Photonic integrated circuits (PICs) have experienced significant advancements over the past few decades, particularly in the last 10 years [1–4]. This progress is attributed to significant advancements in micro-nano fabrication techniques, enabling the realization of large-scale, multifunctional PICs [5–7]. This notably reduces the complexity and cost of building optical systems, enabling large-scale integration and opening new avenues for generating thousands of optical qubits for quantum computing [8] and communication [9].

However, the rate of single-photon sources generated in current PICs and losses are insufficient for the demands of quantum communication [10]. Furthermore, the distance-capacity limit [11,12] constrains the feasibility of long-distance quantum communication, underscoring the urgent need for an increase in communication rates. One possible method is to use weak coherent pulses (WCPs) generated by attenuated light pulses instead of single photons [13], combined with traditional wavelength-division multiplexing (WDM) techniques, to enhance the capacity of long-distance quantum communication. For example, decoy-state techniques can be used to achieve quantum key distribution (QKD) through multi-path WCPs. However, traditional QKD schemes (i.e., BB84) have vulnerabilities, such as detector-side attacks [14], which can compromise the security of the system. With the proposal and demonstration of measurement-device-independent quantum key distribution (MDI-QKD) schemes, these vulnerabilities can be addressed [15]. MDI-QKD requires the deployment of two independent light sources, with Hong–Ou–Mandle (HOM) interference measurements conducted at a third party. Therefore, photon indistinguishability between the independent sources is a prerequisite for achieving high interference visibility, which is a necessary condition for efficiently extracting secure keys in these protocols [16].

When two indistinguishable photons impinge on a beamsplitter (BS), they exhibit “bunching”, known as HOM interference [17], which is a quantum optical effect with no traditional analog and has been observed in single-photon states [18], continuous-wave weak coherent states based-on SMCs [19], and thermal states [20]. When single photons are replaced by WCPs, HOM interference still occurs, but the visibility is limited to 50% [21]. To achieve high interference visibility, the WCPs must become highly indistinguishable, which requires precise control and monitoring of all degrees of freedom, such as frequency, arrival time, polarization, and the average number of photons per pulse.

Wavelength alignment is a critical challenge for independent laser sources without a common reference, and a simple solution is to reference the laser wavelengths to natural atomic transition lines. However, the limited availability of atomic reference frequencies makes it challenging to lock the frequencies of dozens or even hundreds of lasers. Fortunately, soliton microcombs (SMCs) [22] generated from on-chip integrated micro-ring resonators (MRRs) can effectively address this challenge. In the frequency domain, SMC is composed of a series of strictly equidistant spectral lines, with the comb teeth frequencies $f_n$ represented as $f_n=f_p+\mathit{n}{f}_r$, where n represents integer; $f_p$ and $f_r$ represent the frequency of the pump laser and repetition rate of the SMC, respectively. By locking and adjusting only two parameters, $f_p$ and $f_r$, hundreds of precisely locked frequency comb teeth can be obtained. Compared to free-running SMCs already applied in traditional high-speed parallel communication [23,24], quantum communication schemes based on HOM interference have strict requirements for frequency alignment between independent light sources. Although, both $f-2f$ and atomic reference schemes [25,26] have been developed in SMC frequency locking, and the frequency stability can reach beyond ${10}^{-13}$ at 1 s, it is still a challenge to align all the frequencies of two independent SMCs due to the MRRs’ differences and the interaction between the optical frequency and repetition rate adjustment. The differences in the repetition rate of SMCs generated by different MRRs arise from two main sources: first, the slight discrepancies in the FSR due to fabrication errors (including the nonuniformity of film deposition, lithography, etching, etc.) [27]; second, variations in the soliton generation temperature and pump auxiliary power under pump-locked conditions, which result in different repetition rates [28]. Additionally, pump frequency locking prevents adjusting the repetition rate by changing the pump wavelength.

