  Quasiperiodic lattices have recently been shown to be a non-trivial
topological phase of matter. Charge pumping -- one of the hallmarks of
topological states of matter -- was recently realized for photons in a
one-dimensional (1D) off-diagonal Harper model implemented in a photonic
waveguide array. The topologically nontrivial 1D Fibonacci quasicrystal (QC) is
expected to facilitate a similar phenomenon, but its discrete nature and lack
of pumping parameter hinder the experimental study of such topological effects.
In this work we overcome these obstacles by utilizing a family of topologically
equivalent QCs which ranges from the Fibonacci QC to the Harper model.
Implemented in photonic waveguide arrays, we observe the topological properties
of this family, and perform a topological pumping of photons across a Fibonacci
QC.
