Integrated optical frequency combs (OFCs) offer a vast number of time and frequency modes, significantly expanding the dimensionality of photonic systems. Due to their distinctive frequency signatures, OFCs have been extensively studied across both classical and quantum domains, encompassing classical dissipative Kerr soliton (DKS) combs and quantum frequency combs (QFCs). Fully coherent DKS combs have wide applications in ultrafast ranging [1–4], optical communications [5], optical spectroscopy [6,7], frequency synthesis [8,9], and optical computing [10,11]. In addition, QFCs operating at the single-photon level can generate high-dimensional quantum states in the frequency domain, enhancing the complexity and scalability of quantum information processing [12–14]. Recent studies have shown that QFCs enable high-dimensional frequency entanglement [15–18], energy-time entanglement [19–23], and time-bin multiphoton entanglement [24], offering promising applications in quantum communication [25] and quantum computation [26]. It is well known that high-dimensional quantum states are susceptible to decoherence in the presence of sharp bends [27,28]. Therefore, a key challenge in quantum information technology is to achieve robustness of high-dimensional quantum states against sharp bends.

In parallel, topological photonics introduces new capabilities for photonic devices, including unidirectional light transport, along with immunity to structural defects [29–37]. Due to the topological protection properties, topological photonics is soon introduced into nonlinear optics and quantum optics, inspiring many significant advances, including the topological harmonic generation [38,39], topological nonlinear imaging [40], topological single-photon and biphoton states [41–43], topological quantum emitters [28], topological frequency combs [44–47], topological quantum interference [48,49], and even topological quantum logic gates [50]. However, the topological protection of high-dimensional quantum states against sharp bends remains unexplored. Valley photonic crystal (VPC) waveguides, which emulate the quantum valley Hall (QVH) effect, exhibit promising characteristics such as broad bandwidth, low transmission loss, and ultra-compact chip sizes [51]. These features open exciting possibilities for achieving robust on-chip topological transport of high-dimensional quantum states with large bandwidths.

Here we experimentally demonstrate the on-chip topological transport of both QFCs and DKS combs in VPC waveguides. Our VPC topological waveguide supports topologically protected kink states with a significant topological bandgap of approximately 25 THz, enabling the topological transmission of OFCs with extremely wide frequency ranges. Using a Si3N4 micro-resonator with a free spectral range (FSR) of 100 GHz, we can access QFCs and DKS combs at different pump powers. We measure the coincidence-to-accidental ratio (CAR) and joint spectral intensity (JSI) (in a 5×5 mode subspace) of QFCs, and find that the quantum correlations and frequency entanglement are topologically protected. Additionally, we demonstrate that fully coherent DKS combs, including single-soliton states, multisoliton states, and soliton crystals, maintain their spectral envelope and low-noise features even after traveling through the topological interfaces. Our findings show the potential for topologically protected unidirectional transport of high-dimensional quantum states and phase-locked soliton states, promising new approaches to high-dimensional information processing utilizing topology in classical and quantum optics.

Our topological device supporting the on-chip transport of OFCs is illustrated in Fig. 1(a). The topological waveguides are fabricated using a silicon-on-insulator (SOI) wafer with a silicon layer thickness of 220 nm (see Section 4). The VPCs are composed of a graphene-like lattice with a lattice constant of a0=433nm. Each unit cell comprises two triangular holes with side lengths of d1 and d2. For unperturbed unit cells (d1=d2=216nm), the C6 symmetry leads to a degenerated Dirac point at the K and K′ valleys. Breaking the inversion symmetry (d1≠d2) results in a complete bandgap in the Brillouin zone (see Appendix B). It is noted that the eigenmodes at the K and K′ valleys exhibit opposite polarization. The valley Chern numbers of VPC1 (d1=122nm and d2=295nm) and VPC2 (d1=295nm and d2=122nm) are calculated as Cv1=−1/2 and Cv2=1/2, respectively (see Appendix A). At the interface between VPC1 and VPC2, the interchange of VPCs reverses the sign of the valley Chern numbers. Note that the difference between the valley indices at the K point across the interface is ΔCedge1K=Cv2−Cv1>0 [34,52], which determines the propagation direction of the edge state around each valley.

