Introduction

For decades, integrated photonics has played a transformative role in advancing the development and exploration of the quantum information processing. It is not only essential for ensuring the scalability and compactness of quantum systems, but also for deepening our understanding of fundamental quantum principles1,2,3,4,5,6,7. The evolution of on-chip systems has been driven by the pursuit of more advanced or practical solutions, evolving from basic straight waveguides to all-encompassing platforms8,9,10,11. Among the distinct configurations within quantum circuits, an integrated photon-pair source stands as a crucial element12,13,14,15. Owing to its pivotal role in quantum systems16,17,18,19,20,21, it is indispensable for quantum information processing22,23,24,25.

In integrated systems, quantum photon sources are typically generated via parametric processes. There have been various photonic platforms demonstrating intrinsic advantages in previous studies10,26,27,28,29. Among them, silicon devices are the most widely used due to their high third-order susceptibility and the compatibility with mature complementary metal-oxide semiconductor (CMOS) fabrication techniques30,31,32. They have been the foundation for many exploratory works requiring intricate structures25,33,34.

A prevailing belief in quantum state generation is that the coherence of the pump dictates the properties of the resulting sources. Incoherent light is predominantly utilized in fields such as imaging, computing, and remote sensing, with limited use in quantum information processing35,36,37,38,39,40. However, the intrinsic decoherence effect arising from laser linewidth broadening poses a fundamental limitation for coherent light applications in long-distance quantum communication and sensing. This phenomenon leads to a progressive accumulation of quantum noise as the system exceeds the coherence time limit41,42,43. For monolithic integrated systems, the fabrication of high-performance on-chip lasers also remains a non-trivial challenge. Currently, there are increasing interests in utilizing more natural light sources, such as incoherent or partially coherent beams, to generate quantum resources44,45,46,47. Considerable progress has been made in understanding the role of coherence in Spontaneous Parametric Down-Conversion (SPDC), with notable achievements in both the theoretical derivation and experimental characterization of quantum states pumped by incoherent light46,48,49,50,51.

Despite these promising developments, the role of temporal incoherence remains underexplored, particularly in integrated systems. Previous studies have focused on χ(2) processes in bulk materials, where strict phase-matching conditions inherently constrain the investigation of frequency-domain incoherence44,52. Additionally, the quantum correlations of generated photon pairs have not been systematically characterized. This gap not only hinders the optimization of quantum light sources, but also obscures the fundamental physics of temporal incoherence and quantum properties in nonlinear processes.

To address the issue, using an Amplified Spontaneous Emission (ASE) source as pump for photon-pair generation via on-chip Spontaneous Four-Wave Mixing (SFWM) process presents a promising approach. ASE source has high spatial coherence and low temporal coherence38,53,54. In SFWM, loose phase-matching conditions result in a broadband emission spectrum. Thus, a spectrally multimode ASE source can contribute efficiently to photon-pair generation. The Gaussian spatial mode also allows for low-loss propagation of light within the waveguide.

In this work, we reveal a counterintuitive phenomenon where temporal incoherence plays a beneficial role in quantum state generation. Based on the SFWM process in silicon waveguides, we demonstrate the enhanced photon source preparation by incoherent light, challenging the conventional paradigm that relies on coherent lasers. Temporal incoherence can improve pump utilization in the nonlinear process. The pump’s temporal incoherence produces a less-correlated joint spectral amplitude (JSA) and yields higher state purity under the same filtering conditions. We analyze the effects theoretically and predict the results based on analytical expressions and a numerical model. Beyond silicon waveguides, our theoretical framework is universally applicable to SFWM processes in alternative photonic structures, such as bulk χ(3) nonlinear crystals, demonstrating broad adaptability across material systems.

