38].

Figure 8

Measured core brightness as a function of final rf evaporative cut. Within our ability to measure, $T∕T_c$ decreases continuously with the rf frequency. For the black squares brightness $\left(\mathcal{B}\right)$ is defined as the 2D atom density at the vortex core divided by the 2D atom density of the overall smoothed condensate plus thermal cloud profile at the same point. For the open triangles a simplistic brightness $\left({\mathcal{B}}_{\mathit{\text{simple}}}\right)$ is calculated from the ratio of the 2D atom density of the thermal cloud to the 2D atom density of the overall smoothed condensate and thermal cloud profile. At high temperatures $\mathcal{B}$ and $B_{\mathit{\text{simple}}}$ exhibit a clear dependence on the final rf cut. At lower temperatures it is encouraging that as ${\mathcal{B}}_{\mathit{\text{simple}}}$ begins to fail, $\mathcal{B}$ is still continuing a smooth trend downward. Disappointingly at very low temperatures, $\mathcal{B}$ plateaus at about 0.14.