## Synchronized operation of s-SPDC and UPC

Figure 1 shows a conceptual diagram of the s-SPDC generator and UPC detector. Measuring the correlation of photon pairs requires temporally well separated photons in excess of the dead time of photon counting detectors. It follows that pulsed, not continuous-wave, pumps are suitable for exciting the s-SPDC process. An UPC detector pumped by a pulse laser is a significant technique to achieve high detecting efficiency of infrared photons. Moreover, a fiber-based mode-locked laser system provides an effective way to prepare stable and high-power pulse sources for both processes. Therefore, we prepared repetition-synchronized lasers which can be used for generating and detecting the SPDC pairs.

The s-SPDC process takes place with the pump wavelength in the range of 600–900 nm in the PPLN13. Especially, a spectral range of 2–5 µm for the idlers, which is of a great importance for infrared spectral measurements, can be covered by a pump at around 780 nm. Therefore, we used a frequency-doubled erbium-doped fiber laser (EDFL) as a s-SPDC pump. In contrast, the UPC pump pulse was generated by an ytterbium-doped fiber laser (YDFL) oscillating at 1030 nm in order to amplify efficiently the pulse intensity with high-gain ytterbium-doped double-clad fibers.

The output wavelengths of the s-SPDC pairs were simulated under the quasi-phase-matching (QPM) condition:

$$\varDelta K=\left|\overrightarrow{{K}_{1}}-\overrightarrow{{K}_{2}}-\overrightarrow{{K}_{3}}-\overrightarrow{{G}_{\varLambda }}\right|=0 . \left(1\right)$$

Here, \(\left|\overrightarrow{{K}_{i}}\right|=2\pi n\left({\lambda }_{i},T\right)/{\lambda }_{i}\) denotes the wave vectors of the pump (\(i\)=1), idlers (\(i\)=2), and signals (\(i\)=3). \(\left|\overrightarrow{{G}_{\varLambda }}\right|=2\pi /\varLambda\) is the superlattice vector obtained from the periodicity Λ for compensating the phase mismatch. \(n\left({\lambda }_{i},T\right)\) is the refractive index of the crystal at wavelength \({\lambda }_{i}\) and temperature \(T\). Figure 2 shows the phase-matched wavelengths of signals and idlers for various periodicities in the s-SPDC process in PPLN pumped at 780 nm and at T = 25℃. The plotted values were calculated using the temperature-dependent Sellmeier Eq. 21. In the case of a periodicity of 21.5 µm (the dashed line), two pairs, 980 nm and 3810 nm (pair 1) and 1013 nm and 3390 nm (pair 2), were simultaneously generated in a collinear direction. In order to detect the signals in Si photon counting detector (400–1100 nm), we chose a periodicity of 21.5 µm. Note that the non-collinearly phase-matched signals (idlers) covered the spectral range between the two collinear signals (idlers)13. For the periodicity of 21µm, the non-collinear idlers within the radiated angle of 5 degrees covered the infrared range of 2–5 µm. These results show that the non-collinear SPDC pairs can function as a broadband quantum light source with multi-dimensional correlation.

## Specifications of the UPC and SPDC pumps

We constructed a synchronized laser system composed of two fiber-based oscillators. The experimental setup is illustrated in Fig. 3. The two fiber-based lasers were repetition-synchronized by injecting the EDFL pulse into the YDFL cavity22. The EDFL system for pumping the s-SPDC is combined with the YDFL system for UPC by injecting the EDFL pulse into the YDFL oscillator. Cross-phase modulation between the pulses enabled synchronous repetition at 79.6 MHz. (The details of the laser sources are explained in Supplementary 1.) The EDFL pulse is divided into two portions, which respectively serve as the master laser for synchronization and as the s-SPDC pump after SHG. Figure 4(c) show the specifications of the SPDC pump. The s-SPDC pump is generated at 780 nm with 10 mW average power and a 0.4-nm FWHM. Note that the s-SPDC pump was attenuated to less than 1 mW in coincidence measurements in order to suppress the accidental events. The details of optimizing the pump power are explained in Supplementary 2. The spectral and temporal full width at half maxima (FWHM) of the UPC pump are 0.6 nm and 4.3 ps as shown in Fig. 4(a) and (b), which are 1.7 times those of a Fourier-transformed pulse. Moreover, we measured cross-correlation between the UPC pump and idlers in order to estimate the temporal profile of the idlers, with an FWHM of 6.0 ps. It means that a pulse duration of the incident idlers should be 4.1 ps, which is comparable to the temporal width of the UPC pump. According to the theoretical analysis, the conversion efficiency reduces by 30% because of such a temporal mismatch between the UPC pump and idlers. (The estimation details are shown in Supplementary 3.)

