## Abstract

A high visibility Hong-Ou-Mandel (HOM) interference between two independently prepared photons plays an important role in various photonic quantum information processing. In a standard HOM experiment using photons generated by pulse-pumped spontaneous parametric down conversion (SPDC), larger detection time windows than the coherence time of photons have been employed for measuring the HOM visibility and/or drawing the HOM dip. If large amounts of stray photons continuously exist within the detection time windows, employing small detection time windows is favorable for reducing the effect of background noises. Especially, such a setup is helpful for the HOM experiment using continuous wave (cw)-pumped SPDC and the time-resolved coincidence measurement. Here we argue that the method for determining the HOM visibility used in the previous cw experiments tends to suffer from distortion arising from biased contribution of the background noises. We then present a new method with unbiased treatment of the cw backgrounds. By using this method, we experimentally demonstrate a high visibility HOM interference of two heralded telecom photons independently generated by SPDC with employing cw pump light. An observed HOM visibility is 0.87 ± 0.04, which is as high as those observed by using pulse-pumped SPDC photons.

© 2017 Optical Society of America

## 1. Introduction

Indistinguishability of photons is one of the most important prerequisites for many applications of photonic quantum information processing [1–3]. The Hong-Ou-Mandel (HOM) interference [4] is a quantum phenomenon of two indistinguishable photons, and has been widely used as a criterion for testing the indistinguishability of photons [5–11]. A conventional HOM experiment is performed by injecting two photons into two input ports of a half beamsplitter (HBS) and detecting coincidence counts of photons coming from the output ports as shown in Fig. 1(a). When arrival time difference of the two photons is sufficiently large, the detected photons are completely distinguishable as if they are classical particles. As a result a coincidence probability becomes the product of a single detection probabilities of the two detectors. On the other hand, when they simultaneously arrive at the HBS, the coincidence probability will be decreased by bunching effect of bosonic photons while the single detection probability at each detector do not change as shown in Fig. 1(b) and 1(c). Due to such an effect, a dip known as the HOM dip can be observed in the coincidence counts by sweeping the arrival time difference of the input photons. The depth of the HOM dip reflects the degree of indistinguishability of the two input photons which is evaluated by the so-called HOM visibility.

Such HOM experiments have been performed by using two heralded photons produced by a spontaneous parametric down conversion (SPDC) employing pulsed pump, in which detection time windows are usually set to be much larger than the wavepacket duration (also see Fig. 1(a)). On the other hand, when there are non-negligible amounts of temporally-continuous stray photons within the detection time windows, as in the case of the HOM experiment using continuous wave (cw) pump, it is favorable to open detection windows only when the signal photon wavepackets come for reducing the contribution of stray photons. Recently, such measurement has been performed with the use of two photons heralded by one halves of photon pairs produced by cw-pumped SPDC [12, 13]. In [13], Halder *et al.* reported that an observed visibility was 0.77 ± 0.03. But as we will explain later, the visibility determined in [12,13] does not completely reflect the conventional HOM visibility commonly used in pulsed regime and tends to result in an artificial increase in the observed value.

In this paper, we present a new method to determine the HOM visibility with the use of detection windows smaller than the coherence time of the photons. We experimentally observed a HOM visibility of 0.87 ± 0.04 using two heralded single photons in separated time bins prepared by detecting halves of telecom photon pairs. The observed HOM visibility is as high as those observed by using pulsed pump [5–11]. If we evaluate our experimental result by using the same method used in [12,13], we observe the visibility of 0.93 ± 0.03, which is clearly much higher than that observed in [12,13].

## 2. Method

We first review the standard method for observing the HOM interference experiments. Then we discuss an extended method that reduces an effect of stray photons or accidental coincidences by shortening a detection time window even below the durations of light wavepackets. As shown in Fig. 1(a), the light wavepackets in modes 1 and 2 are mixed by a HBS followed by photon detectors D3 and D4. We assume that a relative delay (timing difference) Δ*t* between two incoming light wavepackets is precisely determined in advance. The delay line is introduced in mode 2 for the adjustment of Δ*t*.

As is well known, by counting the two-fold coincidence events between D3 and D4 with detection time windows much larger than the light wavepacket duration for various values of Δ*t*, the HOM dip will be observed, while the single counts of D3 and D4 take constant values [4–11] as shown in Fig. 1(b) and 1(c). The visibility of the HOM interference is represented by *V* = 1 − *P*_{0}/*P*_{∞}, where *P*_{0} and *P*_{∞} are the two-fold coincidence probabilities with Δ*t* = 0 and Δ*t* = *T*, respectively, where *T* is chosen to be much larger than the wavepacket duration of input light.

