1. Introduction

Heralded single photons (HSPs) and entangled photon pairs (EPPs) obtained via twin-photon generation in spontaneous parametric down-conversion (SPDC) [1,2] have been used for wide range of photonic quantum technologies [3]. Boosting generation and detection rates for such nonclassical photons while ensuring synchronizability, should contribute to further implementation of challenging quantum-photonic protocols, which were often subjected to low success probability and heavy attenuation, e.g., multiphoton entanglement [4], all-photonic quantum repeater [5], nonlinear frequency conversion using single photons [6], and long distance quantum communication using a satellite [7]. Because SPDC is a probabilistic process, it is necessary to suppress multi-photon emission, otherwise unwanted higher-order photons would cause detrimental effects to quantum protocols using on-off type single-photon detectors [8]. To make the generation rates of SPDC photons higher with keeping multi-photon emission suppressed, one may increase a repetition rate of excitation pulses ($f_{\mathrm {rep}}$) for SPDC with their pulse energies maintained as small as possible. This direction has been researched to the pump rate from $f_{\mathrm {rep}}=$152 MHz to 10 GHz [9–15], and hence the heralding rate ($R_{\mathrm {h}}$) of single photons has successfully reached to the several Mega counts per second (Mcps) to date [11,14]. However, the pump repetition rate exceeding 10 GHz has not been reported so far. There are two reasons that have made it challenging: first, there is a lack of useful ultra-high-rate pump source. Second, $R_{\mathrm {h}}$ is also limited by saturation of photon count rate (CR) of single-photon detectors which usually occurs in the Mcps range.

In this paper, we overcome these obstacles by the following new techniques: first, we achieve $f_{\mathrm {rep}}$ to be 50 GHz via second harmonic generation (SHG) of an interleaved frequency comb at 1550 nm. This few-frequency-mode-locked pump pulses allows us to obtain an ultra-high excitation rate, which would be close to that with a continuous-wave (CW) pump, as well as to ensure synchronizability of generated photons at telecom wavelengths. Second, we develop a multiplexing scheme of superconducting nanowire single-photon detectors (SNSPDs or SSPDs), which overcomes the CR limitation for a single detector, almost reaching 77 Mcps. Combining these technologies with an efficient photon-pair source, consisting of a waveguide nonlinear crystal placed in a free-space Sagnac-loop interferometer, we demonstrate HSP and polarization-EPP experiments. The presence of the single-photon state in the heralded photons was indicated by the second-order intensity correlation $g^{(2)}(0) < 1/2$ at the heralding rate over 20 Mcps. The nonclassicality of $g^{(2)}(0) < 1$ for the heralded photons still survived even beyond the rate of 50 Mcps. For the polarization-EPPs, we obtained the maximum coincidence rate of 1.6 Mcps with the fidelity of $\displaystyle {0.881 \pm (0.254 \times 10^{-3}})$ to the maximally entangled state.

2. Experimental setup

Figure 1(a) (the upper half) shows an experimental setup of a frequency comb generator at the fundamental wavelength based on electro-optical (EO) modulation technology [16]. A CW laser diode (CWLD, RIO) emits 20 mW of narrow-linewidth single-frequency light at 1550.15 nm, which is guided to a conventional LiNbO${}_3$ dual-drive Mach-Zehnder modulator (DDMZM, SOC). Two RF electrodes of DDMZM (RF1 and RF2) are driven by a 25 GHz ($=f_{\mathrm {mod}}$) sinusoidal wave, which was generated with a microwave signal generator (SG), and amplified before DDMZM. The input and output optical signals of DDMZM are partially ($\sim 1\,\%$) tapped and monitored by two photodetectors. A self-made feedback circuit locks the output-to-input ratio to a constant point without dithering by tracking the DC bias to stabilize the relative phase between the two arms in DDMZM. The 1550 nm comb can be optimized by the lock point in addition to tuning the phase and amplitude of the microwave signal to RF2 by using the phase shifter (PS) and variable attenuator (VAT), respectively. We implemented the optimization by monitoring a frequency spectrum and a temporal duration of SHG pulses shown later. This configuration only requires a single-stage MZM to generate ultraflat frequency combs [17]. The similar setups can be found in Refs. [18–20]. The 1550 nm comb from DDMZM is then guided into a waveshaper (Finisar) used as an interleaver to double $f_{\mathrm {mod}}$ to 50 GHz. The 1550 nm comb consisted of only five frequency modes, is shown in Fig. 1(b), which was monitored by an optical spectrum analyzer (OSA, Finisar) connected with a 10 % fiber tap from the main line after the interleaver. All the other modes were filtered out by the interleaver. Then a 1.3 km-length single-mode optical fiber (SMF) is exploited for chirp compensation. Since the optical power was attenuated to $\sim 0.1$ mW at this point, we amplified it to $>30$ mW by an erbium-doped fiber amplifier (EDFA (omitted), Alnair Labs) after SMF. State of polarization for the 1550 nm comb being drifted after propagating the fiber components was recovered and stabilized by a polarization stabilizer (General Photonics). Finally, the 1550 nm comb was amplified from $>20$ mW up to $\sim 1$ W at maximum by a polarization-maintaining (PM) EDFA (PriTel) and guided to a SHG setup.