Experimentally, we characterize the correlation and entanglement properties of states generated by filtered ASE sources (incoherent) and continuous-wave (CW) lasers (coherent).In addition to a more than 40% higher photon generation rate (PGR), incoherently pumped source shows noticeable advantages in the coincidence-to-accidental ratio (CAR) and heralded second-order autocorrelation \({g}_{H}^{(2)}(\tau )\) at low power. For the prepared polarization entanglement, we achieve Bell value S = 2.64 ± 0.02 and fidelity of 95.7% ± 0.1% for incoherently pumped state. By demonstrating a novel approach to generate high-quality quantum states using incoherent light, this work systematically establishes a framework for bridging incoherent photonic excitation with quantum state preparation. Incoherent light reduces pump-module complexity and relaxes stability requirements. Unlike quasi-monochromatic CW lasers, which demand precise wavelength and polarization control, ASE delivers a stable, polarization-insensitive broadband output. Compared with on-chip lasers that pose substantial fabrication and stabilization requirement, using ASE significantly lowers the barrier to fabricating on-chip source. Our research not only relaxes the technical constraints on laser sources, but also paves the way for monolithic quantum photonic integration.

Results

Coherently and incoherently pumped χ (3) processes

In this study, we investigate the generation of a two-photon state via SFWM process. Assuming the pump power is sufficiently low to avoid two-photon absorption and multiphoton events, the process is governed solely by the following Hamiltonian:

$$H=\frac{3}{4}{\epsilon }_{0}{\chi }^{(3)}\int\,{d}^{3}{{\boldsymbol{r}}}{\hat{E}}_{1}^{(+)}({{\boldsymbol{r}}},t){\hat{E}}_{2}^{(+)}({{\boldsymbol{r}}},t)\cdot {\hat{E}}_{s}^{(-)}({{\boldsymbol{r}}},t){\hat{E}}_{i}^{(-)}({{\boldsymbol{r}}},t),$$
(1)

where ϵ0 and χ(3) are the vacuum permittivity and third-order susceptibility. The spatial integrals represent the light propagation in the medium, with the superscripts ’+’ and ’-’ in the electric field operator corresponding to the annihilation of pump fields and the creation of signal and idler fields, respectively. Due to the intensities of the generated and pump fields, the electric fields can be expressed as follows:

$${\hat{{{\boldsymbol{E}}}}}^{(-)}({{\boldsymbol{r}}},t) \,= \, {f}^{*}(x,y)\int\sqrt{\frac{\hslash \omega }{2{\epsilon }_{0}n(\omega )c}}{a}^{{\dagger} }(r,t)\frac{{e}^{-i[k(\omega )z-\omega t]}}{\sqrt{2\pi }}d\omega,\\ {{{\boldsymbol{E}}}}_{C}^{(+)}({{\boldsymbol{r}}},t) \,= \, Af(x,y)\int\,d\omega {\alpha }_{C}(\omega ){e}^{-i(\omega t-kz)},\\ {{{\boldsymbol{E}}}}_{I}^{(+)}({{\bf{r}}},t) \,= \, Af(x,y){\sum }_{n=0}^{\infty }{\alpha }_{I}({\omega }_{n}){e}^{-i\phi ({\omega }_{n})}{e}^{-i({\omega }_{n}t-kz)},$$
(2)

where a†(r, t) is the creation operator, n is the refractive index as a function of frequency, A is a constant, and f(x, y) is the normalized transverse spatial distribution describing the profile of the guided mode. The pump field \({{{\boldsymbol{E}}}}_{C,I}^{(+)}\) is expressed in classical form for the high intensity compared to the states excited from vacuum fluctuations, and the subscript represents the coherence of the optical field.