The two-fold coincidence probability *P* can be understood as follows: As shown in Fig. 2, we assume that a common fixed time window of width 2*T* is adopted for D3 and D4, regardless of the amount of delay Δ*t*. Let us divide the window into two and denote the two halves by *x* and *y*. Figs. 2(a) and 2(b) show the sketches of the light wavepackets with Δ*t* = 0 (a) and that with Δ*t* = *T* (b), respectively. The light from mode 1 is always detected at the center of *x*, while the one from mode 2 is detected in *x* for Δ*t* = 0 and in *y* for Δ*t* = *T*. Let us denote by 3*x* a detection event at D3 in period *x*, which may be caused by the input signal light, stray background photons, or dark counting, and define 4*x*, 3*y*, 4*y* similarly. Then, the coincidence probability *P* determined in this conventional setup corresponds to the rate of the sum of event (3*x* & 4*x*), (3*y* & 4*x*), (3*x* & 4*y*) and (3*y* & 4*y*). An underlying assumption in this method is that the contribution of backgrounds should be the same for Figs. 2(a) and 2(b), which is satisfied if the statistical properties of stray photons are time-invariant.

If background photons are not negligible, shortening the widths of time windows *x* and *y* is favorable. In this case, each width of the time window may be close to or even much shorter than the wavepacket of signal light width as shown in Fig. 2(c) and 2(d). In this paper, we adopt this regime for suppressing stray photons. The coincidence probability *P* is calculated from the rate of the sum of event (3*x* & 4*x*), (3*y* & 4*x*), (3*x* & 4*y*) and (3*y* & 4*y*), just as in the conventional setup. The contribution of stray photons should be the same for Figs. 2(c) and 2(d) if the statistical properties of stray photons are time-invariant, which is also normally assumed in the conventional setup.

We emphasize that in order to observe a HOM dip, one may change delay Δ*t* of the input signal light coming from mode 2 with sweeping the coincidence windows of modes 3*y* and 4*y* synchronously, which was indeed the method used in the previous experiments [12, 13]. *P*_{∞} corresponds to the rate of the sum of event (3*x* & 4*x*), (3*y* & 4*x*), (3*x* & 4*y*) and (3*y* & 4*y*) as in the conventional HOM experiment as shown in Fig. 2(b) and 2(d). However, when the time difference of the two signal light becomes small, the tail of the signal light wavepacket outside of the coincidence window in mode *x* (*y*) invades the inside of the window in mode *y*(*x*), which leads to the increase of the detection probability at D3(D4). To make matters worse, when the two coincidence windows are completely overlapped (Δ*t* = 0), the coincidence windows in modes (3*y* & 4*y*) in Fig. 2(c), which include only background stray photons, are removed. As a result, the amount of background photons becomes half of that with Δ*t* = *T*, which fakes a higher HOM visibility than the proper one in the original method or ours.

## 3. Experiment

#### 3.1. Photon pair source

Our experimental setup is shown in Fig. 3(a). An initial beam from an external cavity diode laser (ECDL) working at 1560 nm with a linewidth of 1.8 kHz is frequency doubled by using a periodically-poled lithium niobate waveguide (PPLN/W) [14]. The frequency of the ECDL is stabilized by the saturated absorption spectroscopy of rubidium atoms. The obtained cw light at 780 nm is set to be vertically (V) polarized and is coupled to a 40-mm-long and type-0 quasi-phase-matched PPLN/W. It generates non-degenerate photon pairs at 1541 nm and 1580 nm by SPDC. The V-polarized photons at 1541 nm and 1580 nm are separated into different spatial modes by a dichroic mirror (DM). The photons at 1541 nm are flipped to horizontal (H) polarization by a half-wave plate (HWP), and they are coupled to a polarization maintaining fiber (PMF) followed by a fiber-based Bragg grating (FBG). The photons at 1580 nm are also coupled to a PMF followed by a FBG.

