  Quantum Hall system possesses topologically protected edge states which have
enormous theoretical and practical implications in both fermionic and bosonic
systems. Harnessing the quantum Hall effect in optical platforms with lower
dimensionality is highly desirable with synthetic dimensions and has attracted
broad interests in the photonics society. Here, we introduce an alternative way
to realize the artificial magnetic field in a frequency dimension, which is
achieved in a pump-probe configuration with cross-phase modulations in a
one-dimensional four waveguide array. The dynamics of the topological chiral
edge state has been studied and the influence from the crosstalk of the pump
fields has been explored. Our work shows an all-optical way to simulate the
quantum Hall system in a photonic system and holds potential applications in
manipulating light in waveguide systems.
