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Topology is central to phenomena that arise in a variety of fields, ranging from quantum field theory to quantum information science to condensed matter physics. Recently, the study of topology has been extended to open systems, leading to a plethora of intriguing effects such as topological lasing, exceptional surfaces, as well as non-Hermitian bulk-boundary correspondence. Here, we show that Bloch eigenstates associated with lattices with dissipatively coupled elements exhibit geometric properties that cannot be described via scalar Berry phases, in sharp contrast to conservative Hamiltonians with non-degenerate energy levels. This unusual behavior can be attributed to the significant population exchanges among the corresponding dissipation bands of such lattices. Using a one-dimensional example, we show both theoretically and experimentally that such population exchanges can manifest themselves via matrix-valued operators in the corresponding Bloch dynamics. In two-dimensional lattices, such matrix-valued operators can form non-commuting pairs and lead to non-Abelian dynamics, as confirmed by our numerical simulations. Our results point to new ways in which the combined effect of topology and engineered dissipation can lead to non-Abelian topological phenomena.

Topological phases feature robust edge states that are protected against the effects of defects and disorder. These phases have largely been studied in conservatively coupled systems, in which non-trivial topological invariants arise in the energy or frequency bands of a system. Here we show that, in dissipatively coupled systems, non-trivial topological invariants can emerge purely in a system’s dissipation. Using a highly scalable and easily reconfigurable time-multiplexed photonic resonator network, we experimentally demonstrate one- and two-dimensional lattices that host robust topological edge states with isolated dissipation rates, measure a dissipation spectrum that possesses a non-trivial topological invariant, and demonst rate topological protection of the network’s quality factor. The topologically non-trivial dissipation of our system exposes new opportunities to engineer dissipation in both classical and quantum systems. Moreover, our experimental platform’s straightforward scaling to higher dimensions and its ability to implement inhomogeneous, non-reciprocal and long range couplings may enable future work in the study of synthetic dimensions.