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.

 

Fig. 1 (a) The schematic diagram of the HOM interference. The relative delay is adjusted by changing the timing of the photon in mode 2. The widths of the coincidence windows at D3 and D4 are set to be infinitely larger than the wavepacket duration. The HOM interference is observed when the photons in mode 1 and mode 2 arrive at the HBS simultaneously. (b) The sketch of the single detection probability at D3 (D4). (c) The sketch of the coincidence probability between D3 and D4.

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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) Δt between two incoming light wavepackets is precisely determined in advance. The delay line is introduced in mode 2 for the adjustment of Δt.

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 Δt, 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 − P₀/P_∞, where P₀ and P_∞ are the two-fold coincidence probabilities with Δt = 0 and Δt = 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 2T is adopted for D3 and D4, regardless of the amount of delay Δt. 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 Δt = 0 (a) and that with Δt = 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 Δt = 0 and in y for Δt = T. Let us denote by 3x 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 4x, 3y, 4y similarly. Then, the coincidence probability P determined in this conventional setup corresponds to the rate of the sum of event (3x & 4x), (3y & 4x), (3x & 4y) and (3y & 4y). 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.

 

Fig. 2 The configurations for the detection time windows. Each time window is divided into the two temporal modes x and y. The solid and dashed wavepackets correspond to the light wavepackets coming from mode 1 and mode 2, respectively. (a,b) Each of the time windows is set to be much larger than the wavepacket duration. Two incoming wavepackets at D3(D4) are shown without time delay (a) and with time delay T (b). (c,d) Each of the time windows is set to be shorter than the wavepacket duration for suppressing stray photon detection. Two incoming wavepackets at D3(D4) are shown without time delay (c) and with time delay T (d).

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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 (3x & 4x), (3y & 4x), (3x & 4y) and (3y & 4y), 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 Δt of the input signal light coming from mode 2 with sweeping the coincidence windows of modes 3y and 4y synchronously, which was indeed the method used in the previous experiments [12, 13]. P_∞ corresponds to the rate of the sum of event (3x & 4x), (3y & 4x), (3x & 4y) and (3y & 4y) 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 (Δt = 0), the coincidence windows in modes (3y & 4y) 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 Δt = 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.

 

Fig. 3 (a) The experimental setup of a photon pair source. The cw pump beam at 780 nm is obtained by the second-harmonic generation based on a periodically-poled lithium niobate waveguide (PPLN/W) pumped by an external cavity diode laser working at 1560 nm. Non-degenerate photon pairs are generated by SPDC by another PPLN/W. (b) The experimental setup of the HOM interference. We performed the experiment of the HOM interference between photons at 1541 nm which are heralded by the photon detection at D1 and D2 with a time difference of Δt.

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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 FBG₁₅₄₁ and FBG₁₅₈₀ to be 1 nm. For preventing the saturation of the detectors, we set the pump power to be p = 105 μW. The observed coincidence count rate is C = 1.8×10⁵ counts/(s·nm) and single count rates are S₁₅₄₁ = 2.0×10⁶ counts/(s·nm) for 1541 nm and S₁₅₈₀ = 3.0 × 10⁶ counts/(s·nm) for 1580 nm. The single count rates and the coincidence count rate are represented by S₁₅₄₁ = γpη₁₅₄₁, S₁₅₈₀ = γpη₁₅₈₀ and C = γpη₁₅₄₁η₁₅₈₀, 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 = S₁₅₄₁S₁₅₈₀/(Cp) just after the PPLN/W to be about 3.2 × 10⁸ 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 FBG₁₅₄₁ to be 30 pm and FBG₁₅₈₀ 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 × 10³ 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 FBG₁₅₄₁ and FBG₁₅₈₀ 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 FBG₁₅₄₁(30 pm) heralded by the half of the photon pairs which is filtered by FBG₁₅₈₀(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.

 

Fig. 4 The relations between two-fold coincidence counts and detection timings. The red solid curves are the Gaussian fit to the obtained data. (a) The result of the measurement of the timing jitter. The pump power coupled to PPLN/W is set to be 3.5 μW, and we directly connected the output of the FBGs (3 nm) to D1 & D3. The other combinations of detectors (D1, D2), (D1, D4) and (D2, D3) are almost the same as (a). (b) The measurement result of the coherence time τ. The pump power coupled to PPLN/W is set to be 2.5 mW, and we used FBG₁₅₈₀(10 pm) and FBG₁₅₄₁(30 pm).

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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 FBG₁₅₈₀ 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 Δt heralds two photons at 1541 nm with the same time difference Δt. They are filtered by FBG₁₅₄₁ 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 Δt, 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 Δt = t₁ 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 t₁, (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 Δt = t₀ 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 C₀ by selecting the stop signals with the timing of t₀ 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 − C₀/C_∞.

 

Fig. 5 The coincidence counts between (a) D1 & D2 and (b) D1 & D3, respectively. (c) The three-fold coincidence among D1, D2 and D3 for the start signal of D1. The delay time of the stop signal of D2 is chosen to be Δt = 2.6 ns. (d) The three-fold coincidence among D1, D2 and D3 for the start signal of D1. The delay time is chosen such that Δt is equal to the time lag between photons coming from long and short paths. During the experiment, the pump power coupled to PPLN/W is set to be 2.5 mW.