We first characterized our photon pair source by measuring the single counts and the coincidence counts just after the FBGs in Fig. 3(b). In order to evaluate the photon pair generation efficiency, we set the bandwidths of FBG_{1541} and FBG_{1580} to be 1 nm. For preventing the saturation of the detectors, we set the pump power to be *p* = 105 *μ*W. The observed coincidence count rate is *C* = 1.8×10^{5} counts/(s·nm) and single count rates are *S*_{1541} = 2.0×10^{6} counts/(s·nm) for 1541 nm and *S*_{1580} = 3.0 × 10^{6} counts/(s·nm) for 1580 nm. The single count rates and the coincidence count rate are represented by *S*_{1541} = *γpη*_{1541}, *S*_{1580} = *γpη*_{1580} and *C* = *γpη*_{1541}*η*_{1580}, where *γ* is a photon pair generation efficiency and *η _{λ}* is a overall transmittance of the system for the photon at

*λ*nm. We estimated

*γ*=

*S*

_{1541}

*S*

_{1580}/(

*Cp*) just after the PPLN/W to be about 3.2 × 10

^{8}pairs/(s·mW·nm). When we performed the HOM experiment described in detail later, we connected the output of the FBGs to the fiber-based optical circuit and set the bandwidths of FBG

_{1541}to be 30 pm and FBG

_{1580}to be 10 pm, respectively. In addition, we set the coincidence window to be 80 ps and the pump power to be 2.5 mW. In this setup, the observed coincidence count rate between detectors D1 and D3 was 1.0 × 10

^{3}counts/s.

#### 3.2. Detection system

Next, we measured the timing jitter of the overall timing measurement system with superconducting nanowire single photon detectors (SNSPDs) [15]. For the measurement of the timing jitter, we used bandwidths (3 nm) for FBG_{1541} and FBG_{1580} which correspond to the temporal width of 1.2 ps, and measured the arrival time of the picosecond photons by directly connecting the output of the FBGs to the detectors. In this setup, since the temporal widths of the photons are sufficiently shorter than the timing jitter of the system, the distribution of arrival time difference of the photons reflects the timing jitter of the system. The typical experimental result is shown in Fig. 4(a). By fitting the experimental data by Gaussian, the deconvolved timing jitter is estimated to be 85 ps FWHM for all of the four detectors. We also estimated the coherence time *τ*, which is defined by the temporal width of the photon after FBG_{1541}(30 pm) heralded by the half of the photon pairs which is filtered by FBG_{1580}(10 pm). In our experiment, the temporal distribution obtained by the coincidence detection reflects the convolution of *τ* and timing jitters of the photon detectors. Fig. 4(b) shows the two-fold coincidence count between D1 and D3, and the fitted Gaussian curve. By subtracting the effect of the timing jitter of each detector from the FWHM of the fitted Gaussian, *τ* is estimated to be 231 ps FWHM.

#### 3.3. HOM interference

The HOM interference is performed by using a fiber-based optical circuit as shown in Fig. 3(b). The photons at 1580 nm are filtered by FBG_{1580} with a bandwidth of 10 pm. After the FBG, the photons are split into two different spatial modes by a fiber-based half beamsplitter (FHBS) followed by SNSPDs. The photon detections by D1 and D2 with a time difference of Δ*t* heralds two photons at 1541 nm with the same time difference Δ*t*. They are filtered by FBG_{1541} with a bandwidth of 30 pm as shown in Fig. 3(b). After passing through a fiber-based polarizing beamsplitter (FPBS), the two H-polarized photons propagate in a single-mode fiber (SMF), and they are separated into a long path and a short path by a FHBS. The photons are reflected by a Faraday mirror (FM) at the end of each path, at which the polarizations of the photons are flipped. After going back to the FHBS, the two photons are split into two paths and they are detected by D3 and D4. We note that each of the two photons receives polarization fluctuation in the SMF. In the optical circuit, such fluctuations are much slower than the round-trip time of the photon propagation in each path and thus the birefringence effect in the SMF is automatically compensated after passing through each path by using the FM [16].

We collect the four-fold coincidence events by using a time-digital converter (TDC). The electric signal from D1 is used as a start signal, and the electric signals from D2, D3 and D4 are used as stop signals. When the photons are detected at D1, the histograms of the delayed coincidence counts in the stop signals at D2 and D3 are obtained as shown in Figs. 5(a) and 5(b), respectively. In Fig. 5(b), left peak S and right peak L indicate the events where the photons at 1541 nm passed through the short and long paths, respectively. When we postselect the events where the stop signal of D2 has a delay Δ*t*, additional two peaks S′ and L′ appear in the histogram of D3 as in Fig. 5(c), which shows the three-fold coincidence among D1, D2 and D3. The two-fold coincidence between D1 and D4, and the three-fold coincidence among D1, D2 and D4 show almost the same histograms as Fig. 5(b) and 5(c), respectively. We chose a value of Δ*t* = *t*_{1} such that the two peaks S′ and L are separated, and defined the timing of the peak S′ as *y* and that of peak L as *x*. Then we determined the four-fold coincidence count *C*_{∞} from the events where the stop signals of D2, D3 and D4 were recorded at the timings *t*_{1}, (*x* or *y*) and (*x* or *y*), respectively, with all of the coincidence windows having a width of 80 ps. This corresponds to the situation shown in Fig. 2(d). If we chose the delay for D2 to be Δ*t* = *t*_{0} such that peak S′ is overlapped on L, the post-selected histogram of the three-fold coincidence among D1, D2 and D3 becomes as shown in Fig. 5(d). In this case, we determined the four-fold coincidence count *C*_{0} by selecting the stop signals with the timing of *t*_{0} for D2, and (*x* or *y*) for D3 and D4. This corresponds to the situation shown in Fig. 2(c). The visibility is obtained by *V* = 1 − *C*_{0}/*C*_{∞}.