In this paper, we experimentally demonstrated HOM interference between 16 sets of WCPs based on comb tooth pairs generated by two independent on-chip SMCs. The frequency alignment and long-term stable operation of comb teeth between two independently generated SMCs are attributed to three operational aspects: first, the pump laser frequencies for generating the two independent SMCs are locked to same ${}^{87}\mathrm{Rb}$ atomic transition from $\mathit{F}=2$ to ${\mathit{F}}^{\prime }=3$ using modulation transfer spectroscopy technology [29]; second, by employing pump power and temperature perturbations, we achieve a tuning of the repetition rates of two independently generated SMCs over 100 kHz based soliton steps exceeding 3 GHz; third, long-term stability of the repetition rate is achieved through microwave injection locking (MIL) technology [30]. Thanks to the above operations, the interference visibility of all channels’ WCPs HOM interference exceeds 46%, laying the foundation for long-distance, high-efficiency secure key extraction. Simultaneously, the development of quantum WDM technology at telecom wavelengths based on the MDI-QKD protocol becomes feasible, promising to further enhance the rates and user capacity of long-distance quantum communication.

WCPs can be obtained by attenuating a laser. For two independently generated WCPs $a$ and $b$ with the same polarization state, their states can be expressed as $${|\alpha ⟩}_a=\mathit{D}(\alpha ){|0⟩}_a,{|\beta ⟩}_b=\mathit{D}(\beta ){|0⟩}_b,$$(1)where $\mathit{D}(\alpha )$ and $\mathit{D}(\beta )$ represent displace operators. By adding a delay time $\tau$ to the input of path $a$ at times $t$, as shown in Fig. 1(a), the states with the delay time before entering the BS can be expressed as $$e^{-i{\omega }_a(t+\tau )}{e}^{-i{\omega }_{b}t}\mathit{D}(\alpha )\mathit{D}(\beta )|0⟩.$$(2)

Furthermore, in our experiment, the generated pulses are Gaussian weak coherent states, and their spectral function ${\varphi }_{a(b)}(\omega )$ is expressed as $${\varphi }_{a(b)}(\omega )=\frac{1}{{(\pi )}^{\frac{1}{4}}\sqrt{{\sigma }_{a(b)}}}{e}^{-\frac{{(\omega -{\overline{\omega }}_{a(b)})}^2}{2{\sigma }_{a(b)}^2}},$$(3)where ${\overline{\omega }}_{a(b)}$ represents the central frequency of photons $a$ and $b$; ${\sigma }_{a(b)}$ is its spectral width. Assuming the detectors have a flat frequency response and the BS is an ideal 50:50 BS, the coincidence probability at the BS outputs $c$ and $d$ is given by [31] $$p_{\text{cd}}(\tau )=\frac{1}{2}-V\frac{{\sigma }_a{\sigma }_b}{{\sigma }_a^2+{\sigma }_b^2}{e}^{-\frac{{\sigma }_a^2{\sigma }_b^2{\tau }^2+{({\overline{\omega }}_a-{\overline{\omega }}_b)}^2}{{\sigma }_a^2+{\sigma }_b^2}},$$(4)where the $V$ represents the HOM interference visibility, which is limited to 50% for WCPs between the independent input states $a$ and $b$ with $\tau =0$, as shown in Fig. 1(b-iii). With the increasing delay time $\tau$, $p_{\text{cd }}$ will increase and interference visibility will decrease, as shown in Figs. 1(b-i) and 1(b-ii). At $\tau =0$, the interference visibility $V$ decreases as the photon distinguishability increases. For WCPs generated by on-chip SMCs, the spatio-temporal mode and output polarization are ensured through chip design and coupling to a single-mode polarization-maintaining (PM) fiber. To align the pulses’ center frequency without a common reference, it is necessary to achieve independent full locking of the SMC and tuning of the repetition rate in the experiment.