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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 C₀ 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 ${\widehat{a}}_{ik}^{†}$ and ${\widehat{b}}_{jk}^{†}$, respectively, where i = 1, 2, j = 3, 4 and k = x, y. The above operators satisfy the commutation relations $\left[{\widehat{a}}_{ik},{\widehat{a}}_{i^{\prime }{k}^{\prime }}^{†}\right]={\delta }_{i{i}^{\prime }}{\delta }_{k{k}^{\prime }}$ and $\left[{\widehat{b}}_{jk},b_{j^{\prime }{k}^{\prime }}^{†}\right]={\delta }_{j{j}^{\prime }}{\delta }_{k{k}^{\prime }}$. As in our experiment, we assume that the transmittance η_i of the system including the detection efficiency of Di 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 = η₃η₄〈: (n̂_3x + n̂_3y)(n̂_4x + n̂_4y) :〉, where ${\widehat{n}}_{jk}={\widehat{b}}_{jk}^{†}{\widehat{b}}_{jk}$ is the number operator for output modes j = 3, 4 and k = x, y. P₀ and P_∞ are calculated by using P for the input signal light wavepackets coming from 1x & 2x and 1x & 2y, 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, $〈{\widehat{a}}_{ik}^{†}{\widehat{a}}_{ik}〉=n$ and $〈:{\left({\widehat{a}}_{ik}^{†}{\widehat{a}}_{ik}\right)}^2〉/n^2=g_n^{(2)}$ holds. Similarly, we define s and $g_s^{(2)}$ such that, when mode ik includes the signal light as well as stray photons, $〈{\widehat{a}}_{ik}^{†}{\widehat{a}}_{ik}〉=s$ and $〈:{\left({\widehat{a}}_{ik}^{†}{\widehat{a}}_{ik}\right)}^2:〉/s^2=g_s^{(2)}$ 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 $\widehat{U}{b}_{3k}^{†}{\widehat{U}}^{†}=\left({\widehat{a}}_{1k}^{†}+{\widehat{a}}_{2k}^{†}\right)/\sqrt{2}$ and $\widehat{U}{b}_{4k}^{†}{\widehat{U}}^{†}=\left({\widehat{a}}_{1k}^{†}+{\widehat{a}}_{2k}^{†}\right)/\sqrt{2}$. Thus V_theory is represented by

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 S₃₍₄₎ at D3(4) and coincidence count probability C₃₄ 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

 

Fig. 6 (a) The setup for $g_{\mathit{ex}}^{(2)}$ measurement. The signal photons which include temporally-continuous stray photons from the long arm and the stray photons from the short arm are mixed at a HBS. $g_{\mathit{ex}}^{(2)}$ is obtained by the single counts of D3(4) and coincidence counts between D3 & D4 conditioned on photon detection at D1. (b) The single counts at D3 conditioned on photon detection at D1. χ is estimated from the single count probability of the signal and stray photons (S₃ = η₃(s + n)/2) and that of stray photons (S_3,∞ = η₃n) conditioned on photon detection at D1.

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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 S₃ = η₃(s + n)/2 (S_3,∞ = η₃n). Hence χ is estimated from the observed values S₃ and S_3,∞ as

Table 1. The pump power dependence of V, V_theory, χ and $g_{\mathit{ex}}^{(2)}$.

View Table

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 1x are decomposed into mutually orthogonal modes labeled by index l, and its creation operator is denoted by ${\widehat{a}}_{l,1x}^{†}$. The signal mode is orthogonal to every noise mode, namely, $\left[{\widehat{a}}_{1x}^{†},a_{l,1x}^{†}\right]=0$. We further define the creation operators ${\widehat{a}}_{l,2x}^{†}$, ${\widehat{b}}_{l,3x}^{†}$, and ${\widehat{b}}_{l,4x}^{†}$ 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 ${\widehat{a}}_{l,iy}^{†}$, ${\widehat{b}}_{l,jy}^{†}$. Let us define sums of the photon number operators by ${\widehat{N}}_{ik}={\sum }_l{\widehat{a}}_{l,ik}^{†}{\widehat{a}}_{l,ik}$ and ${\widehat{N}}_{jk}={\sum }_l{b}_{l,jk}^{†}{\widehat{b}}_{l,jk}$. Since the noise photons are assumed to be stationary, we have 〈N̂_ix〉 = 〈N̂_iy〉 and 〈:N̂_3xN̂_4x:〉 = 〈:N̂_3yN̂_4y:〉. Define N := (〈N̂_1x〉 + 〈N̂_2x〉)/2 = (〈N̂_1y〉 + 〈N̂_2y〉)/2.

 

Fig. 7 The left figure shows the photon distribution considered in the main text. The right figure presents the sketch in the case where additional stray photons in other modes are detected.

