Ultrametricity and energy landscape in photonic spin glasses

Hierarchy is the architecture of complex systems. While Euclidean metric describes homogeneous and static systems, ultrametric geometry is for heterogeneous ensembles, hierarchic relations, and dynamic contexts. Applications spread across all fields: communication technologies, information retrieval, clusterwise regression, analysis, and synthesis of narratives and emotions. The paradigmatic model describing complex systems is the spin glass theory, according to which the topology of the many metastable energy states is intrinsically ultrametric, as a consequence of the breaking of their symmetry. Despite the importance, a real experimental observation of ultrametricity in spin glasses with a large number of spins has not yet been reported, due to the difficulty of finding a physical system in which the order parameter can be easily obtained from direct measurements. Random lasers have recently been proposed as excellent photonic counterparts of spin glasses with robust replica symmetry breaking features. Here, we report the ultrametric structure of the replica space with clustered domains in random lasers, as predicted in the Parisi ansatz. We show that the number of states forming isosceles triangles in the metric space increases through the laser threshold giving rise to the construction of a genealogical tree. Moreover, from the hierarchical topology of the states, we obtain a direct observation of the complex energy landscape with evident breaking of ergodicity in the glassy regime. These results mark a key advance in complexity science and invigorate current efforts to harness physical systems for the realization of optical computers and quantum optimization processors.