As a temporally highly incoherent source, ASE is generated without a resonant cavity, resulting in a broadband, smoothed, and stable spectrum. Compared with coherent laser, an additional phase factor ϕ(ω), randomly distributed in the range [0,2π], illustrates the temporal incoherence of the ASE field. Due to the lack of correlation between each component in frequency, the spectral distribution α(ω) carries different implications. For coherent light, it represents the normalized amplitude density function, where ∫∣αC(ω)∣2dω = 1. However, uncorrelated independent components lead to the result  < E(ωi)*E(ωj) > = 0, i ≠ j. The electric field of ASE illumination needs to be expressed in discrete form. This distinction is also reflected in the subsequent photon pair generation process. In this scenario, the power of coherent and incoherent light are respectively given by

$${P}_{C} \,= \, \frac{1}{2}n{\epsilon }_{0}c{A}^{2}{\left\vert \int{\alpha }_{C}(\omega )d\omega \right\vert }^{2},\\ {P}_{I} \,= \, \frac{1}{2}{\epsilon }_{0}nc{A}^{2}{\Sigma }_{n=1}^{\infty }| {\alpha }_{I}({\omega }_{n}){| }^{2}.$$
(3)

By applying first-order perturbation theory, the two-photon state can be written as55,56,57,58.

$$\left\vert \psi \right\rangle=\left\vert 0\right\rangle+\iint F({\omega }_{s},{\omega }_{i})I({\omega }_{s},{\omega }_{i}){a}^{{\dagger} }({\omega }_{s}){a}^{{\dagger} }({\omega }_{i})\left\vert 0\right\rangle d{\omega }_{s}d{\omega }_{i},$$
(4)

where \(I({\omega }_{s},{\omega }_{i})=i\gamma L\,{\mbox{sinc}}\,(\Delta kL/2){e}^{-i{{\Phi }}({\omega }_{s},{\omega }_{i})}\) represents the nonlinear effects in the waveguide and the phase-matching conditions. It is determined by the pump frequency and the detuning frequency. For a non-monochromatic pump, the generated two-photon state should be described by a joint distribution function, which is illustrated by the two photon amplitude F(ωs, ωi) = ξC∫ dωpαC(ωp)αC(ωi + ωs − ωp). For a coherent laser, \({\xi }_{C}=\frac{1}{2}{\epsilon }_{0}nc{A}^{2}\int{\alpha }_{C}(\omega )d\omega\) is a constant determined by the line-shape and power. The integral term ∫ dωpαC(ωp)αC(ωi + ωs − ωp) represents the self-convolution of the pump field, reflecting the overlap of the optical field with itself when the generated photons are asymmetric around ω0. In order to provide an analytical solution, we assume that the coherent pump field satisfies Gaussian distribution centered at ω0 with bandwidth σp, and its normalized form is given by \({\alpha }_{C}(\omega )=C\cdot exp\{-\frac{{(\omega -{\omega }_{0})}^{2}}{2{\sigma }_{p}^{2}}\}\). Integrating over the given range frequency yields the production rate of photon pairs. We assume symmetric distribution of signal and idler photons about ω0, with a detuning unit interval ΔΩ significantly larger than σp. The detuning range is defined as Ω(m) = m*ΔΩ (ωi,s = ω0 ± Ω(m)), where m represents the number of intervals and ΔΩ is the frequency shift per interval. In SFWM, the phase-matching function varies slowly with frequency, so within a detuning interval, I(ωs, ωi) can be treated as constant. Therefore, the PGR ΔNC(Ω(m)) is given by:

$$\Delta {N}_{C}(\Omega (m))=\frac{\Delta \Omega }{2\pi }{\gamma }^{2}{L}^{2}{P}_{C}^{2}sin{c}^{2}\left(\frac{\Delta {k}_{m}L}{2}\right)\frac{\sqrt{2}}{2}.$$
(5)

The entire generating rate can be obtained through direct summation. See more details in Supplementary Information. For incoherent pump, the characteristic of each component being mutually uncorrelated alters the interaction mode for SFWM. The inner product of the two-photon amplitude \({{{\mathscr{F}}}}_{I}\) can be expressed as

$${{{\mathscr{F}}}}_{I}=\left| {\xi }_{I}{\sum }_{p=1}^{\infty }{\alpha }_{I}({\omega }_{p}){e}^{-i\phi ({\omega }_{p})}{\alpha }_{I}({\omega }_{s}+{\omega }_{i}-{\omega }_{p}){e}^{-i\phi ({\omega }_{i+s-p}) }\right|^{2}.$$
(6)

