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The engineering of synthetic dimensions allows for the construction offictitious lattice structures by coupling the discrete degrees of freedom of aphysical system, such as the quantized modes of an electromagnetic cavity orthe internal states of an atom. This method enables the study of static anddynamical Bloch band properties in the absence of a real periodic latticestructure. So far, the vast majority of implementations have focused on linearand conservative processes, with the potentially rich physics and opportunitiesoffered by nonlinearities and dissipation remaining largely unexplored. Here,we theoretically and experimentally investigate the complex interplay betweenBloch band transport, nonlinearity, and dissipation, exploring how a syntheticdimension realised in the frequency space of a coherently-driven opticalresonator influences the dynamics of nonlinear waves of the system. Inparticular, we observe and study nonlinear dissipative Bloch oscillationsoccurring along the synthetic frequency dimension, sustained by localizeddissipative structures (solitons) that persist endlessly in the resonator. Theunique properties of the dissipative soliton states can extend the effectivesize of the synthetic dimension far beyond that achieved in the linear regime,as well as enable long-lived Bloch oscillations and high-resolution probing ofthe underlying band structure. Besides representing the first experimentalstudy of the interplay between Bloch oscillations and dissipative solitons, ourwork establishes Kerr resonators as an ideal platform for the study ofnonlinear dynamics in long-scale synthetic dimensions, with promisingapplications in topological photonics.

Bloch Oscillations of Driven Dissipative Solitons in a Synthetic Dimension