Figure 1(b) shows the calculated edge dispersion of the VPCs. There exists a pair of valley kink states (blue curves) with a bandwidth of approximately 25 THz (highlighted in yellow), spanning from 175 to 200 THz. Importantly, such a large bandwidth allows the topological transport of OFCs with a bandwidth of approximately 200 nm at telecommunication wavelengths. The two valley kink states with opposite group velocities are locked to different valleys, referred to as “valley-locked” chirality [33,34]. The right panel of Fig. 1(b) displays the field distribution for the valley kink state at kx=0.68π/a0, showing strong localization of electric field around the interface.

To verify robustness against sharp turns, we designed two types of waveguides: straight and Z-shaped topological waveguides [see scanning electron microscopy (SEM) images in Figs. 1(c) and 1(d)]. The in-and-out coupling of the topological waveguides is achieved using lensed fibers, with an insertion loss of 6 dB/face. As shown in Fig. 1(e), the measured transmission spectrum of the Z-shaped waveguide closely matches that of the straight waveguide, indicating that the valley kink states are robust against sharp bends. It can clearly be seen that the measured transmission spectra align well with the simulation results (see Appendix B). Due to the cutoff excitation wavelengths of the pump laser, we are only able to access the topological bandgap from 1490 to 1640 nm.

Thanks to the large topological bandgap of the valley kink states, we are able to achieve on-chip topological transport of broadband QFCs with frequency entanglement. Our QFC is generated using a Si3N4 micro-resonator with a high-quality factor (Q-factor) of 1.68×106 (see Appendix C). To manipulate the broadband phase matching for spontaneous four-wave mixing (FWM) processes, we carefully design the waveguide cross-section to achieve weak anomalous group-velocity dispersion [53].

Figure 2 shows the experimental setup for the topological transport of OFCs, with the details provided in Section 4. We pump the Si3N4 micro-resonator at below-threshold power around 1550 nm. As a result of the spontaneous FWM process, a two-photon high-dimensional frequency-entangled state, also referred to as a biphoton frequency comb, is generated [14]. The QFCs with a high-dimensional quantum state could significantly increase the capability of quantum information processing [15,24]. Generally, the quantum state of the QFC can be written as [15] |Ψ⟩=∑k=1Nαk|k,k⟩si,with∑|αk|2=1,(1)where |k,k⟩si is the signal and idler photons for the kth comb line pair, k=1,2,…,N is the mode number, and the complex element αk represents the amplitude and phase of the signal-idler photon pair. The |k,k⟩si is given by |k,k⟩si=∫dΩΦ(Ω−kΔω)|ωp+Ω,ωp−Ω⟩si,(2)in which Ω is the frequency deviation from the pump frequency ωp, Δω is the FSR of the resonator, and Φ(Ω) is the Lorentzian spectrum function. The quantum state |k,k⟩si is a superposition of multiple frequency modes, symmetrically distributed around the pump mode.

Figures 3(a)–3(c) show the measured single-photon spectra at the outputs of the micro-resonator, straight, and Z-shaped topological waveguides, respectively. We observe comb-like spectra with a bandwidth of 80 nm and a mode spacing of approximately 0.8 nm. In our QFC, an individual signal or idler photon forms a coherent superposition of 48 frequency modes, signifying the realization of a quantum system with at least 48 dimensions. Notably, the spectra of the QFCs detected at the output ports of the topological waveguides are nearly identical, indicating that the QFC experiences no significant loss after traversing sharp bends. This observation is consistent with the transmission spectra of the topological waveguides shown in Fig. 1(e).