Unlike the case where the power is calculated by summing up the amplitudes of coherent light, the presence of ϕ(ω) characterizing the incoherence ensures independent interactions between the each two components. Even for the photons generated in same mode, the contributions of different combinations Am = αI(ωm)αI(ωl + ωk − ωm) are still irrelevant, i.e.

$$\left\langle {\left(\Sigma {A}_{m}{e}^{-i{\phi }_{m}}\right)}^{2} \right\rangle=\Sigma \left({\left\vert {A}_{m}{e}^{-i{\phi }_{m}}\right\vert }^{2}\right).$$
(7)

Based on this relationship, we apply the same transformation as in Eq 5, the PGR for incoherent pump is denoted as

$$\Delta {N}_{I}(\Omega (m))=\frac{\Delta \Omega }{2\pi }{\gamma }^{2}{L}^{2}{P}_{I}^{2}sin{c}^{2}\left(\frac{\Delta {k}_{m}L}{2}\right).$$
(8)

See more deduction details in Supplementary Information. This conclusion is consistent with the expression for monochromatic pump55, as both cases share the same underlying physical interpretation. Neglecting the phase-matching term, frequency components satisfying ωs + ωi = 2ω0 exhibit the maximum joint intensity; as ωs + ωi deviates further from 2ω0, the corresponding joint intensity decreases. This is reflected in the optical field’s overlap, represented by the self-convolution. ASE light can be viewed as a linear combination of N independent monochromatic light sources. These components interact in the medium to generate photon pairs. The total arrangement (\({A}_{N}^{n}=\frac{N!}{(N-n)!}\)) is given by \({A}_{N}^{1}+{A}_{N}^{2}={N}^{2}\), where \({A}_{N}^{1}\) corresponds to degenerate pump, and \({A}_{N}^{2}\) represents the combinations of non-degenerate pump. Their sum equals to \({\sum }_{i=1}^{\infty }{P}_{i}\mathop{\sum }_{j=1}^{\infty }{P}_{j}={P}^{2}\).

In addition to the effect on PGR, pump coherence also alters the joint frequency distribution of the generated state, i.e. JSA. Coherent lasers have fixed phase relation. The JSA can therefore be written as the self-convolution of the pump amplitude. In contrast, incoherent light exhibits spectral uncorrelation, the original amplitude convolution between the phase-fixed field components reduces to an intensity-product integral over the individual frequency modes. For the generated two-photon state in the mode \(\left\vert {1}_{{\omega }_{s}},{1}_{{\omega }_{i}}\right\rangle\), the JSAs for coherent and incoherent pumps can be expressed as

$${f}_{C}({\omega }_{s},{\omega }_{i})\, \propto \, \int{E}_{C}({\omega }_{p})*{E}_{C}({\omega }_{s}+{\omega }_{i}-{\omega }_{p})d{\omega }_{p}\\ {f}_{I}({\omega }_{s},{\omega }_{i})\, \propto \, \sqrt{\int_{-\infty }^{\infty }| {E}_{I}({\omega }_{p}){| }^{2}| {E}_{I}({\omega }_{s}+{\omega }_{i}-{\omega }_{p}){| }^{2}d{\omega }_{p}}.$$
(9)

We provide numerical simulations in Fig. 1 to illustrate the differences between these two cases. Figures 1a, b show the calculated JSA using a filtering bandwidth and pump bandwidth of 200 GHz with a rectangular line shape. The amplitude convolution in the coherent case represents a coherent superposition, concentrating energy in central spectral regions (ωs + ωi = 2ω0). The incoherent case corresponds to a linear intensity sum, resulting in a more uniform energy distribution across the entire spectral band.

