Measuring topology in a synthetic dimensions lattice
Topology in 2-D materials is important and cool; it explains the quantum Hall effect, and in the limit of really high magnetic fluxes (of order 10^4 Tesla for crystalline materials) gives rise to the fractal Hofstadter butterfly. These high fluxes are inaccessible in traditional condensed matter settings, but we engineered them in our effective 2-D lattice of ultracold 87Rb atoms. We created this lattice using a synthetic dimensions approach: a real 1-D lattice defined sites along the first, 'real', dimension, while the internal spin states of the atoms served as sites along a second, 'synthetic', dimension. We then took advantage of the hybrid imaging that occurs in this lattice during time of flight: momentum is measured along the 'real' axis, while position is measured along the 'synthetic' axis with single site resolution. This allowed us to map out the position of the atoms in the synthetic direction for every point in the lowest band, i.e. for every value of the real axis crystal momentum. We then levered a Diophantine equation derived by Thouless, Kohomoto, Nightingale, and den Nijs to extract the topological invariant of our system, in 2-D called the Chern number, and provide an intuitive picture of how this equation arises in our system.