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We first calculate the two-fold coincidence probability P = η₃η₄〈:(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_jk. When the input signal light wavepackets come from 1x & 2x, one of the four terms in P is calculated as $〈:{\widehat{n}}_{3x}^{\prime }{\widehat{n}}_{4x}^{\prime }:〉=〈:\left({\widehat{n}}_{3x}+{\widehat{N}}_{3x}\right)\left({\widehat{n}}_{4x}+{\widehat{N}}_{4x}\right):〉=\left(〈:{\widehat{n}}_{1x}^2:〉+〈:{\widehat{n}}_{2x}^2:〉+2〈{\widehat{n}}_{1x}+{\widehat{n}}_{2x}〉〈{\widehat{N}}_{1x}+{\widehat{N}}_{2x}〉\right)/4+〈:{\widehat{N}}_{3x}{\widehat{N}}_{4x}:〉=s^2{g}_s^{(2)}/2+2sN+〈:{\widehat{N}}_{3x}{\widehat{N}}_{4x}:〉$, where 〈n̂_1x〉 = 〈n̂_2x〉 = s and $〈:{\widehat{n}}_{1x}^2:〉=〈:{\widehat{n}}_{2x}^2:〉=s^2{g}_s^{(2)}$. Here we used 〈n̂_ikN̂_ik〉 = 〈n̂_ik〉〈N̂_ik〉 from the assumption, and a unitary transformation by the HBS as $\widehat{U}{\widehat{b}}_{l,3k}^{†}{\widehat{U}}^{†}=\left({\widehat{a}}_{l,lk}^{†}+{\widehat{a}}_{l,2k}^{†}\right)/\sqrt{2}$ and $\widehat{U}{\widehat{b}}_{l,4k}^{†}{\widehat{U}}^{†}=\left({\widehat{a}}_{l,1k}^{†}-{\widehat{a}}_{l,2k}^{†}\right)/\sqrt{2}$ in addition to the transformation of the signal modes. Similarly, 〈:n̂′_3yn̂′_4y:〉 is calculated as $n^2{g}_n^{(2)}/2+2nN+〈:{\widehat{N}}_{3x}{\widehat{N}}_{4x}:〉$, where 〈n̂_1y〉 = 〈n̂_2y〉 = n and $〈:{\widehat{n}}_{1y}^2:〉=〈:{\widehat{n}}_{2y}^2:〉=n^2{g}_n^{(2)}$. 〈n̂′_3xn̂′_4y〉 and 〈n̂′_4xn̂′_3y〉 are calculated as sn + N (s + n) + N². As a result,

(4)$$\begin{array}{r}\hfill P_0={\eta }_3{\eta }_4(\frac{1}{4}\left(s^2{g}_s^{(2)}+n^2{g}_n^{(2)}\right)+2N^2+4N(s+n)\\ \hfill +2sn+2〈:{\widehat{N}}_{3x}{\widehat{N}}_{4x}:〉).\end{array}$$

On the other hand, when the input signal light wavepackets come from 1x & 2y, 〈:n̂′_3xn̂′_4x:〉 and 〈:n̂′_3yn̂′_4y:〉 are calculated as $\left(s^2{g}_s^{(2)}+n^2{g}_n^{(2)}\right)/4+N(s+n)+〈:{\widehat{N}}_{3x}{\widehat{N}}_{4x}:〉$. Similarly, 〈n̂′_3xn̂′_4y〉 and 〈n̂′_3yn̂′_4x〉 are calculated as (s + n)²/4 + N (s + n) + N². As a result, we obtain

(5)$$\begin{array}{r}\hfill P_{\infty }={\eta }_3{\eta }_4(\frac{1}{2}\left(s^2{g}_s^{(2)}+n^2{g}_n^{(2)}\right)+2N^2+4N(s+n)\\ \hfill +\frac{1}{2}{\left(s+n\right)}^2+2〈:{\widehat{N}}_{3x}{\widehat{N}}_{4x}:〉).\end{array}$$

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 S₃₍₄₎ at D3(4) and coincidence count C₃₄ between D3 & D4 conditioned on the photon detection at D1 are described by S₃₍₄₎ = η₃₍₄₎(s + n + 2N)/2, and $C_{34}={\eta }_3{\eta }_4\left(s^2{g}_s^{(2)}+n^2{g}_n^{(2)}+4N(s+n)+4〈:{\widehat{N}}_{3x}{\widehat{N}}_{4x}:〉\right)/4$. The observed correlation function is expressed as

(6)$$\begin{array}{lll}{g}_{\mathit{ex}}^{(2)}\hfill & =\hfill & \frac{C_{34}}{S_3{S}_4}\hfill \\ \hfill & =\hfill & \frac{s^2{g}_s^{(2)}+n^2{g}_n^{(2)}+4N(s+n)+4〈:{\widehat{N}}_{3x}{\widehat{N}}_{4x}:〉}{{\left(s+n+2N\right)}^2}.\hfill \end{array}$$

By combining equation (4), (5) and (6), the visibility V_theory = 1 − P₀/P_∞ is described by

(7)$$V_{\text{theory}}=\frac{{\left(1-\chi \right)}^2}{{\left(1+\chi \right)}^2\left(1+g_{\mathit{ex}}^{(2)}\right)},$$

where 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.

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