Fig. 1: Joint spectral amplitude and purity.
Fig. 1: Joint spectral amplitude and purity.
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a Joint spectral amplitude with coherent pump mode. b Joint spectral amplitude with incoherent pump mode. c, d Purity as a function of filtering and pump bandwidth. e Purity versus pump bandwidth. The filtering bandwidth is 100 GHz. f Purity versus filtering bandwidth. The pump bandwidth is 100 GHz.

In general, the JSA retains the frequency anti-correlation for the weak phase mismatch in SFWM. Nevertheless, after finite filtering the incoherent behavior typically demonstrates a higher purity. Fig. 1c–f present comprehensive comparisons of the purity \({{\mathscr{P}}}\), which is defined as \({{\mathscr{P}}}={{\rm{Tr}}}({\rho }_{s}^{2})\), where \({\rho }_{s}={{{\rm{Tr}}}}_{i}(| \Psi \rangle \langle \Psi | )\). For coherent lasers, achieving the same purity requires either broader pump filtering or narrower photon filtering.

In “Methods’ and Supplementary Information, we develop a numerical model to statistically analyze outcomes, such as coincidence counts and single side counts, as a function of bandwidth or center frequency detuning. These results provide a guidance for selecting the bandwidth of the incoherent pump to simultaneously ensure both the brightness and purity of the source.

Measurements of quantum correlation properties

In the experiment, we record the single-side count rate and coincidence count rate. The uncertainty of photon counting in the -Results- section stems from Poissonian photon-counting statistics. Figures 2a–c present the quantitative characteristics of quantum correlation properties for photon pairs under CW laser and 200 GHz ASE pump. Figure 2a shows the coincidence counts of signal and idler photons at 1559.79 nm (C22) and 1540.56 nm (C46), which closely align with numerical predictions (solid line). Since coincidence counts scale quadratically with pump power, the proportional difference between coherent and incoherent pumps becomes more pronounced at higher power levels. The single-side count, measured to estimate accidental coincidences, can be expressed as Nsc = NSFWM + a ⋅ P + N0, where NSFWM denotes contributions from photon pairs. a represents the intensity of linear noise from Raman scattering in the fiber and ASE sideband photons. N0 accounts for background noise such as detector dark counts. When the coincidence bin is Δτ, the accidental coincidences and the theoretical CAR can be expressed as NBC = ΔτN1N2 and CAR = NCC/NBC. At low pump power, noise photons significantly contribute to the single-side count. Compared to coherent pump, incoherent pump generates more photon pairs, achieving a higher peak CAR-up to 345 for 200 GHz ASE, whereas the maximum CAR for a CW laser is approximately 160. At higher pump powers, accidental coincidences dominate the noise count, causing the CAR curve for coherent pump to fall behind that of incoherent pump due to its lower efficiency. In addition, we investigate the bandwidth of incoherent pump and the multiwavelength correlation properties of photon pairs. See Supplementary Information for more details.

Fig. 2: Experimental results of the correlated two-photon states.
Fig. 2: Experimental results of the correlated two-photon states.
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a Coincidence count rate versus pump power; The insert figure is coincidence histogram of the signal and idler photons at 1559.79 nm (C22) and 1559.79 nm (C46). b Single side count rate of the idler photon from 1559.79nm (C22) versus pump power. c CARs versus pump power. The lines are theoretically calculated by coincidence and single side count. The error bars of the CAR are obtained by Poissonian photon-counting statistics of coincidence count and background noise fluctuation.

