  We propose a realistic scheme to detect the 4D quantum Hall effect using
ultracold atoms. Based on contemporary technology, motion along a synthetic
fourth dimension can be accomplished through controlled transitions between
internal states of atoms arranged in a 3D optical lattice. From a
semi-classical analysis, we identify the linear and non-linear quantized
current responses of our 4D model, relating these to the topology of the Bloch
bands. We then propose experimental protocols, based on current or
center-of-mass-drift measurements, to extract the topological 2nd Chern number.
Our proposal sets the stage for the exploration of novel topological phases in
higher dimensions.

Quantum Gases