Observation of ultrametricity in photonic spin glasses

Hierarchy is the architecture of complex systems [1]. 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 [2]. 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 [3]. Despite the importance, a real experimental observation of the universal behaviour has not yet been reported, due to the difficulty of finding a physical system that displays all the characteristics of spin glasses. Random lasers have recently been proposed as excellent photonic counterparts of spin glasses with robust replica symmetry breaking features [4–6]. Here, we report the ultrametric structure of the replica space with clustered domains in random lasers, as predicted in the Parisi Ansatz [7]. We show that the number of the 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. Our results, as well as being a milestone in complexity science, fuel the lively research into physical systems which in recent years has been leading to the creation of optical computers and quantum annealers [8–12].