The heralded second-order autocorrelation function \({g}^{(2)}(\tau )=\frac{{N}_{1}{N}_{123}}{{N}_{12}{N}_{13}}\) is a key indicator for characterizing the single-photon properties of the source. In the measurements, the TDC (UQDevices Logic16) recording three-fold coincidence events use a fixed 2 ns coincidence bin. When heralded signal photons at 1561.42 nm (C20) Channel equals to 0.35 MHz, the incoherent pumped antibunching dip of \({g}_{H}^{(2)}(\tau )\) for the heralded single photons is shown in Fig. 3a. The error bars are estimated by Poissonian photon counting statistics. Figures 3b, c show the \({g}_{H}^{(2)}(0)\) as a function of pump power. The second-order autocorrelation function is positively correlated with the photon PGR. As the single side counts increase, larger accidental coincidences negatively affect the purity of the heralded photons. In the condition, ASE pumped source exhibits higher brightness at the same power, resulting in a faster growth rate. However, the single side counts N1 and threefold coincidence events N123 are primarily dominated by system noise at low power. In this case, a higher PGR leads to a lower second-order autocorrelation. For incoherent pump, an increase in bandwidth reduces the efficiency of collecting correlated photons. See more details in the Supplementary Information.

Fig. 3: Measurement of the single-photon and spectral properties.
Fig. 3: Measurement of the single-photon and spectral properties.
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a Heralded second-order autocorrelation \({g}_{H}^{(2)}(\tau )\) of heralded photon at 1561.42 nm (C20). b Coherently and Incoherently pumped heralded second-order autocorrelation \({g}_{H}^{(2)}(0)\) versus pump power. c Incoherently pumped heralded second-order autocorrelation \({g}_{H}^{(2)}(0)\) with different bandwidth. d Joint spectral intensity of coherent pump. e Joint spectral intensity of incoherent pump with 100 GHZ bandwidth. f Joint spectral intensity of incoherent pump with 200 GHZ bandwidth.

Figures 3d–f show the joint spectral intensity (JSI) for two types of pumps, with the center pump frequency set at 1550.12 nm (C34). A 50 GHz dense wavelength division multiplexing (DWDM) is used for dividing frequency-correlated photons and filtering. The broadening of the pump light, detuning of the pump center frequency, and overlap between channels result in coincident events being detected in the asymmetric channels. We compare the JSI under different pump and filtering bandwidth conditions; see more details in the Supplementary Information.

Characterization of polarization entanglement

In addition to correlation, entanglement is also a pivotal property of quantum states and a key resource in quantum information. We use a Sagnac interferometer loop to prepare polarization-entangled photons (see “Supplementary Information” for details). By adjusting the pump’s polarization state, we can generate the maximally entangled Bell state \(\left\vert {\Phi }^{+}\right\rangle=\left\vert HH\right\rangle+\left\vert VV\right\rangle\).

We characterize the degree of entanglement using several methods. Figure 4 shows the two-photon polarization interference fringes and the reconstructed experimental density matrix ρex.The net fringe visibility for coherent pump is 99.6%  ± 0.1% (H basis) and 95.3%  ± 0.2% (D basis). For incoherent pump, the net fringe visibility is 97.2%  ± 0.1% (H basis) and 97.1%  ± 0.1% (D basis). The estimated net fidelity, calculated as \(F={\left[Tr\left(\sqrt{\sqrt{{\rho }_{th}}{\rho }_{ex}\sqrt{{\rho }_{th}}}\right)\right]}^{2}\), equals to 94.5%  ± 0.2% for coherent pump and 95.7%  ± 0.1% for incoherent pump, where Tr is the trace and ρth is the ideal density matrix.

Fig. 4: Experimental results of the polarization entanglement.
Fig. 4: Experimental results of the polarization entanglement.
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a–c Characterization of Polarization Entanglement with coherent pump. (a) Two-photon polarization interference fringes under H and D bases. b, c Experimentally measured density matrices. d–f Characterization of Polarization Entanglement with incoherent pump. (d) Two-photon polarization interference fringes under H and D bases. e, f Experimentally measured density matrices.

Furthermore, we measure the S parameters of the Clauser-Horne-Shimony-Holt (CHSH) inequality. For two polarization settings (θs = −22.5°, 67.5°, 22.5°, 112.5°; θi = −45°, 45°, 0°, 90°), the S values are 2.59  ± 0.03 (coherent) and 2.64  ± 0.02 (incoherent), with no background noise. These results confirm that the generated state is nearly an ideal Bell state, and the temporal coherence of the pump does not adversely affect polarization entanglement preparation.

