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My current research concerns exploring the dynamical and topological properties of quasicrystals (a novel form of condensed matter that is non-periodic, but long range ordered) in the novel context of optical lattices.
In particular, I have been studying coherent quantum transport phenomena in a 2D arrangement with five-fold rotational symmetry. While a general theoretical understanding is made difficult due to the lack of periodicity, a simplification can be made by considering the weak-coupling limit. In this limit, we've shown that the dynamics can be closely approximated by the semiclassical equations of motion and have also found the surprising appearance of a non-Abelian Berry connection at the corner of the so-called `pseudo-Brillouin-zone'.
Future directions of this work include exploring ways of breaking time-reversal symmetry and applying the semiclassical dynamics approach to quantum oscillations in incommensurately modulated electronic materials. The former direction is closely related to the generation of artificial gauge fields for ultracold atomic gases, with approaches including optical flux lattices and time-periodic shaking.
In Plain English
After the discovery of X-ray diffraction it was generally considered that ordered materials (those which displayed sharp Bragg peaks in their diffraction patterns) were periodic and a simple theorem known as the `crystallographic restriction theorem' ensured that the only allowed rotational symmetries of a periodic structure are 1-,2-,3-,4- and 6-fold. Despite this, in 1983, Dan Schectman discovered that the X-ray diffraction pattern of an aluminium alloy remarkably displayed both sharp Bragg peaks but with a crystallographically disallowed 5-fold rotational symmetry. These and similar materials were soon termed `quasicrystals', and their structure was found to be related to aperiodic tilings such as the well-known Penrose tiling.
In recent years there has been a growing interest in the `simulation' of models from condensed matter by using systems of so-called `ultracold atomic gases'. Here the atoms play the role of electrons and an `optical lattice' potential created from the interference of overlapping laser beams, models the background ionic lattice. The key advantage of these systems is that the optical lattice is highly controllable and free from defects that can obscure various quantum phenomena. By simply arranging the lasers to have a rotational symmetry that is disallowed for a periodic structure one can then extend the control available in cold-atom systems to simulate quasicrystals. In my research I have been exploring novel theoretical questions that were previously less relevant in condensed matter quasicrystals (and therefore remained uninvestigated), but which are now highly relevant for their optical lattice analogues.