To confirm the quantum correlation of our QFCs, we record the relative arrival time between correlated photon pairs (S7I7) and uncorrelated photon pairs (S7I8) as coincidences. Any events involving uncorrelated photon pairs are considered accidental counts, which include dark counts, background counts, and uncorrelated photon counts [20]. We fit the three coincidence peaks using the second-order Glauber correlation function gsi(Δt)∝exp(−Δt/τ), where τ represents the coherence time of the correlated photons. The fitted coherence times for the three QFCs are 2.90 ns, 2.69 ns, and 2.45 ns, respectively, which align with the resonance linewidth (around 115 MHz). Due to the decoherence of quantum states at sharp corners, the coherence time of the photon pairs is slightly reduced. After transmission through the Z-shaped waveguide, the coherence time decreased by approximately 8.92% compared to that in the straight waveguide. This small degradation indicates the topological protection of time correlation in the QVH systems. In addition, we obtain CAR values of 9.7, 10.7, and 10.2 from the correlation peaks, respectively. It is worth noting that high on-chip power (6 mW) increases accidental counts, leading to low CAR values. Furthermore, we demonstrate that our topological QFCs can function as wavelength-multiplexed heralded signal-photon sources (see Section 4).

Furthermore, to assess the spectral correlations across the photon-pair spectrum, we measure the JSI distributions for mode-by-mode photon counting, covering five sideband pairs (S4−8I4−8). The JSI, which demonstrates spectral correlations, arises from energy and momentum conservation [54]. The results shown in Figs. 4(a)–4(c) clearly reveal the frequency correlation of the signal-idler modes generated through the FWM processes. Only photon pairs that satisfy the energy conservation relation (2ωp=ωs+ωi) exhibit strong frequency correlation. We implement a correction for the three JSIs by subtracting the accidental coincidence counts (induced by dark counts, background noise, and after-pulse detection of InGaAs SPDs).

We use Schmidt decomposition to evaluate the frequency entanglement of QFCs [55] (see Appendix E). Figures 4(d)–4(f) show the distributions of normalized Schmidt coefficients λn and entanglement entropy Sk for three QFCs, respectively. The Schmit coefficients λn represent the possibility of obtaining the nth quantum state, where nonzero Schmidt coefficients (greater than one) indicate the frequency entanglement [55]. The Schmidt numbers K=(∑λn2)−1 and entanglement entropy Sk=−∑λnlog2λn are used to demonstrate the presence of two-photon frequency entangled states, where a larger K value indicates higher-quality entanglement [55]. We obtain the entanglement entropy of 1.41, 1.16, and 1.07 for three QFCs, respectively. Moreover, the Schmidt numbers K are calculated as 1.89, 1.63, and 1.53, respectively, which proves the existence of frequency entanglement. By comparing the Schmidt numbers of QFCs at the outputs of straight and Z-shaped topological waveguides, we demonstrate the robustness of frequency entanglement in the presence of sharp bends. Notably, using superconducting nanowire single-photon detectors (SNSPDs) could significantly improve the Schmidt numbers. By filtering out the off-diagonal elements in the measured JSI, we can achieve a Schmidt number greater than 4.0. Such a significant Schmidt number indicates the presence of a quantum state with effective dimensions of D=4. Consequently, this can lead to a topological high-dimensional quantum entangled state, paving the way for intricate, large-scale quantum simulations and computations.

Alternatively, one can evaluate the single-photon purity, which is associated with the factorizability of two-photon quantum states via Schmidt decomposition. Typically, single-photon purity is quantified by Tri(ρ^s2), where ρ^s=Tri(|Ψ⟩⟨Ψ|) denotes the density operator of the heralded single photon, and Tri is the partial trace over the idler mode. Note that the heralded single-photon purity is given by Tri(ρ^s2)=K−1 [54]. For our QFCs, the single-photon purities are calculated to be 0.53, 0.61, and 0.65, respectively, indicating the emergence of high-purity quantum states.