Discussion

In this study, we report on-chip quantum states generation using both coherent and incoherent pumps in standard silicon strip waveguides. Contrary to conventional assumptions that temporal incoherence degrades quantum states, our study reveals its unexpected capacity to improve sources performance in integrated systems. The enhancement effect on photon-pair generation has been theoretically analyzed, and numerical simulations demonstrate the transfer of the spectrally uncorrelated property. Experimentally, the photon-pair generation rate with ASE pump is over 40% higher than coherently pumped situation. The increased brightness mitigates the impact of background noise at low power, improving quantum correlation properties, including coincidence and \({g}_{H}^{(2)}(0)\). Furthermore, we demonstrate high-quality polarization entanglement properties of the photon source.

From an engineering viewpoint, incoherent pump balances simplicity and performance. Compared with coherent lasers that require high system stability, it markedly reduces fabrication and packaging complexity and lowers per-device cost. These improvements shorten prototyping time and lower engineering risk as designs move toward monolithic integration.

In conclusion, our work presents novel quantum resources without stringent coherence constraints. This challenges the traditional view that high-quality photon sources are only achievable with coherent pump.Remarkably, the interplay between optical incoherence and quantum properties revealed in this work represents a universal feature of parametric nonlinear processes,making our findings applicable to diverse physical systems where SFWM occurs. The discovery also significantly expands the selectivity of pump sources. ASE, often seen as background noise, can be effectively utilized for photon source generation and entanglement preparation. The enhanced performance and reduced coherence requirements of our approach advance the development of monolithic integrated systems, paving the way for scalable quantum photonics technologies.

Methods

Numerical analysis

In the section of correlation measurement, the impact of temporal incoherence is primarily reflected in the photon yield. In theory, we consider the pump to be a narrowband source, resulting in the generated photon pairs being symmetrically counted. However, as a temporally incoherent pump, the bandwidth σA of the ASE source is comparable to the DWDM channel bandwidth. It is also difficult to ensure that the central frequency ω0 of the pump exactly matches the central frequency ωd of the DWDM. Therefore, we need to discretize both the pump spectrum and DWDM channels using numerical methods for accurate calculation.

Here, we present a universal model for quantitatively analyzing the collection of broadband photon pairs generated in SFWM. Signal and idler photons are counted separately, reflecting the limitations imposed by the finite collecting bandwidth of DWDM channels on asymmetrically distributed pump components. In simulations, ideal parameters, such as smoothing rectangular pumps and lossless DWDM, are used to numerically study the single-side and coincidence counts as a function of bandwidth and detuning frequency. Considering the actual experimental setup and combining the measured spectral line-shape and channel transmittance, the calculated results closely match the experimental data. The numerical simulation process is outlined in four steps. Refer to the Supplementary Information for details.

For comparison with experiment results, we should take into account losses during transmission, division, and detection. The actual detectable coincidence count rate can be expressed as Ncc = μtiμtsμdiμdsN0, where N0 is the number of generated photons, μt and μd represent the transmission and detection efficiencies, respectively. Interactions between the pump components lead to the separate generation of idler and signal photons within the discretized DWDM channels.

Experimental setups

The schematic diagram of the experimental setup is shown in Supplementary Information. In the pump source, a tunable CW laser (Toptica DL100) operating at 1550.17 nm serves as the coherent seed laser, while an ASE source with a spectrum spanning 1528 nm to 1563 nm provides the incoherent seed laser. The linewidth of coherent laser is about 100 MHz. To improve amplification efficiency, a pre-filtering procedure is implemented before the Erbium-Doped Fiber Amplifier (EDFA) for the ASE source. Due to the fully random polarization of the incoherent light, the amplified pump beam then passes through a polarizer to ensure that the pump propagates with a single polarization in the waveguide. The tunable attenuator (TA) is used to control input power, and four identical band-pass filters (BPF) are set to suppress sideband noise.