We set the pump power coupled to PPLN/W to be 2.5 mW. The total measurement time is 156 hours. By the experimental data, we obtained *C*_{∞} and *C*_{0} as 87 counts and 11 counts, respectively. The observed visibility was 0.87 ± 0.04. This value is as high as those observed by using pulsed sources.

We mention that if we employ the same method used in [12,13] instead of the present method, the visibility was calculated to be 0.93. This value is clearly much higher than that observed in [12,13].

## 4. Discussion

We consider the reason for the degradation of the observed visibility of the HOM interference. In our experiment, since the effect of the timing jitters of the detectors is estimated to be small due to the large coherence time, we only consider the effect of stray photons including multiple pair emission. Below we derive a simple relationship among the visibility and the intensity autocorrelation functions of the signal light wavepackets and stray photons. For simplicity, we assume that the signal photons and stray photons observed in a time window *x* or *y* are in a single mode. We define creation operators of the input and the output light of the HBS as
${\widehat{a}}_{ik}^{\u2020}$ and
${\widehat{b}}_{jk}^{\u2020}$, respectively, where *i* = 1, 2, *j* = 3, 4 and *k* = *x*, *y*. The above operators satisfy the commutation relations
$\left[{\widehat{a}}_{ik},{\widehat{a}}_{{i}^{\prime}{k}^{\prime}}^{\u2020}\right]={\delta}_{i{i}^{\prime}}{\delta}_{k{k}^{\prime}}$ and
$\left[{\widehat{b}}_{jk},{b}_{{j}^{\prime}{k}^{\prime}}^{\u2020}\right]={\delta}_{j{j}^{\prime}}{\delta}_{k{k}^{\prime}}$. As in our experiment, we assume that the transmittance *η _{i}* of the system including the detection efficiency of D

*i*for

*i*= 3, 4 is much less than 1 such that the events where two or more photons are simultaneously detected at the single detector are negligible, and the detection probabilities are proportional to the photon number in the detected mode. The two-fold coincidence probability

*P*is expressed as

*P*=

*η*

_{3}

*η*

_{4}〈: (

*n̂*

_{3x}+

*n̂*

_{3y})(

*n̂*

_{4x}+

*n̂*

_{4y}) :〉, where ${\widehat{n}}_{jk}={\widehat{b}}_{jk}^{\u2020}{\widehat{b}}_{jk}$ is the number operator for output modes

*j*= 3, 4 and

*k*=

*x*,

*y*.

*P*

_{0}and

*P*

_{∞}are calculated by using

*P*for the input signal light wavepackets coming from 1

*x*& 2

*x*and 1

*x*& 2

*y*, respectively. As is often the case with practical settings, we assume that the signal light wavepackets and stray photons have no phase correlation and are statistically independent. We characterize the property of stray photons alone by the average photon number

*n*and the normalized intensity correlation function ${g}_{n}^{(2)}$. That is to say, if mode

*ik*does not include the signal light, $\u3008{\widehat{a}}_{ik}^{\u2020}{\widehat{a}}_{ik}\u3009=n$ and $\u3008:{\left({\widehat{a}}_{ik}^{\u2020}{\widehat{a}}_{ik}\right)}^{2}\u3009/{n}^{2}={g}_{n}^{(2)}$ holds. Similarly, we define

