Hierarchical self-organization of highly-ordered granular ensemble of optical solitons through collective motions

Self-organizations of ordered patterns in far-from-equilibrium many-body systems host fundamental importance in many disciplines. Meanwhile, complex systems often feature hierarchical structures with distinct scales for different layers, enabling high-level effective dynamics without exhaustive tracking of all possible degrees of freedoms. In this work, we report a study of the self-organization dynamics of highly-ordered soliton ensembles in a high-harmonic mode-locked fiber lasers through collective motions driven by nonlocal optomechanical interactions and local collisions, which exhibit a series of universal characteristics reminiscent of phase transitions. Moreover, the multi-soliton laser-field can be coarsely grained as a granular ensemble of limit-cycle oscillators with simple interaction rules derived from fine-scale physics. The self-organization of the multitude of solitons in the mode-locked laser cavity can then be mapped into a low-dimensional dynamic model that essentially reproduced the emergent process. Our work affords a conceptual framework for understanding the complex structure formation in nonlinear laser systems, and may help to design ultrafast lasers by exploiting universal principles of collective motions.