 

Fig. 1. (a) Experimental setup for 1550 nm frequency comb, second harmonic generation, and Sagnac-loop photon source. In the upper half, the red and black lines denote optical fiber and electrical components. Frequency spectra of (b) fundamental and (c) SHG combs. (d) Autocorrelation trace of SHG pulses. (e) Wavelength spectra of unfiltered SPDC, filtered SPDC by BPF1 at 1535 nm, and that by BPF2 at 1565 nm.

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The lower half of Fig. 1(a) shows the free-space setup for SHG followed by a photon source with a Sagnac loop using periodically-poled lithium niobate waveguides (PPLN-WGs). PPLN-WG1 is a self-made, type-0, MgO-doped PPLN waveguide [21]. The length, core diameter, and grating period are 10 mm, 8 $\mu$m, and 17.9$\,\mu$m, respectively. PPLN-WG1 is temperature-stabilized to 35.2 $^\circ$C with a thermoelectric cooler. The maximum SHG power over 100 mW was obtained by single-path excitation with a coupled pump power $\sim 400$ mW (coupling efficiency $> 90$ %) of the 1550 nm comb. Total attenuation for the unconverted 1550 nm comb after PPLN-WG1 to the Sagnac loop was estimated to be at least −150 dB using harmonic separators, etc. A laser power stabilizer (BEOC) removed residual power fluctuation and drift of SHG pulses. Fig. 1(c) shows a SHG spectrum obtained by another OSA (Advantest), and its fitted envelope with a full width at half maximum (FWHM) of $\sim 135$ GHz ($\sim 0.27$ nm) using a Gaussian. An autocorrelation trace of the SHG pulses is shown in Fig. 1(d) obtained with an autocorrelator (Femtochrome). The pulse duration was 3.6 ps (Gaussian FWHM), which was slightly longer than that for the transform-limited duration estimated from the envelop width of the SHG spectrum. The small side wings are cross-correlation between the main and the adjacent pulses in the pulse train. The five modes with 50 GHz spacing was chosen to maximize the pump $f_{\mathrm {rep}}$ by taking advantage of a phase-matching bandwidth of our SHG crystal, filtering bandwidths of BPFs for our photon source explained later. Additionally, four sidebands were observed aside with the main five modes in Fig. 1(c). Although it is obvious in the time domain picture, it would be noteworthy that the frequency mixing mainly based on SHG $\left (\sum _{j=1}^{5} 2\omega _j\right )$ and sum frequency generation $\left (\mathrm {SFG,}\,\sum _{j=1}^{5}\sum _{k=j+1}^{5}\omega _j+\omega _k\right )$, which was visualized here using the limited number of fundamental modes $\omega _{j(k)}$, remains $f_{\mathrm {rep}}$ unchanged after the frequency-doubling process. After passing through an optical VAT consisted with a half-wave plate and a polarization beamsplitter, a mode-matching lens, and a pair of quartz plates (Quartz) to adjust relative delay between horizontal and vertical polarization, the SHG pulses are introduced to our photon source. Here, DM1 is a dichroic mirror which reflects 775 nm and transmits the telecom wavelength.