In the -Correlation- section, a polarization controller (PC) is used to optimize the polarization state for maximum coupling efficiency before the light enters the waveguide. The SOI waveguide, a single silicon strip waveguide with grating couplers, is used to generate photon-pair state in the system. It is 1 cm long with transverse dimensions of 220 nm in height and 450 nm in width, and has a total insertion loss of 20 dB. Residual pump light is filtered by cascaded band-stop fiters (BSF), achieving a total rejection exceeding 120 dB. The generated photon pairs have a continuous broadband emission spectrum. The idler and signal photons are separated by a 10-channel 200 GHz DWDM and detected by two free-running InGaAs avalanche photon detectors (APD1, APD2, ID220, detection efficiencies 10%, dead time 5 μs). Detection signals are processed by a board (PicoQuant TimeHarp 260) to record coincidence events, with a time bin width of 0.8 ns, as shown in Figure 3a. Further details on system parameters are provided in Supplementary Information.

The heralded second-order autocorrelation function \({g}_{H}^{(2)}(\tau )\) involves threefold coincidence events, which imposes requirement on the detection efficiency. Therefore, the heralded signal photons (N1) at 1561.42 nm (C20) are detected using a three-channel SNSPD (detection efficiency 80%), while the heralding idler photons at 1538.98 nm (C48) channel are detected with a delay time τ after passing through a 50:50 fiber beam splitter (FBS) (N2, N3). The coincidences (N12, N13) and threefold coincidence events (N123) and measured single side counts are recorded simultaneously by a time-to-digital converter (TDC, UQDevices Logic16).

In the -Polarization Entanglement- section, we employed a Sagnac interferometer configuration, which is a widely used and self-stabilizing method for generating polarization-entangled states. The PC before entering the Sagnac interferometer modulates the output pump after passing through a circulator to a 45° linear polarization.

The loop includes two half-wave plates (HWPs), two quarter-wave plates (QWPs), a polarization beam splitter (PBS), and an approximately 1 cm long end-face coupling silicon strip waveguide with a 5 dB insertion loss. The PBS splits the pump into clockwise (\(\left\vert H\right\rangle\)) and counterclockwise (\(\left\vert V\right\rangle\)) directions. The HWP and QWP in the Sagnac loop are adjusted to align the polarization of both clockwise and counterclockwise pumping along the horizontal axis (TE polarization) in the silicon waveguide. This configuration allows only the TE polarization mode to propagate along the silicon waveguide on the chip, with no polarization rotation occurring during propagation, as confirmed by three-dimensional finite-difference time-domain simulations.

Therefore, in the clockwise direction, the generated photon pair is in the state \(\left\vert HH\right\rangle\). Upon passing through the wave plates, the polarization state is rotated to \(\left\vert VV\right\rangle\). Conversely, in the counterclockwise direction, the \(\left\vert HH\right\rangle\) polarized state remains unchanged at the PBS. After the photon pairs from both counterpropagating directions recombine at the PBS, the resulting entangled state at the output port of the Sagnac loop can be expressed as.

$$\Phi=\left\vert HH\right\rangle+\eta {e}^{i\delta }\left\vert VV\right\rangle,$$
(10)

where the η2 represents the ratio of the pump powers for the V and H polarizations, while δ denotes the phase difference that arises due to the birefringence experienced by the signal and idler photons in the \(\left\vert HH\right\rangle\) and \(\left\vert VV\right\rangle\) states. The birefringence here originates from components other than the silicon chip in the setup, as only the TE polarization state interacts with the chip.

The frequency non-degenerate signal and idler photons are separated by the same 200 GHz DWDM. PCs are used to compensate for the polarization changes introduced during the transmission in fibers. By adjusting the fiber polarization rotator and polarizer (FPRP), we can perform quantum state tomography and measure biphoton interference.