Fig. 3: Physical mechanism of the breathing dynamics of soliton crystals (SCs).

a Schematic of the 1-defect breather with a breathing period of ≈1080 roundtrips, during which the soliton on one side of the defect moves to the other side. Left inset: The intracavity positions of the solitons near the defect are aligned and superimposed over multiple roundtrips to better illustrate the spatial motion. Right inset: the experimental RF spectrum with a breathing frequency close to the breathing frequency extracted from the cumulative intensity variation of the simulated crystal. b The experimental OSA spectrum of a 1-defect breather with the strong comb lines indicating crystal spacing and structure marked in yellow. The pump and AMX positions are also shown. c The simulated temporal waveform extracted from (a) after removing the corresponding prominent lines in the spectrum of 1-defect breather. Inset: the phase relation between three pulses. d The experimental time-trace of breathing intensity measured by a high-speed PD and oscilloscope, shown in black. Corresponding simulations of the breathing time trace closely match the experiment and are plotted with and without noise in yellow and magenta respectively. e The interaction plane limit cycle of the 1-defect breather plotted with and without noise in yellow and magenta respectively corresponding to the plots in d. Inset: Simulated interaction plane limit cycle of all solitons in the crystal. f The breathing frequency as a function of the third-order dispersion D₃ at different detunings. The shaded area is the experimentally accessible region, with the results closely matched by simulations. The gray star indicates the experimental conditions in (d–e). g The breathing frequency as a function of the AMX strength and pump power. The gray star indicates the conditions in (d–e). Due to thermal dependence of the AMX strength, most area in (g) is experimentally accessible by varying the detuning or chip temperature