Figure 2(a) shows the experimental setup for multi-path weak coherent pulse HOM interference based on independently generated SMCs. The two SMCs are generated in two independent high-index doped silica glass MRRs based on the well-developed AOM frequency-shifting laser-assisted intracavity thermal balance scheme. The MRRs [Fig. 2(e)] are packaged in 14-pin butterfly packages with thermo-electric coolers (TECs). The two pump lasers are independently referenced to the same atomic transition line to ensure frequency alignment of the pump lasers. The auxiliary lights are generated by AOM frequency shifting, which helps to further enhance the long-term stability of the SMCs (see Appendix A). The SMCs are coupled out through fiber arrays and phase-randomized by adding Gaussian white noise via PMs. The different frequency comb teeth are then separated using a commercial DWDM. Subsequently, the comb teeth are modulated into Gaussian pulses with a full width at half maximum (FWHM) of $\sim 200\mathrm{ps}$ using an IM. While the pulses are attenuated to single-photon level and coupled into a 5:5 BS, HOM interference dip can be observed by adjusting relative delay time with optical delay lines between two-path WCPs.

The frequency alignment without common reference and long-term stability between the two independently generated SMCs are prerequisites for achieving multi-path WCPs HOM interference. Fully frequency-stabilized SMCs are achieved through atomic referencing and MIL. The pump lasers are frequency referenced to the transition spectrum line of ${}^{87}\mathrm{Rb}:\mathit{F}=2$ to ${\mathit{F}}^{\prime }=3$ through the modulation transfer spectroscopy method. To appraise the frequency stability of the two locked lasers, we recorded the beat frequency by a frequency counter between the two locked 1560.492044 nm lasers. As shown in Fig. 3(b), the peak-to-peak frequency fluctuation of the two laser-locked pumps is 0.3487 MHz, meaning that each pump laser frequency drifts by $\sim 0.2476\mathrm{MHz}$, whose frequency fluctuation is much smaller than the spectral width ($\sim 3.75\mathrm{GHz}$) of the pulse generated from the comb tooth. The repetition rate of the free-running SMC fluctuates by $\sim 5\mathrm{kHz}$ due to dynamic changes in intracavity power and temperature. To further reduce the impact of repetition rate fluctuation on the instantaneous frequency of the comb teeth and the long-term stability of the SMC, the repetition rate of the SMC is locked to an ultra-stable external microwave signal through MIL. In our experiment, the pump laser generates a fourth-order sideband $\sim 48.98\mathrm{GHz}$ away from the pump laser using an IM driven by a microwave signal with frequency $\sim 12.245\mathrm{GHz}$, aligning it with the repetition rate of the SMC. By comparing Fig. 3(c-left) with Fig. 3(c-right), the FWHM of the repetition rate signal is compressed by three orders of magnitude through MIL, which indicates successful transfer of the ultra-stable microwave signal stability to the repetition rate. Once the pump frequency and repetition rate are locked, the SMC source works at a rather stable state and can stably survive for a long time (over 5 days), which is suitable for future practical applications.

Moreover, due to fabrication errors, the intrinsic FSRs of the two MRRs used in the experiment differ by $\sim 200\mathrm{kHz}$. This results in a frequency difference of $\sim \mathit{n}\cdot 200\mathrm{kHz}$ between the nth comb tooth, generated by the two MRRs. This accumulation of frequency differences is highly detrimental to channels far from the pump, leading to a reduction in HOM interference visibility. This poses a challenge for using SMC-based DWDM systems in quantum communication. The AOM frequency-shifting laser-assisted intracavity thermal balance scheme used in our experiment for SMC generation allows for soliton steps exceeding 3 GHz, corresponding to a repetition rate tunable by more than 700 kHz theoretically under pump frequency locking conditions. As shown in Fig. 3(c-left), when reaching the single-soliton state in a microcavity, the repetition rates of the SMCs generated in the two different MRRs [as shown in Fig. 3(a)] differ by 108.8 kHz. By adjusting the beat signal between pump and auxiliary, and cavity’s temperature and pump power, we successfully shift the repetition rate of MRR1 to match that of MRR2 [Fig. 3(c-right)].

