In the quantum Hall effect conductivity is quantized, having either integer or certain fractional values. The integer values are topological quantum numbers — also known in mathematics as Chern numbers — that characterize different topological insulating phases. This well-known phenomenon is now also being explored in the physics of ultracold atoms, where the quantum Hall effect can be reproduced in a Fermi gas trapped in a two-dimensional optical lattice.
Topological phases can be explored in ultracold atom experiments using artificial magnetic fields. To characterize the phases, Alexandre Dauphin and Nathan Goldman have proposed a method of extracting the Chern numbers from the time evolution of the centre of mass of the atomic cloud: the ultracold gas is initially prepared in a topological phase and is driven by a constant force; tracking the dynamics of the atomic gas through its sudden release from the trapping potential then reveals information about the topological order.
The idea is a general one, so in principle it could be applied to any cold atom experiment and would also mean that different higher-order topological phases could be distinguished.
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Georgescu, I. What's in a number?. Nature Phys 9, 692 (2013). https://doi.org/10.1038/nphys2809