Fig. 1. (a) Scheme of VPC waveguides supporting on-chip topological transport of OFCs. (b) Edge dispersion of the VPCs, where the yellow region denotes the operation bandwidth of the valley kink state. Right panel: Hz field distributions for the topological edge state. SEM images of the (c) straight and (d) Z-shaped topological waveguides. (e) Measured transmission spectra of the straight (orange curve) and Z-shaped topological waveguides (green curve).

Fig. 2. (a) Experimental setup for topological transport of OFCs. To generate QFCs, the pump laser is actively tuned by a proportional-integral-differential (PID) controller, while the auxiliary laser is not used. To generate DKSs, both lasers are utilized to pump the resonator. EDFA, erbium-doped fiber amplifier; FPC, fiber polarization controller; CIRC, optical circulator; OPM, programable optical power meter; FBG, fiber Bragg grating; OSA, optical spectrum analyzer; ESA, electrical spectrum analyzer; TBPF, tunable bandpass filter; SPD, single-photon detector. (b) Experimental setup for the DKS spectrum measurements. (c) Experimental setup for time correlations and JSI measurements of QFCs. (d) Experimental setup for RF beat notes of the single-soliton states and a CW reference laser.

Fig. 3. (a)–(c) Measured single-photon spectra and (d)–(f) signal-idler coincidence histograms of the QFCs at the outputs of the original micro-resonator, straight, and Z-shaped topological waveguides, respectively. The pink, purple, and yellow marked regions denote the selected signal modes, idler modes, and several modes eliminated by the FBG, respectively.

Fig. 4. Quantum properties of the QFCs detected at the outputs of the micro-resonator, straight, and Z-shaped topological waveguides. (a)–(c) Measured JSI distributions. (d)–(f) Distributions of normalized Schmidt coefficients λn and entanglement entropy Sk.

Fig. 5. Measured optical spectra of DKS combs at the outputs of the original micro-resonator, straight, and Z-shaped topological waveguides. (a)–(c) Single-soliton states. (d)–(f) Multisoliton states. (g)–(i) Perfect soliton crystals.

Fig. 6. (a) Band diagram of the VPC slab with inversion symmetry (red-dotted curves) compared with inversion symmetry breaking (black-diamond curves), where Γ, K, and M denote the high-symmetry points in the first Brillouin zone. (b) Simulated transmission spectra of the straight interface (orange curve) and Z-shaped interface (green curve). (c), (d) Simulated field profiles of the valley kink states at the frequency of 193 THz (around 1550 nm) at different interfaces.

Fig. 7. (a) Microscopy image of the Si3N4 micro-ring with W=1.8  μm, H=0.8  μm, θ=89°, and gap width of 0.45 μm. (b) Simulated GVD curves for TE and TM modes, where the inset denotes a diagram of the waveguide cross-section. (c) Simulated mode profiles of TE and TM modes.

Fig. 8. (a) Measured transmission spectrum of the Si3N4 micro-resonator. (b) Dispersions of the micro-resonator extracted from the measured transmission. (c) Lorentzian fitting of the resonant dip.

Fig. 9. Prototype for generating both QFCs and DKS combs.

Fig. 10. (a) Simulated JSI of QFCs for the Si3N4 resonator. (b) Schmidt coefficients λn and (c) entropy of entanglement for the QFC.

Fig. 11. Numerical simulation results of DKS combs evolution.

Fig. 12. (a) Calculated group velocity (vg) and GVD (β2) as a function of angular frequency. (b) Evolution of the single-soliton temporal profile along the propagation distance z.

Fig. 13. RF beat notes of the single-soliton states for the (a) original DKS combs, and DKS combs after the transport of the (b) straight and (c) Z-shaped topological waveguides.