We also perform on-chip topological transport of DKS combs using the same Si3N4 micro-resonator. Before the experimental scheme, we numerically simulate the dynamic evolution of DKS combs described by the Lugiato–Lefever equation (LLE) (see Appendix F). However, accessing stable DKS combs in micro-resonators is challenging due to limitations imposed by laser tuning precision, frequency stability, and short thermal lifetime [53]. To address these challenges, we use a dual-pump method to extend the region of soliton existence (see Section 4). Also, the beat between the auxiliary and pump lasers could facilitate the generation of soliton crystals [56].

By scanning the pump laser from the blue to the red side of the resonator mode, one can clearly observe the comb evolution processes, including Turing rolls, chaotic states (breathing soliton states), multisoliton states, and single-soliton states [57]. Typically, a sharp decrease in intracavity power implies the arrival of a bistable state (referred to as a soliton step), which also demonstrates the arrival of a soliton state [53]. By stopping the pump laser scan at soliton step regions, DKS states can be easily accessed.

Figures 5(a)–5(c) show the measured optical spectra of the single-soliton states at the outputs of the micro-resonator, straight, and Z-shaped topological waveguides, respectively. These spectra show the smooth sech2-shaped spectral envelope with a bandwidth of about 20 THz. The single-soliton spectrum also exhibits a 3 dB bandwidth of 29.3 nm, corresponding to a soliton pulse width of 87.5 fs. To further verify the noise performance of the single-soliton states, we use a CW laser to beat with one of the spectral lines of the combs. The resulting beat notes exhibit distinct frequency lines with resolution bandwidths (RBWs) of 100 kHz and signal-to-noise ratios of 30 dB (see Appendix H), indicating excellent low-noise characteristics.

Due to the inherently stochastic nature of intracavity dynamics, multisoliton states with a random soliton number N can be accessed during the frequency-tuning process [53]. The measured spectra of the multisoliton states in three cases are shown in Figs. 5(d)–5(f), where the spectrum of a multisoliton state is caused by interference among several solitons. As the number of solitons increases, the spectral characteristics become progressively more complex [57]. In other words, the intracavity solution for a multisoliton state can be described as the sum of several distinct, independent soliton solutions located at different positions. Interestingly, the adiabatic backward tuning method can effectively reduce the soliton number, allowing the transition from a multisoliton state to a single-soliton state [53].

We also access perfect soliton crystals using the forward tuning method with relatively low pump powers (100 mW). The formation of perfect soliton crystals results from the collective self-organization of multiple copropagating solitons. Figures 5(g)–5(i) show the measured spectra of perfect soliton crystals in three cases. The combs exhibit several evenly distributed supermodes separated by 13 FSRs, leading to a mode spacing of 1.24 THz. Additionally, our perfect soliton crystals can be regarded as a single soliton with a larger FSR, which can be used to generate ultra-high-repetition-rate DKS combs [58]. In the time domain, 13 DKSs with constant pulse separations of 2π/13 are present in the resonator, reaching the maximum allowed number of solitons for the given pump. Since the comb power is distributed across 13 supermodes, the power of each supermode is amplified by a factor of 132, and the energy conversion efficiency is increased 13-fold compared to that of the single soliton state [57].

Benefiting from the large bandwidth of topological waveguides, we demonstrate the topological transport of DKS combs over a spectrum of approximately 20 THz. Remarkably, all DKS combs pass smoothly through sharp bends without noticeable loss. Despite traveling through sharp turns, the DKSs retain their specific spectral envelope and low-noise characteristics, demonstrating the topological protection of soliton properties.

We have experimentally demonstrated the on-chip topological transport of QFCs and DKS combs at telecommunication wavelengths. In our fabricated VPC structures, we observe topologically protected kink states with a bandwidth of 25 THz. Using a Si3N4 micro-resonator, we access both frequency-entangled QFCs and mode-locked DKS combs. We show that the quantum correlations and frequency entanglement of high-dimensional quantum states exhibit no significant decoherence in the presence of structural perturbations, owing to the topological protection of the QVH effect. Furthermore, we demonstrate that mode-locked DKS combs retain their perfect spectral envelope and low-noise characteristics even after passing through topological interfaces. Topologically protected OFCs provide robust, complex, highly controllable, and scalable quantum resources, offering promising advances in quantum communication and information processing.

