The devil’s staircase of breathing soliton molecules

• First demonstration on devil's staircase in breathing soliton molecule mode-locked laser is reported. • The staircases in breathing soliton molecules have winding number with larger denominator, enabling denser frequency comb generation. • Lag synchronization is revealed to induce frequency-locking in interacting breathing solitons for the first time. • Breathing soliton molecules show stronger chaotic tendency than single breathing solitons, providing a new route to optical chaos. • The study confirms the universality of fractal dynamics across different pulse regimes. The devil’s staircase is a particularly fascinating synchronization phenomenon widely presented in science and engineering, manifesting as an infinite number of synchronized state when tuning parameters. Optical breathing soliton oscillator has attracted significant interest in studying devil’s staircase phenomena within a single cavity. However, the devil’s staircase of multiple breathing solitons under interaction is remained unexplored. Here, we study the synchronization dynamics of breathing soliton molecules under soliton interaction, and present the first evidence of devil’s staircase in breathing soliton molecules within a mode-locked fiber laser. The devil’s staircase of breathing soliton molecule exhibits a larger winding number than a single breathing soliton, which is of great significance to densify the frequency comb. Interestingly, the devil’s staircase of breathing soliton molecules exhibits a stronger tendency toward chaotic behavior, which may indicate a new method to access a chaotic state. Supported by numerical simulations, our conclusions lay a foundation for future studies on fractal, synchronization, and chaos dynamics in the regime of interactional breathing solitons.