Vortex Reconnections and Rebounds in Trapped Atomic Bose-Einstein Condensates

Reconnections and interactions of filamentary coherent structures play a fundamental role in the dynamics of fluids, redistributing energy and helicity among the length scales and inducing fine-scale turbulent mixing. Unlike ordinary fluids, where vorticity is a continuous field, in quantum fluids vorticity is concentrated into discrete (quantized) vortex lines turning vortex reconnections into isolated events, making it conceptually easier to study. Here, we report experimental and numerical observations of three-dimensional quantum vortex interactions in a cigar-shaped atomic Bose-Einstein condensate. In addition to standard reconnections, already numerically and experimentally observed in homogeneous systems away from boundaries, we show that double reconnections, rebounds, and ejections can also occur as a consequence of the nonhomogeneous, confined nature of the system.

Popular Summary

Long, almost one-dimensional structures known as filaments are ubiquitous. Chains of atoms such as DNA molecules, ropelike magnetic fields in electrically conducting plasmas, strings in liquid crystals, and narrow beams of light all exhibit filamentary behavior. As these filaments evolve, they interact with one another and may reconnect, exchanging neighboring strands and modifying the structure of the underlying field. These interactions play a key role in the dynamics of many types of fluids. Cigar-shaped Bose-Einstein condensates (BECs)—a type of quantum fluid cooled to nearly absolute zero—offer a useful platform for studying filament interactions. Because the circulation of the fluid is quantized, reconnections are isolated dramatic events that are easier to study than in other systems.

We have developed an innovative experimental imaging technique that allows us to track quantum vortices with unprecedented resolution, providing direct evidence of vortex reconnections in quantum fluids. Combining experiments and numerical modeling, we unambiguously identify novel vortex interactions—such as rebounds, double reconnections, and ejections—arising from the confined and inhomogeneous nature of trapped atomic BECs. These newly identified interactions go beyond the simple standard reconnections already observed in superfluid helium, and they are likely to be important in the dynamics of turbulent BECs, where more than two vortices are present.

Our imaging technique sets up BECs as an ideal experimental platform for testing the recent suggestion that some properties of filament reconnections may be universal in nature. The processes that we identify could be a key starting point for better understanding the behavior of superfluids near their boundaries.

Figure 1

Sketch of an imaging sequence. A trapped condensate (smaller, light blue ellipsoid) contains a transverse vortex line that moves and rotates around the trap center; the direction of the atomic flow around the vortex filament is indicated by the yellow arrow. A small fraction of atoms is repeatedly extracted, typically every 12 ms; these atoms expand and fall in the gravity field, and are imaged in absorption by a probe laser beam after they are spatially separated from the trapped condensate. Each absorption image contains the essential features associated with the vortex lines.

Figure 2

(a) Examples of absorption images of the outcoupled atoms (the same as in Fig. 1). The vortex axial position is clearly visible. (b) After integrating radially and fitting the absorption images, we determine the residuals, which exhibit minima (pink) and maxima (green) due to interference effects among atoms that are outcoupled from the trapped condensate at different places and times. (c) Full temporal sequence of residuals for a given condensate, showing the real-time evolution of a vortex which moves axially and rotates around the $x$ axis, from an initial orientation along $y$ (green-pink) at $t_1$ to an orientation along $z$ (green-pink-green) in $t_2$ and then along $-y$ (pink-green) at $t_3$. The relation between the shape of the residuals and the orientation of the vortex is extracted from numerical simulations.

Figure 3

Examples of different interaction mechanisms observed in the case of two approaching vortices. Each temporal sequence is shown twice with two different color palettes; the red palette enhances the contrast, so that also vortices close to the edges can be seen, whereas the pink-green palette better illustrates the vortex orientation in the radial plane. (a),(b) Vortices approach and bounce back, (c) their axial trajectories intersect preserving visibility and orientation, (d) they cross, producing sudden changes of visibility, and (e),(f) the visibility of one vortex is almost completely lost after interacting with the other.

Figure 4

The first six columns show radial and axial snapshots from the GP simulations of two interacting vortex lines. On the right, the axial coordinate $x$ (in units of $R_x$) of the center of vorticity of each vortex is plotted versus normalized time $\tau =t/T_0$. Initial line colors (red and blue) help identify vortices in the snapshots until they reconnect. After the first reconnection, line colors switch to orange and green and again to red and blue if a second reconnection occurs. Line transparency indicates how visible vortices are expected to be, given their orbit amplitude (see Appendix B for further details on line transparency). The gray region highlights the interaction interval where the minimum distance between the vortices is smaller than $R_{\perp }$. (a)–(c) Perpendicular vortices are imprinted on opposite radial planes with corresponding orbit parameters $\chi =0.22$, 0.25, 0.375, respectively: (a) illustrates a vortex rebound, (b) shows the double reconnection interaction, with reconnections occurring at $\tau =0.208$ and $\tau =0.221$ (see Appendix pp2 for a zoom on the double reconnection event), and (c) depicts a single reconnection occurring at $\tau =0.179$, with the consequent triggering of Kelvin waves. Panel (d) illustrates a nonsymmetrical reconnection (at $\tau =0.130$) between a vortex imprinted on the central plane of the condensate through its center (blue) and a vortex (red) imprinted orthogonally to the first one with a large orbit parameter $\chi =0.7$. One of the reconnected vortices lies on an even wider orbit (larger $\chi$), where the BEC density is lower and its visibility becomes consequently greatly reduced. Panel (e) describes the orbiting dynamics between two parallel vortices imprinted on different orbits ($\chi =0.33$, 0.5). Notice that in (a)–(c) the first snapshot corresponds to $\tau =0$, whereas in (d) and (f) the snapshots are all later in time.

Figure 5

Statistical analysis of experimental observations. (a) Occurrence of rebound events (purple) as a function of the vortex-vortex relative velocity, within the ensemble of all collision events (gray). The velocity $v_{\mathrm{rel}}$ is normalized to the speed of sound $c$ evaluated at the center of the BEC. The inset shows the relative occurrence for each bin. (b) Fraction of rebound events as a function of the relative angle just before the interaction. (c) Occurrence of events (green) in which one vortex line disappears after the interaction, as a function of the largest orbit parameter of the vortex pair ${\chi }_{\max }$, i.e., the amplitude of the outer vortex orbit in the BEC. The inset shows the relative occurrence per bin.

Figure 6

Schematic picture of the outcoupling technique. In the upper part of the figure, energy levels are reported as a function of the vertical coordinate. In the lower part, the modulus of the trapping magnetic field is reported. The sketched outcoupled atoms in red and cyan are not to scale.

Figure 7

Temporal sequence of radial (top) and axial (bottom) snapshots from GP simulations showing in detail two interacting vortex lines undergoing a double reconnection. The snapshots refer to the numerical simulation reported in Fig. 4(b), here illustrated employing a finer temporal resolution.

Figure 8

Residual phase $\delta$ of the extracted portion, after a 13-ms expansion, for various in situ vortex orientations $\theta$, obtained by using the function [Eq. (c1)] to fit the shape of the residuals in GP simulations. These numerical results are for a solitonic vortex stationary state that passes through the condensate center; i.e., even though the vortex is considered for numerous orientations, it is always a straight line that remains in the $x=0$ plane. The (+) symbols are the data points while the solid line guides the eye. For reference, the vertical dashed lines indicate examples of the in situ vortex orientation as indicated by the lower arrows.