*s*and ${g}_{s}^{(2)}$ such that, when mode

*ik*includes the signal light as well as stray photons, $\u3008{\widehat{a}}_{ik}^{\u2020}{\widehat{a}}_{ik}\u3009=s$ and $\u3008:{\left({\widehat{a}}_{ik}^{\u2020}{\widehat{a}}_{ik}\right)}^{2}:\u3009/{s}^{2}={g}_{s}^{(2)}$ holds. From the definitions, we obtain ${P}_{0}={\eta}_{3}{\eta}_{4}\left({s}^{2}{g}_{s}^{(2)}+{n}^{2}{g}_{n}^{(2)}+4sn\right)/2$ and ${P}_{\infty}={\eta}_{3}{\eta}_{4}\left({s}^{2}{g}_{s}^{(2)}+{n}^{2}{g}_{n}^{(2)}+{\left(s+n\right)}^{2}\right)/2$. Here we used a unitary operator

*Û*of the HBS satisfying $\widehat{U}{b}_{3k}^{\u2020}{\widehat{U}}^{\u2020}=\left({\widehat{a}}_{1k}^{\u2020}+{\widehat{a}}_{2k}^{\u2020}\right)/\sqrt{2}$ and $\widehat{U}{b}_{4k}^{\u2020}{\widehat{U}}^{\u2020}=\left({\widehat{a}}_{1k}^{\u2020}+{\widehat{a}}_{2k}^{\u2020}\right)/\sqrt{2}$. Thus

*V*

_{theory}is represented by

*χ*=

*n*/

*s*. When

*n*≪

*s*, the visibility is approximated by ${V}_{\text{theory}}=1/\left(1+{g}_{s}^{(2)}\right)$, and it takes maximum of

*V*

_{theory}= 1 for the input of genuine single photons [17].

In our experimental setup in Fig. 3, the intensity correlation
${g}_{s}^{(2)}$ of the signal light could be measured by using the coincidence between D3 and D4 under the heralding of D1 if we run an additional experiment with the short arm detached from the FHBS. Instead, we calculated the same quantity from the same experimental data gathered for the main result. This implies that the determined value
${g}_{\mathit{ex}}^{(2)}$ includes the contribution of the stray photons coming into FHBS from the short arm as shown in Fig. 6(a). The single count probability *S*_{3(4)} at D3(4) and coincidence count probability *C*_{34} between D3 & D4 conditioned on the photon detection at D1 are represented by *S _{i}* =

*η*(

_{i}*s*+

*n*)/2 and ${C}_{34}={\eta}_{3}{\eta}_{4}\left({s}^{2}{g}_{s}^{(2)}+{n}^{2}{g}_{n}^{(2)}\right)/4$. The measured quantity is then given by ${g}_{\mathit{ex}}^{(2)}\equiv {C}_{34}/\left({S}_{3}{S}_{4}\right)=\left({g}_{s}^{(2)}+{\chi}^{2}{g}_{n}^{(2)}\right)/{\left(1+\chi \right)}^{2}$, which is equal to ${g}_{s}^{(2)}$ for

*χ*= 0. Using this measured quantity, the visibility in Eq. (1) is represented by

The number of the single counts at D3 conditioned on the photon detection at D1 is shown in Fig. 6(b). The single count probability with (without) the signal within the time window is represented by *S*_{3} = *η*_{3}(*s* + *n*)/2 (*S*_{3,∞} = *η*_{3}*n*). Hence *χ* is estimated from the observed values *S*_{3} and *S*_{3,∞} as

*χ*= 1.4 × 10

^{−2}. The visibility is calculated to be

*V*

_{theory}= 0.90 from Eq. (2), which is in good agreement with the observed visibility of

*V*= 0.87 ± 0.04. Furthermore, we performed the HOM experiment for three different values of the pump power, and compared

*V*and

*V*

_{theory}. The experimental results are shown in Table 1.

*χ*and ${g}_{\mathit{ex}}^{(2)}$ increase according to the pump power because the amount of the temporally-continuous stray photons increases. We see that the visibilities

*V*

_{theory}estimated from

*χ*and ${g}_{\mathit{ex}}^{(2)}$ are in good agreement with directly-observed visibilities

*V*for the pump powers. These results indicate that the degradation of the visibilities is mainly caused by the stray photons. Since the main cause of the stray photons is the multiple emission of photon pairs from SPDC, a higher visibility will be obtained by the suppression of the multiple emission with the use of a lower pump power and a shorter timing selection for the photon detection.

## 5. Conclusion

In conclusion, we have performed an experiment of the HOM interference between two photons produced by the two independent SPDC processes with cw pump light. In order to evaluate the unbiased interference visibility of the heralded signal light wavepackets, we introduced a new method to observe the HOM visibility. We observed the visibility of 0.87 ± 0.04, which is comparable to those observed by using pulsed sources. We also presented a simple relation between the visibility and the second order intensity correlation function, and showed that it holds with a good approximation in this experiment. The relation is convenient for the estimation of the possible visibility from the second order intensity correlation functions of the various input light sources. The results presented here will be useful for many applications such as quantum relays [18,19], quantum repeaters [20,21], mesurement-device-independent quantum key distribution [22–24] and distributed quantum computation [25] without the necessity of performing active synchronizations of the photon sources.

