  Topological phases feature robust edge states that are protected against the
effects of defects and disorder. The robustness of these states presents
opportunities to design technologies that are tolerant to fabrication errors
and resilient to environmental fluctuations. While most topological phases rely
on conservative, or Hermitian, couplings, recent theoretical efforts have
combined conservative and dissipative couplings to propose new topological
phases for ultracold atoms and for photonics. However, the topological phases
that arise due to purely dissipative couplings remain largely unexplored. Here
we realize dissipatively coupled versions of two prominent topological models,
the Su-Schrieffer-Heeger (SSH) model and the Harper-Hofstadter (HH) model, in
the synthetic dimensions of a time-multiplexed photonic resonator network. We
observe the topological edge state of the SSH and HH models, measure the SSH
model's band structure, and induce a topological phase transition between the
SSH model's trivial and topological phases. In stark contrast with
conservatively coupled topological phases, the topological phases of our
network arise from bands of dissipation rates that possess nontrivial
topological invariants, and the edge states of these topological phases exhibit
isolated dissipation rates that occur in the gaps between the bulk dissipation
bands. Our results showcase the ability of dissipative couplings to enable
time-reversal symmetry broken topological phases with nonzero Chern numbers,
which have proven challenging to realize in the optical domain. Moreover, our
time-multiplexed network, with its ability to implement multiple synthetic
dimensions, dynamic and inhomogeneous couplings, and time-reversal symmetry
breaking synthetic gauge fields, offers a flexible and scalable architecture
for future work in synthetic dimensions.