We fabricate the topological devices on an SOI wafer with a 220 nm thick top silicon layer and 3 μm thick buried silicon layer. The edge coupler, silicon waveguide, and VPC structures are etched to a depth of 220 nm. We then deposit 1 μm thick SiO2 cladding using plasma-enhanced chemical vapor deposition (PECVD). The entire chip is deeply etched and diced into multiple individual chips. The devices are fabricated using electron beam lithography (Vistec EBPG 5200+) and an inductively coupled plasma etching process (SPTS DRIE-I). For details on the fabrication process of the structures used in this work, see Refs. [59,60].

In this experimental setup, we use a compact CW laser (Pure Photonics) with a wavelength of 1550.78 nm to pump the Si3N4 micro-resonator, and an EDFA to amplify the pump. Two cascaded bandpass filters are used to suppress the amplified spontaneous emission (ASE) noise generated by the EDFA. The power effectively coupled into the resonator is 6 mW. A temperature controller (TEC) is used to control the thermal instability of the mirco-resonator. The output QFC is coupled into topological waveguides after the polarization control. A 99:1 BS is connected to the output port, where 1% of the output power is detected by a programmable OPM (Joinwit) to monitor the pump power, allowing active control of the laser wavelength using a PID algorithm. The use of PID control ensures a stable resonant state for long-time coincidence measurement. The remaining 99% of the output power is filtered by an FBG, and the remaining QFCs are split by a 50:50 BS. The signal and idler modes are selected by two TBPFs (WL Photonics) with a bandwidth of 0.11 nm, and detected by two InGaAs SPDs (Aurea Technology).

For the JSI measurement, the SPDs are set to gated mode with a 20 MHz repetition rate, 20% quantum efficiency, and 10 μs dead time. The signals are then sent to a time analyzer to record coincidence events.

In this experimental setup, we use the dual-pump method to generate DKS combs by a computer-controlled soliton generation system [61]. The pump laser is excited by the same CW laser and amplified to 0.4 W by an EDFA. An additional auxiliary laser of 1 W power is used to stabilize the intracavity energy to extend the soliton steps. To reduce the crossed interaction of the two pumps, they are changed to orthogonal polarization modes by two independent FPCs. The two lasers are injected into the bus waveguide in opposite directions, with two circulators separating the input and residual pump light. The output comb is split by a 99:1 BS, with 1% of the output power detected by an OPM. The other 99% of the output power is split by a 90:10 BS, where 10% of the remaining power is sent to an OSA to measure the spectrum of the generated Kerr combs. After the polarization control, the remaining 90% of the residual output is coupled into the topological waveguides. The output of the topological waveguide is then sent to another OSA to monitor its spectrum.

In the computer-controlled soliton generation system, the auxiliary laser is tuned to the resonance wavelength to generate primary FWM sidebands. Note that the frequency modes of the pump laser and the FWM sidebands generated by the auxiliary laser are removed from the optical spectrum. The pump laser is automatically tuned from the blue to the red side to access the soliton states. The script assesses the intracavity state by monitoring the measured output power in the frequency-tuning process. Once the soliton state is accessed, the automation script gives the “stop” command for the pump laser (see Appendix D). In this case, the soliton state will hold for several hours.

We also calculate the heralded efficiency ηh, which is the probability of detecting an idler photon when the signal photon is detected. In general, the heralded efficiency can be given by ηh=cc/csignalηdet [62], where cc and csignal are the coincidence and signal count rates, respectively, and ηdet is the detection efficiency of the SPD for the idler mode (20%). Based on the obtained measurement, we derive a heralding efficiency of ηh=6% without considering the losses of the experimental setup.