## Appendix

We show that the relation among *V*, *χ* and
${g}_{\mathit{ex}}^{(2)}$ in the main text holds true even under the presence of stationary photon noise in modes different from the signal modes, as long as noise photons in different modes are statistically independent. The situation is shown in Fig. 7. The additional noises accompanying the input signal wavepacket of mode 1*x* are decomposed into mutually orthogonal modes labeled by index *l*, and its creation operator is denoted by
${\widehat{a}}_{l,1x}^{\u2020}$. The signal mode is orthogonal to every noise mode, namely,
$\left[{\widehat{a}}_{1x}^{\u2020},{a}_{l,1x}^{\u2020}\right]=0$. We further define the creation operators
${\widehat{a}}_{l,2x}^{\u2020}$,
${\widehat{b}}_{l,3x}^{\u2020}$, and
${\widehat{b}}_{l,4x}^{\u2020}$ such that the four modes with a common index *l* form the input and output modes of HBS. We define the modes in window *y* similarly as
${\widehat{a}}_{l,iy}^{\u2020}$,
${\widehat{b}}_{l,jy}^{\u2020}$. Let us define sums of the photon number operators by
${\widehat{N}}_{ik}={\sum}_{l}{\widehat{a}}_{l,ik}^{\u2020}{\widehat{a}}_{l,ik}$ and
${\widehat{N}}_{jk}={\sum}_{l}{b}_{l,jk}^{\u2020}{\widehat{b}}_{l,jk}$. Since the noise photons are assumed to be stationary, we have 〈*N̂ _{ix}*〉 = 〈

*N̂*〉 and 〈:

_{iy}*N̂*

_{3x}

*N̂*

_{4x}:〉 = 〈:

*N̂*

_{3y}

*N̂*

_{4y}:〉. Define

*N*:= (〈

*N̂*

_{1x}〉 + 〈

*N̂*

_{2x}〉)/2 = (〈

*N̂*

_{1y}〉 + 〈

*N̂*

_{2y}〉)/2.

We first calculate the two-fold coincidence probability *P* = *η*_{3}*η*_{4}〈:(*n̂′*_{3x} + *n̂′*_{3y})(*n̂′*_{4x} + *n̂′*_{4y}):〉 between D3 and D4 in Fig. 1 in the main text, where *n̂′ _{jk}* =

*n̂*+

_{jk}*N̂*. When the input signal light wavepackets come from 1

_{jk}*x*& 2

*x*, one of the four terms in

*P*is calculated as $\u3008:{\widehat{n}}_{3x}^{\prime}{\widehat{n}}_{4x}^{\prime}:\u3009=\u3008:\left({\widehat{n}}_{3x}+{\widehat{N}}_{3x}\right)\left({\widehat{n}}_{4x}+{\widehat{N}}_{4x}\right):\u3009=\left(\u3008:{\widehat{n}}_{1x}^{2}:\u3009+\u3008:{\widehat{n}}_{2x}^{2}:\u3009+2\u3008{\widehat{n}}_{1x}+{\widehat{n}}_{2x}\u3009\u3008{\widehat{N}}_{1x}+{\widehat{N}}_{2x}\u3009\right)/4+\u3008:{\widehat{N}}_{3x}{\widehat{N}}_{4x}:\u3009={s}^{2}{g}_{s}^{(2)}/2+2sN+\u3008:{\widehat{N}}_{3x}{\widehat{N}}_{4x}:\u3009$, where 〈

*n̂*

_{1x}〉 = 〈

*n̂*

_{2x}〉 =

*s*and $\u3008:{\widehat{n}}_{1x}^{2}:\u3009=\u3008:{\widehat{n}}_{2x}^{2}:\u3009={s}^{2}{g}_{s}^{(2)}$. Here we used 〈

*n̂*〉 = 〈

_{ik}N̂_{ik}*n̂*〉〈

_{ik}*N̂*〉 from the assumption, and a unitary transformation by the HBS as $\widehat{U}{\widehat{b}}_{l,3k}^{\u2020}{\widehat{U}}^{\u2020}=\left({\widehat{a}}_{l,lk}^{\u2020}+{\widehat{a}}_{l,2k}^{\u2020}\right)/\sqrt{2}$ and $\widehat{U}{\widehat{b}}_{l,4k}^{\u2020}{\widehat{U}}^{\u2020}=\left({\widehat{a}}_{l,1k}^{\u2020}-{\widehat{a}}_{l,2k}^{\u2020}\right)/\sqrt{2}$ in addition to the transformation of the signal modes. Similarly, 〈:

