  When a two-dimensional electron gas is exposed to a perpendicular magnetic
field and an in-plane electric field, its conductance becomes quantized in the
transverse in-plane direction: this is known as the quantum Hall (QH) effect.
This effect is a result of the nontrivial topology of the system's electronic
band structure, where an integer topological invariant known as the first Chern
number leads to the quantization of the Hall conductance. Interestingly, it was
shown that the QH effect can be generalized mathematically to four spatial
dimensions (4D), but this effect has never been realized for the obvious reason
that experimental systems are bound to three spatial dimensions. In this work,
we harness the high tunability and control offered by photonic waveguide arrays
to experimentally realize a dynamically-generated 4D QH system using a 2D array
of coupled optical waveguides. The inter-waveguide separation is constructed
such that the propagation of light along the device samples over
higher-dimensional momenta in the directions orthogonal to the two physical
dimensions, thus realizing a 2D topological pump. As a result, the device's
band structure is associated with 4D topological invariants known as second
Chern numbers which support a quantized bulk Hall response with a 4D symmetry.
In a finite-sized system, the 4D topological bulk response is carried by
localized edges modes that cross the sample as a function of of the modulated
auxiliary momenta. We directly observe this crossing through photon pumping
from edge-to-edge and corner-to-corner of our system. These are equivalent to
the pumping of charge across a 4D system from one 3D hypersurface to the
opposite one and from one 2D hyperedge to another, and serve as first
experimental realization of higher-dimensional topological physics.