## Specifications of the up-converter

In order to verify that the UPC detector is useful for detecting the idlers, we theoretically estimated the conversion efficiency and experimentally demonstrated its validity. According to the theoretical simulation of s-SPDC, the total flux of the idlers is around \({10}^{7}\) photons/s in the collinear SPDC process with the pump average power 10 mW. It is difficult to detect such a microscopic output flux by a broad-bandwidth photodetector. Moreover, the conversion efficiency must be sufficiently high to preserve the quantum information after UPC. Using an intense pulse with a narrowband and short-temporal duration as the UPC pump attains high conversion efficiency20,23 and allows the number of the weak SPDC idler photons to be counted with sensitive detectors such as a Si-based single photon avalanche photodiode (Si-SPAD). The conversion efficiency \({\Phi }\) from the infrared idler with a center frequency at \({\omega }_{2}\) to the visible photon at \({\omega }_{3}\) is expressed as follows23:

$${\Phi }=2{Z}_{0}{\gamma }^{2}{I}_{1}{L}^{2}{\text{sinc}}^{2}\left[L{\left\{{\left(\frac{\varDelta k}{2}\right)}^{2}+2{Z}_{0}{\gamma }^{2}{I}_{1}\right\}}^{\frac{1}{2}}\right] \left(2\right),$$

where

$$\gamma =\frac{{d}_{eff}}{c}\sqrt{\frac{{\omega }_{2}{\omega }_{3}}{{n}_{2}{n}_{3}}} \text{a}\text{n}\text{d} \varDelta k=\left|\overrightarrow{{k}_{1}}+\overrightarrow{{k}_{2}}-\overrightarrow{{k}_{3}}+\overrightarrow{{G}_{\varLambda }}\right|,$$

and where \(\left|\overrightarrow{{k}_{i}}\right|=2\pi n\left({\lambda }_{i},T\right)/{\lambda }_{i}\) denotes the wave vector of the pump (\(i\)=1), infrared (\(i\)=2) and up-converted (\(i\)=3) light. \({Z}_{0}\) is the impedance of free space, \({I}_{1}\) is the peak intensity of the UPC pump, and \(L\) is the interaction length in the nonlinear crystal. \({d}_{eff}\) is the effective nonlinear coefficient reduced by the periodic structure. The simulated efficiency at the maximum UPC pump power 190 mW was estimated to be 61%. In addition, as Eq. (2) indicates, the phase-matched wavelengths cover a certain range depending on the pump spectral width and the crystal length. For a spectral width of 0.6 nm, the UPC windows are calculated to be 29.3 nm for translating the infrared light at around 3810 nm, which corresponds to 1.95 nm for the accompanying signals. We describe the spectral and temporal windows of the UPC detector in Supplementary 4. In our experimental setup, the UPC efficiency at the maximum UPC pump power 190 mW is estimated to be 29% in a 29.3 nm spectral window. Here, the maximum pump power is restricted by self-phase-modulation (SPM) in the amplifying fiber because SPM broadens the spectral and temporal profiles of the UPC pulse. Therefore, SPM reduces the peak intensity and the conversion efficiency in turn. The details of the efficiency estimation are explained in Supplementary 3.

## Spectral filtering for extracting accompanying pairs

To measure the correlation of the s-SPDC photons, the temporal timing and spectral width of the signals must be identical to those of the up-converted idlers. For the temporal timing, the group velocity mismatch between the signals and idlers in the PPLN should be less than the repetition period 12.5 ns of the UPC pump pulse train. The temporal delay between 980 nm and 3810 nm induced in a 20-mm-long crystal is calculated to be 3.1 ps, which is much shorter than the repetition period 12.5 ns. However, the spectral range of the signals needs post-filtering due to the non-collinear phase-matched SPDC photons and the tolerant bandwidth of the up-converter. As described above, the UPC process also functions as a bandpass filter with a 29.3-nm-wide converting band. Therefore, extracting only the corresponding signals is required.

First, we compared simulated and experimental spectra of the up-converted idlers. Figure 5(a) shows the spectrum of idlers simulated using the equation of the total flux \({F}_{2}\) of the idlers as follows13:

$${F}_{2}=\int d{\omega }_{2}d{\theta }_{2}\frac{{\omega }_{2}^{3}{\omega }_{3}{n}_{2}^{2}{L}^{2}{d}_{eff}^{2}{P}_{1}}{2{\pi }^{2}{c}^{5}{\epsilon }_{0}{n}_{1}{n}_{3}}\frac{\text{sin}{\theta }_{2}}{{\text{cos}}^{3}{\theta }_{2}} \text{S}\text{i}\text{n}{\text{c}}^{2}\left[\frac{\varDelta KL}{2}\right] \left(3\right)$$

Here \({\theta }_{2}\) is the output angle of the idlers against the collinear direction, and \({P}_{1}\) is the SPDC pump power. The non-collinear components within the radiated angle of 0.8 degree covered a range of 3390–3810 nm. Considering such broadband coverage, we extracted the spectra at the different crystal temperatures and periodicities in which the idlers were phase-matched at different output angles, as shown in Fig. 5(b). The two outermost peaks (conditions a and h) corresponded to the collinear pairs indicated by the green markers in Fig. 2. The intensity of the non-collinear idlers (conditions b-g) was smaller than that of the collinear ones because of the mismatch of the spatial mode and shrinking of the interaction length in the nonlinear crystal. The spectral widths of the up-converted collinear idlers in the infrared region were 29.3 nm for 3810 nm and 19.8 nm for 3390 nm; these values are in good agreement with the theoretical UPC window. The accompanying signals corresponded to spectral widths of 1.95 nm for 980 nm and 1.85 nm for 1013 nm.