_{ik}*n̂′*

_{3y}

*n̂′*

_{4y}:〉 is calculated as ${n}^{2}{g}_{n}^{(2)}/2+2nN+\u3008:{\widehat{N}}_{3x}{\widehat{N}}_{4x}:\u3009$, where 〈

*n̂*

_{1y}〉 = 〈

*n̂*

_{2y}〉 =

*n*and $\u3008:{\widehat{n}}_{1y}^{2}:\u3009=\u3008:{\widehat{n}}_{2y}^{2}:\u3009={n}^{2}{g}_{n}^{(2)}$. 〈

*n̂′*

_{3x}

*n̂′*

_{4y}〉 and 〈

*n̂′*

_{4x}

*n̂′*

_{3y}〉 are calculated as

*sn*+

*N*(

*s*+

*n*) +

*N*

^{2}. As a result,

*x*& 2

*y*, 〈:

*n̂′*

_{3x}

*n̂′*

_{4x}:〉 and 〈:

*n̂′*

_{3y}

*n̂′*

_{4y}:〉 are calculated as $\left({s}^{2}{g}_{s}^{(2)}+{n}^{2}{g}_{n}^{(2)}\right)/4+N(s+n)+\u3008:{\widehat{N}}_{3x}{\widehat{N}}_{4x}:\u3009$. Similarly, 〈

*n̂′*

_{3x}

*n̂′*

_{4y}〉 and 〈

*n̂′*

_{3y}

*n̂′*

_{4x}〉 are calculated as (

*s*+

*n*)

^{2}/4 +

*N*(

*s*+

*n*) +

*N*

^{2}. As a result, we obtain

Next, we introduce the relation among
${g}_{\mathit{ex}}^{(2)}$,
${g}_{s}^{(2)}$ and
${g}_{n}^{(2)}$ in our experimental setup in Fig. 3 in the main text. The single count *S*_{3(4)} at D3(4) and coincidence count *C*_{34} between D3 & D4 conditioned on the photon detection at D1 are described by *S*_{3(4)} = *η*_{3(4)}(*s* + *n* + 2*N*)/2, and
${C}_{34}={\eta}_{3}{\eta}_{4}\left({s}^{2}{g}_{s}^{(2)}+{n}^{2}{g}_{n}^{(2)}+4N(s+n)+4\u3008:{\widehat{N}}_{3x}{\widehat{N}}_{4x}:\u3009\right)/4$. The observed correlation function is expressed as

*V*

_{theory}= 1 −

*P*

_{0}/

*P*

_{∞}is described by

*χ*= (

*n*+

*N*)/(

*s*+

*N*).

## Funding

Core Research for Evolutional Science and Technology, Japan Science and Technology Agency (CREST, JST) JPMJCR1671; Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research JP16H02214, JP25286077, JP26286068, JP15H03704 and JP16K17772; Japan Society for the Promotion of Science (JSPS) Bilateral Open Partnership Joint Research Projects. Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Research Fellow JP16J05093.

## References and links

**1. **J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. **84**, 777–838 (2012). [CrossRef]

**2. **X.-S. Ma, T. Herbst, T. Scheidl, D. Wang, S. Kropatschek, W. Naylor, B. Wittmann, A. Mech, J. Kofler, E. Anisimova, V. Makarov, T. Jennewein, R. Ursin, and A. Zeilinger, “Quantum teleportation over 143 kilometres using active feed-forward,” Nature **489**, 269–273 (2012). [CrossRef] [PubMed]

**3. **M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nature Photonics **7**, 540–544 (2013). [CrossRef]

**4. **C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. **59**, 2044–2046 (1987). [CrossRef] [PubMed]

**5. **H. d. Riedmatten, I. Marcikic, W. Tittel, H. Zbinden, and N. Gisin, “Quantum interference with photon pairs created in spatially separated sources,” Phys. Rev. A **67**, 022301 (2003). [CrossRef]

**6. **T. Yang, Q. Zhang, T.-Y. Chen, S. Lu, J. Yin, J.-W. Pan, Z.-Y. Wei, J.-R. Tian, and J. Zhang, “Experimental synchronization of independent entangled photon sources,” Phys. Rev. Lett. **96**, 110501 (2006). [CrossRef] [PubMed]