The SMC output from the MRR is phase-randomized by a PM loaded with 5 MHz Gaussian white noise, ensuring that the generated WCPs have a phase distribution randomly between 0 and 2π, thereby satisfying the phase randomization requirements for QKD [19,32]. Subsequently, the comb-teeth pairs with identical wavelengths are selected and modulated into pulses by IM with FWHM of $\sim 200\mathrm{ps}$ (corresponding to a spectral width of $\sim 3.75\mathrm{GHz}$), as shown in Fig. 4(a). The repetition rate of the WCPs is set to 1.25 MHz. The photon counts of the two paths are attenuated to $\mathrm{60,000}\pm 1000{\mathrm{s}}^{-1}$ and $\mathrm{59,500}\pm 1000{\mathrm{s}}^{-1}$, as shown in Fig. 4(b). Considering that the detection efficiency of SNSPD1 and SNSPD2 is $\sim 80\%$, the average photon number of each path is estimated to be $\sim 0.06$ per pulse. Finally, the WCPs from two standalone SMCs are interfering at a symmetric polarization-maintaining BS. The two-path output photons are detected by SNSPD1(2) whose gate times are set to 1080 ps and the coincidence counts are analyzed by time-correlated single-photon counting (TCSPC). The trigger signal generated by the arbitrary waveform generator (AWG) is sent to the TCSPC to control the coincidence counting. The relative delay between the two independent WCPs is controlled by electronically controlled adjustable optical delay lines OD1 and OD2. The delay lines offer an adjustment precision of 5 fs, ensuring sufficient accuracy for the interference of two paths with pulse duration of $\sim 200\mathrm{ps}$.

Figure 5(a) shows the experimentally measured normalized coincidence counts for the 16 channels of weak coherent states (points with error bars) and the theoretical fit with a Gaussian curve (colored continuous lines). Figure 5(b) shows the HOM interference fringe of channel 3, with the fringe visibility being estimated to be $46.69\%\pm 0.85\%$. The shape of the HOM “dip” retains a Gaussian curve, and the FWHM of the “dip” matches that of the Gaussian pulse. The experimental results are shown in Fig. 5(c), where the visibilities of HOM interference fringes are distributed from 46.02% to 47.17%. The high interference visibility provides a solid foundation for efficient and secure QKD.

The implementation of multi-channel WCPs HOM interference using fully locked SMCs paves the way for quantum communication systems based on conventional fiber DWDM systems. However, any differences in the degrees of freedom (e.g., arrival time, spectrum, polarization, and the average number of photons per pulse) between the two independently generated sets of multi-channel WCPs will reduce photon indistinguishability, leading to a deterioration in HOM interference visibility. In our experiment, the independent WCPs are generated from two fully locked SMCs. The frequency differences between the comb teeth pairs ($\sim 0.2477\mathrm{MHz}$) are much smaller than the pulse spectral width ($\sim 3.75\mathrm{GHz}$), making the impact of pulse center frequency mismatch on interference visibility minimal. The SMCs are coupled into single-mode fibers, thus minimizing the impact of spatio-temporal mode differences. However, the main sources of error in this experiment arise from the non-ideal 50:50 BS used for interference, unequal detector efficiencies, the limited extinction ratio of the Gaussian pulses generated by the IM, as well as slight variations in pulse intensity and minor differences in modulation spectra [33,34]. In practical applications, it is necessary to further mitigate the effects of the complex environment on pulse polarization, delay, and pulse shape during long-distance transmission by incorporating polarization correction modules [35], time feedback techniques [36], and dispersion compensation [37] into the QKD system.

In our experiment, we successfully achieved 16-channel WCPs HOM interference with all channel visibility over 46% based on two independently generated SMCs. This paves the way for high-capacity, long-distance quantum communication using conventional DWDM systems without sacrificing security by introducing trusted nodes. The SMCs demonstrated in this work, with comb teeth frequency alignment achieved through independent locking and repetition rate tuning, can enable quantum communication protocols based on HOM interference (e.g., MDI-QKD). These protocols not only provide immunity to detection-side attacks, thereby enhancing the security of quantum communication, but also allow all users to share expensive resources (e.g., single-photon detectors), further reducing costs.