  Synthetic dimensions have garnered widespread interest for implementing high
dimensional classical and quantum dynamics on lower dimensional geometries.
Synthetic frequency dimensions, in particular, have been used to experimentally
realize a plethora of bulk physics effects, such as effective gauge potentials,
nontrivial Hermitian as well as non-Hermitian topology, spin-momentum locking,
complex long-range coupling, unidirectional frequency conversion, and
four-dimensional lattices. However, in synthetic frequency dimensions there has
not been any demonstration of boundary effects which are of paramount
importance in topological physics due to the bulk edge correspondence, since
systems exhibiting synthetic frequency dimensions do not support well-defined
sharp boundaries. Here we theoretically elucidate a method to construct
boundaries in the synthetic frequency dimension of dynamically modulated ring
resonators by strongly coupling it to an auxiliary ring, and provide an
experimental demonstration of this method. We experimentally explore various
physics effects associated with the creation of such boundaries in the
synthetic frequency dimension, including confinement of the spectrum of light,
the discretization of the band structure, and the interaction of such
boundaries with the topologically protected one-way chiral modes in a quantum
Hall ladder. The incorporation of boundaries allows us to observe topologically
robust transport of light along the frequency axis, which shows that the
frequency of light can be controlled through topological concepts. Our
demonstration of such sharp boundaries fundamentally expands the capability of
exploring topological physics, and is also of importance for other applications
such as classical and quantum information processing in synthetic frequency
dimensions.