**7. **R. Kaltenbaek, R. Prevedel, M. Aspelmeyer, and A. Zeilinger, “High-fidelity entanglement swapping with fully independent sources,” Phys. Rev. A **79**, 040302 (2009). [CrossRef]

**8. **P. Aboussouan, O. Alibart, D. B. Ostrowsky, P. Baldi, and S. Tanzilli, “High-visibility two-photon interference at a telecom wavelength using picosecond-regime separated sources,” Phys. Rev. A **81**, 021801 (2010). [CrossRef]

**9. **M. Tanida, R. Okamoto, and S. Takeuchi, “Highly indistinguishable heralded single-photon sources using parametric down conversion,” Opt. Express **20**, 15275–15285 (2012). [CrossRef] [PubMed]

**10. **A. McMillan, L. Labonté, A. Clark, B. Bell, O. Alibart, A. Martin, W. Wadsworth, S. Tanzilli, and J. Rarity, “Two-photon interference between disparate sources for quantum networking,” Scientific reports **3**, 2032 (2013). [CrossRef] [PubMed]

**11. **Y. Tsujimoto, Y. Sugiura, M. Ando, D. Katsuse, R. Ikuta, T. Yamamoto, M. Koashi, and N. Imoto, “Extracting an entangled photon pair from collectively decohered pairs at a telecommunication wavelength,” Opt. Express **23**, 13545–13553 (2015). [CrossRef] [PubMed]

**12. **M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon, and H. Zbinden, “Entangling independent photons by time measurement,” Nature Physics **3**, 692–695 (2007). [CrossRef]

**13. **M. Halder, A. Beveratos, R. T. Thew, C. Jorel, H. Zbinden, and N. Gisin, “High coherence photon pair source for quantum communication,” New Journal of Physics **10**, 023027 (2008). [CrossRef]

**14. **T. Nishikawa, A. Ozawa, Y. Nishida, M. Asobe, F.-L. Hong, and T. W. Hänsch, “Efficient 494 mw sum-frequency generation of sodium resonance radiation at 589 nm by using a periodically poled zn:linbo3 ridge waveguide,” Opt. Express **17**, 17792–17800 (2009). [CrossRef] [PubMed]

**15. **S. Miki, T. Yamashita, H. Terai, and Z. Wang, “High performance fiber-coupled nbtin superconducting nanowire single photon detectors with gifford-mcmahon cryocooler,” Opt. Express **21**, 10208–10214 (2013). [CrossRef] [PubMed]

**16. **A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, “Plug and play systems for quantum cryptography,” Applied Physics Letters **70**, 793–795 (1997). [CrossRef]

**17. **R. Ikuta, T. Kobayashi, K. Matsuki, S. Miki, T. Yamashita, H. Terai, T. Yamamoto, M. Koashi, T. Mukai, and N. Imoto, “Heralded single excitation of atomic ensemble via solid-state-based telecom photon detection,” Optica **3**, 1279–1284 (2016). [CrossRef]

**18. **B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Quantum relays and noise suppression using linear optics,” Phys. Rev. A **66**, 052307 (2002). [CrossRef]

**19. **D. Collins, N. Gisin, and H. De Riedmatten, “Quantum relays for long distance quantum cryptography,” Journal of Modern Optics **52**, 735–753 (2005). [CrossRef]

**20. **Z.-S. Yuan, Y.-A. Chen, B. Zhao, S. Chen, J. Schmiedmayer, and J.-W. Pan, “Experimental demonstration of a bdcz quantum repeater node,” Nature **454**, 1098–1101 (2008). [CrossRef] [PubMed]

**21. **N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. **83**, 33–80 (2011). [CrossRef]

**22. **H.-K. Lo, M. Curty, and B. Qi, “Measurement-device-independent quantum key distribution,” Phys. Rev. Lett. **108**, 130503 (2012). [CrossRef] [PubMed]

**23. **S. Pirandola, C. Ottaviani, G. Spedalieri, C. Weedbrook, S. L. Braunstein, S. Lloyd, T. Gehring, C. S. Jacobsen, and U. L. Andersen, “High-rate measurement-device-independent quantum cryptography,” Nat. Photonics **9**, 397–402 (2015).

**24. **A. Scherer, B. C. Sanders, and W. Tittel, “Long-distance practical quantum key distribution by entanglement swapping,” Opt. Express **19**, 3004–3018 (2011). [CrossRef] [PubMed]

**25. **C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, “Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects,” Phys. Rev. A **89**, 022317 (2014). [CrossRef]