In situ imaging of vortices in Bose-Einstein condensates

We present an application of dark-field imaging that enables in situ detection of two-dimensional vortex distributions in single-component Bose-Einstein condensates (BECs). By rotating a ${}^{87}\mathrm{Rb}$ BEC in a magnetic trap, we generate a triangular lattice of vortex cores in the BEC, with core diameters on the order of 400 nm and cores separated by approximately $9\mu \mathrm{m}$. We have experimentally confirmed that the positions of the vortex cores near the BEC center can be determined without the need for ballistic expansion of the BEC. Our imaging method should allow for the determination of arbitrary distributions of vortices and other superfluid density defects in cases where expansion of the BEC is either impractical or would significantly alter the physical characteristics and appearance of vortices or defects. Our method is also a step toward real-time measurements of complex two-dimensional vortex dynamics within a single BEC.

Figure 1

(a) BEC imaging optics (not to scale). 780-nm probe light (shaded in gray) is directed toward the BEC along the vertical (axial) imaging axis. Light refracted by the BEC (represented by dashed lines) is collected with a microscope objective, and imaged onto the CCD camera. A mask placed in the intermediate Fourier plane of the imaging system provides a high-pass spatial filter. (b) Image of a U.S. Air Force resolution test target, obtained with an offline replica of the imaging system using 780-nm laser light and showing group 6 element 1 (bottom row) and group 7 elements 4–6 (top three rows). (c) Zoomed image of group 7 element 6, the features enclosed in the superimposed white box in (b). These features are the smallest features on our test target, with a width of 2.$19\mu \mathrm{m}$ and a center-to-center separation of 4.$38\mu \mathrm{m}$. The image of the target is used to determine a measured magnification of $M=19.7\pm 0.4$, where the error is due to our uncertainty in measuring the periodicity in the test target image. All images obtained with the offline imaging system were taken with a Point Grey Firefly MV CMOS camera with $6\times 6\text{−}\mu \mathrm{m}$ pixels.

Figure 2

Raw 60-$\mu \mathrm{m}$-wide images (right panels) of a section of silica nanofiber with a diameter of $\sim 500$ nm (vertically oriented in each image in the panels on the right), shown without the use of background subtraction or other signal-enhancing techniques; 660-nm imaging light was used for all images. Panels on the left show horizontal $(x$-direction) cross sections through each corresponding image on the right along the white line superimposed on the images; image intensity $I\left(x\right)$ is plotted vs $x$ (arbitrary units are the same for each cross section). (a)–(c) Dark-field images taken with a 2.5-ms exposure and using masks with diameters of $100\mu \text{m},370\mu \text{m}$, and $\sim 1.5$ mm respectively. Circular masks were used for (a) and (c), whereas a wire mask, aligned approximately parallel to the fiber, was used for (b). (d) Bright-field image of the nanofiber with no mask in place, taken with a 0.25-ms exposure. See text for a discussion of the calculated relative signal ratio (RSR) for the cross sections [23].

Figure 3

Images of expanded BECs with a vortex lattice, using an $M=5$, NA = 0.2 imaging system. (a)–(c) show a 135-$\mu \mathrm{m}$-wide field of view, and (d) and (e) show a 200-$\mu \mathrm{m}$-wide field of view. For each image, the BEC was released from the trap and allowed to expand for a variable time $t_{\exp }$ shown on the image. (a)–(d) Raw dark-field images of an expanded BEC with a vortex lattice taken at varying expansion times, with no background subtraction. A circular mask with diameter of $\sim 1.6$ mm was used for all dark-field images. The lattice becomes resolvable between $t_{\exp }$ = 22 and 32 ms. (e) Reference absorption image of an expanded BEC with a vortex lattice, obtained using standard methods of bright-field imaging with background subtraction and grayscale contrast inversion.

Figure 4

(a)–(f) 80-$\mu \mathrm{m}$-wide in situ images of BECs obtained with our $M\sim 20$, NA = 0.25 imaging system. Dark-field images of the BEC are shown (a) without a vortex lattice and (b) with a vortex lattice. The wire mask used for both images had a diameter of $370\mu \text{m}$, and was aligned horizontally with respect to the image. (c) Dark-field image of a BEC with a vortex lattice, but with a 250-$\mu \mathrm{m}$-diameter wire mask; vortices are visible within the superimposed white rectangle, but not as apparent as in (b), as additional refracted light from the BEC reaches the camera. (d) Dark-field image obtained with a rotating BEC with a vortex lattice when the imaging system is not properly focused. All dark-field images have been processed by subtraction of a background image taken in the absence of a BEC. Reference in situ absorption images are shown for (e) a nonrotating BEC and (f) a rotating BEC that is expected to contain vortices. Bright-field images were obtained using standard methods of bright-field imaging with background subtraction and grayscale contrast inversion. (g) Magnified view of the region bounded by the white rectangle in (b), with pixelation due to the $13\times 13\text{−}\mu \mathrm{m}$ camera pixels. Neighboring vortex cores are separated by $a\sim 9$ $\mu \mathrm{m}$. (h) Cross section along the middle row of vortex cores shown in (e); the vertical scale is proportional to image intensity per pixel, calibrated to number of photons; the horizontal plot axis shows the distance $\Delta x$ away from the central vortex core, with the scale corresponding to real distances at the object plane. The FWHM of the central vortex core is $\delta =2.4\pm 0.5\mu \mathrm{m}$. The FWHM was found by fitting a Gaussian to the intensity profile, and the reported error is due to the uncertainty from the fit.

Figure 5

Basic imaging system with the probe beam shown in gray (not to scale). Horizontal dashed lines indicate light refracted by the bulk BEC into low spatial frequencies. Solid lines indicate light refracted by vortex cores into high spatial frequencies.

Figure 6

Detected signal, in the number of photons expected to fall on a $16\times 16\text{−}\mu \mathrm{m}$ region centered at a location $r^{\prime }$ away from the center of the image. The calculated signals due to a BEC with a vortex (filled circles) and one without a vortex (open squares) are shown. The parameters used in this calculation are those described in the text.