Quantum Honeycomb

Gary Ruben, School of Physics
Supervisors: David M. Paganin and Michael J. Morgan

Introduction

A new structure resembling the packed hexagonal form of honeycomb has been predicted and observed [1] in computer models of groups of atoms that are cooled down to a point where they lose their individual identities – a state of matter known as a Bose-Einstein condensate (BEC). The honeycomb in this case is built not from wax but from a regular arrangement of spinning vortices that form holes in the pancake-shaped cloud of atoms. The wavelike behaviour of such BEC clouds has previously been tested by splitting them into two parts and allowing the parts to collide, in experiments that are analogous to Thomas Young's famous 200 year-old experiment with light. Prompted by previous work on the Young's 3-pinhole interferometer [2], in this research we have shown that if the cloud is instead split into three parts, the pattern of dark stripes observed by Young and previous BEC researchers changes dramatically into the lattice of swirling vortices.
Vortices in BECs differ from the kind one normally thinks of, such as hurricanes or water flowing down a sink. The difference lies in the importance of quantum mechanics to their description – in fact they can only rotate at certain defined or quantised speeds, in the same way that sub-atomic particles and their constituents can only acquire energies in quantised amounts. The study of vortices provides insight into Bose-Einstein condensation and analogous condensed matter systems such as cosmological models and superconductors.
The modelled BEC is about 50 μm in diameter, or the width of a human hair. This means that, with suitable optics, just as with a microscope, one can hope to observe and film their behaviour experimentally.

Research question / problem

In a recent experiment by Scherer et al. [3], which has already been numerically modelled [4], vortices were observed as a result of the interference of a three-way segmented BEC, formed by shining a laser light sheet on a pancake-shaped condensate.  However, no lattice was reported in the resulting experimental images, prompting us to model a similar system to demonstrate formation of the predicted lattice. 
Why was no lattice observed experimentally? Our work suggests that if the pieces are not sufficiently well separated due to an insufficiently bright or poorly-focused laser light sheet, the BEC “pushes” the lattice out of the observable region to the dim outer parts of the cloud. We believe that improved optics should help with these problems and have also proposed an experimental modification whereby the transverse magnetic trap component could be removed during merging to allow the lattice and vortex cores to expand, easing their observation. As the vortices are likely to be unstable within a spheroidal cloud, another reason may be that the experimental BEC cloud was not sufficiently flat.

A new type of lattice structure

To date, only one type of structured lattice of vortices has been predicted to exist in BECs. This "Abrikosov lattice" [5] has hexagonal or triangular symmetry and is experimentally produced by spinning a BEC cloud. In the three-piece BEC system, the lattice instead has honeycomb symmetry and is produced without rotating the cloud. Consequently, equal numbers of clockwise and counter-clockwise spinning vortices are produced, whereas the vortices in the Abrikosov lattice rotate in only one direction.

A new mechanism for lattice formation

To date, two mechanisms have been described for the generation of vortices in BECs; the Kibble-Zurek mechanism [6] by which vortices are created from the reconciliation of initial random phases in the BEC, and the "snake instability" of dark stripe solitons [7]. We have described a new type of vortex creation mechanism for BECs which arises naturally from the presence of three expanding sources – the sources in this case being the initial BEC pieces. The generation of a vortex lattice by the 3-pinhole Young's interferometer [2] prompted the study of this analogous BEC system.

Vortex dynamics

For BECs to form requires them to be magnetically confined. We have shown that this confinement leads the lattice to melt as the vortices interact chaotically, clustering into structures that migrate and scatter throughout the cloud. Previous studies of the static behaviour of vortex clusters [8], the dynamic behaviour of vortex-antivortex dipoles, and of vortex-sound interactions are seen in a new light when viewed as movies of interaction dynamics.

Fig. 1. Three-segment Bose-Einstein condensate interference within a harmonic confining trap showing progression from the ground state, through lattice formation to a late-stage characterised by complex vortex–antivortex (VA) dynamics. (Top) The initial BEC state formed in a harmonic trap divided into three parts by a laser light-sheet of intensity kB×180 nK. (Second from top) Two-piece interference fringes between pairs of pieces. (Third frame) Honeycomb vortex-antivortex lattice. (Bottom frame) Complex vortex-antivortex-sound dynamics following melting of the lattice due to harmonic trap confinement.

Videos

Each video frame represents 1.08ms of real time. At 15 fps, the video is approximately 1/62 of the real time evolution rate. The square simulation region is 147 um along each side.

This movie corresponds to Fig. 3(a) in the paper [1]. It shows the probability density of the harmonically trapped, three-segment Bose-Einstein condensate following removal of a light sheet of intensity I_0=0.8.

This movie corresponds to Fig. 5(b) in the paper [1]. It shows the probability density of the harmonically trapped, three-segment Bose-Einstein condensate following removal of a light sheet of intensity I_0=0.3.

This movie is not represented in the paper [1]. It shows the probability density of a harmonically trapped, two-segment Bose-Einstein condensate following removal of a light sheet of intensity I_0=0.8.

References

[1] G. Ruben, D. M. Paganin and M. J. Morgan, Vortex-lattice formation and melting in a nonrotating Bose-Einstein condensate, Phys. Rev. A 78, 013631 (2008).
[2] G. Ruben and D. M. Paganin, Phase vortices from a Young’s three-pinhole interferometer, Phys. Rev. E 75, 066613 (2007).
[3] D. R. Scherer, C. N. Weiler, T. W. Neely, and B. P. Anderson, Phys. Rev. Lett. 98, 110402 (2007).
[4] R. Carretero-González, B. P. Anderson, P. G. Kevrekidis, D. J. Frantzeskakis, and C. N. Weiler, Phys. Rev. A 77, 033625 (2008).
[5] A. A. Abrikosov, Sov. Phys. JETP 5, 1174 (1957).
[6] J. R. Anglin and W. H. Zurek, Phys. Rev. Lett. 83, 1707 (1999).
[7] D. L. Feder, M. S. Pindzola, L. A. Collins, B. I. Schneider, and C. W. Clark, Phys. Rev. A 62, 53606 (2000).
[8] M. Möttönen, S. M. M. Virtanen, T. Isoshima, and M. M. Salomaa, Phys. Rev. A 71, 033626 (2005).