Fig. 1. Schematic diagram of the system. (a) Two detuned and self-sustained optical microcavities with different resonant frequencies, ω10 and ω20, which are directly coupled at strength g. (b)–(d) Frequency spectra of the coupled cavities, showing three different long-term states: unsynchronized, limit cycle (LC), and synchronized (Sync.). Light blue represents the noise backgrounds from which the first- and second-order synchronizations are distinguished.

Fig. 2. Long-term evolutions of the two cavity modes under different coupling strengths. Three different categories are shown: (a) the unsynchronized (g˜=0.3), (b) limit cycle (g˜=0.398), and (c) synchronized states (g˜=0.4). (a1)–(c1) Phase difference; (a2)–(c2) transient frequencies; (a3)–(c3) trajectory encircling types (black cross as the axis); and (a4)–(c4) dynamical potential near the synchrony point. In all figures, the given detuning Δ˜=0.3 and Kerr factor δ˜=0.1.

Fig. 3. Parameter dependence of the synchronization. (a), (b) Maximum of the frequency differences, max|ω1−ω2|, versus the coupling strength g˜, with (Δ˜=0.2, δ˜=0.1) in (a) and (Δ˜=0.3, δ˜=0.1) in (b); inset shows the derivative. (c) Phase diagram in the (Δ˜,g˜) plane with the Kerr factor δ˜=0.1. The inaccessible (gray), limit cycle (dark blue), and synchronized (light blue) regimes are marked. The red cross stands for the triple phase point (Δ˜T,g˜T). (d) The triple phase point (Δ˜T,g˜T) depending on the Kerr factor δ˜.

Fig. 4. Hysteresis behavior in frequency difference. (a), (b) Frequency differences |ω1−ω2| versus the evolution time τ in the first- and second-order transition regimes. Insets: the real-time evolution of the coupling strength g˜(τ). (c), (d) Maxima of the frequency differences, max|ω1−ω2| versus g˜(τ). For each plot, the Kerr factor δ˜=0.1; the detuning Δ˜=0.2 in (a) and (c), and Δ˜=0.3 in (b) and (d).