Second, we controlled the signal spectra. The dashed curve in Fig. 5(c) is the theoretical simulation of a broad non-collinear spectrum with a coverage range of 980–1013 nm. It fairly closely matches the experimental curve indicated by the solid curve. In order to suppress noise counts in the coincidence measurements, we extracted only the signal components corresponding to the up-converted idlers by using a narrow band-pass filter. The spectra of the extracted signals are shown in Fig. 5(d). Again, the two outermost peaks (conditions i and o) correspond to the collinear pairs indicated by the red markers in Fig. 2. The non-collinear components were measured by changing the angle of the filter. The widths of the signals were estimated to be 2.4 nm for 980 nm and 3.3 nm for 1013 nm. This mismatch reduced the purity of the correlation by a few percent.

## Coincidence measurement of simultaneous SPDCs

We performed coincidence measurements on the filtered signals and up-converted idlers. The coinciding events were logged in the TCSPC module as a function of the time delay between the two detectors when both Si-SPADs detected photons within a 50-ps temporal bin, much shorter than the system temporal resolution 1 ns. In order to compare the numbers of coincident events for the correlated and uncorrelated pairs, the temporal window was determined by the active width of the coincidence histogram. We performed the measurements for relevant four pairs taken from either of two signals (980 nm and 1013 nm) and either of two idlers (3810 nm and 3390 nm). Figures 6(a) and (d) shows typical histograms of coincident counts of the two correlated pairs with the SPDC pump power 0.93 mW, in which clear peaks were observed in a temporal window of 1 ns. In contrast, the histograms of the two uncorrelated pairs exhibited little counts in Figs. 6(b) and (c). These four histograms show that the s-SPDC pairs were produced by only one photon, not two independent photons. Therefore, it is suggested that the total quantum state of the s-SPDC pairs\(|\psi ⟩\) should be depicted as follows:

$$|\psi ⟩=\left|{\omega }_{s1}⟩\right|{\omega }_{i1}⟩+\left|{\omega }_{s2}⟩\right|{\omega }_{i2}⟩ \left(4\right)$$

Here \(|{\omega }_{s\left(i\right)j}⟩\) is the single-photon state of the signal (idler) of the pair j (j = 1,2) with center frequency \({\omega }_{s\left(i\right)j}\). Note that a finite number of coincident counts are accidentally logged in the temporal origin even for the uncorrelated pairs as shown in Figs. 6(b) and (c). The origin is attributed to uncorrelated coincidental events following the Poisson distribution and up-converted noise photons from the UPC pump itself24. The characterization of the accidental events is described in Supplementary 2.

Moreover, we measured accidental-coincidence counts that were outside the window to calculate the coincidence to accidental-coincidence ratios, CAR:

$$CAR=\frac{{C}_{tr}}{{C}_{ac}}=\frac{{C}_{to}-{C}_{ac}}{{C}_{ac}} \left(5\right)$$

We investigated accidental events \({C}_{ac}\) between the signals (idlers) and uncorrelated photons by delaying the counting window by a cycle of the laser repetitions lasting 12.5 ns. The number of true coincidence counts \({C}_{tr}\) was calculated by subtracting \({C}_{ac}\) from the total counts \({C}_{to}\). Figure 7 plots CAR as a function of the SPDC pump power. At an average power of 0.93 mW, the maximum CAR values of 6.2 and 6.5 were attained for SPDC pair 1 and 2. These values were limited by the number of the uncorrelated photons produced by the mismatch of filtering or the total quantum efficiency including the detector sensitivity, and are relatively smaller than values determined in coincidence experiments using a pulsed UPC or cavity-type UPC18,19. The purity of the correlation is a fundamental parameter specifying the availability of correlated pairs for QICT or QIS. Here, there is room for improving the CAR by employing detectors that are more sensitive in the near-infrared region covering the signal wavelengths. In the present setup, the quantum efficiency of the Si-SPAD is around 3% for signals at around 1000 nm. We can radically improve the CAR value by replacing the detector with a more sensitive one with an efficiency of 15%. Additionally, the use of a proper band-pass filter for strictly extracting the corresponding SPDC pairs or a single-mode fiber coupling system for introducing the photons into SPAD can reduce uncorrelated photons radiating in non-collinear directions.