The bottom of Fig. 1(a) shows a waveguide/free-space hybrid type of photon source, in which a waveguide nonlinear crystal deployed in a free-space Sagnac-loop interferometer [22–24]. Our configuration consists of PPLN-WG2 (NEL), a dual-wavelength polarization beamsplitter (DPBS) and dual-wavelength half-wave plate (DHWP). DPBS and DHWP are designed for both the telecom and 775 nm wavelengths. PPLN-WG2 is a type-0 ridged PPLN waveguide whose length, core diameter, and grating period are 34 mm, 8 $\mu$m, and 18 $\mu$m, respectively. It is temperature-stabilized to $\sim 58$ $^\circ$C. Both of the facets are angled 6$^\circ$ to the normal of the incident beam to reduce back reflection with standard anti-reflection (AR) coatings for both wavelengths. The pump pulses are split into vertically polarized "Pump1" and horizontally polarized "Pump2" at DPBS to excite PPLN-WG2 from the clockwise and counter-clockwise directions. The splitting ratio was controlled by HWP after VAT. The superposition state of "Pair1" and "Pair2" is thus generated at the output of the Sagnac loop, and can be written as $\displaystyle {| \varphi _{\theta } \rangle = \left ( | HH\rangle + e^{i\theta } |VV\rangle \right )/\sqrt {2}}$. The relative phase $\theta$ can be controlled by Quartz before the Sagnac loop. In this work, the experimental output $\displaystyle {|\varphi _{\mathrm {e}} \rangle }$ became closer to $\displaystyle {|\varphi ^- \rangle = \left ( | HH\rangle - |VV\rangle \right ) / \sqrt {2} }$. The transmitted beam through DM1 is split by another dichroic mirror (DM2) into shorter and longer wavelengths with respect to 1550 nm. Then the photons are filtered by volume holographic gratings (VHGs, Ondax, typically having $>70$ dB stop-band suppression and $\sim 90$ % pass-band transmittance) used as narrow bandpass filters labeled as BPF1 and BPF2, which are centered at 1535 nm and 1565 nm, respectively. The FWHM of the filters are both $\sim 0.9$ nm, corresponding to $\sim 115$ GHz at 1535 nm ($\sim 110$ GHz at 1565 nm). Fig. 1(e) shows spectra of unfiltered SPDC tapped before DM2, filtered SPDC after BPF1 and BPF2, all of which were obtained by a spectrometer (Princeton Instruments). Although the unfiltered SPDC spectrum is as broad as $\sim 80$ nm (FWHM), photons outside the filter was sufficiently blocked by BPFs. However, we observed small side lobes, by which we obtained $\sim 5$ % of total counts in the transmission spectra of BPFs (not shown). With the small discrepancy between the BPF bandwidths, the side lobes could yield a small amount of uncorrelated noise photons. Finally, the frequency-nondegenerate, correlated photons are sent to the photon detection setup described in the following sections.

Note, Pair1 (horizontally-polarized after DHWP) was only generated (i.e., Pump1 was only used) for the HSP experiment shown in the next section.

3. HSP experiment using 8-mux detectors

Figure 2(a) shows a setup for the HSP experiment using eight multiplexed (8-mux) SSPDs. For "Heralding" (subscripted with "h" in this paper), an incident beam at 1565 nm is coupled to the single mode fiber and split by a $1\times 8$ fiber beamsplitter (FBS${}_{1\times 8}$), which was actually a $1\times 2$ FBS (FBS${}_{1\times 2}$) followed by two $1\times 4$ FBSs. Each output of FBS${}_{1\times 8}$ is connected to a SSPD [25] through a fiber polarization controller (omitted). The single-ended, unbalanced detection signal from each SSPD is fed to a self-made Balun transformer to obtain balanced outputs having positive and negative polarity. These signals are electrically combined by a $2\times 1$ power combiner (PC${}_{2\times 1}$), before which we add a fixed-length (30 cm) delay line (DL) in the positive polarity signal. We utilized this self-differencing technique [26], to make detection signals of SSPDs as short as possible and to reduce trailing noise in the raw signals. The temporal timing and voltage magnitude of each PC${}_{2\times 1}$ output are aligned and equalized by using a variable delay line (VDL) and VAT, respectively. In fact, VDL and VAT were collectively controlled on each line using a combination of coaxial attenuators with fixed attenuation selected from 0 to −3 dB with a 0.5 dB step. The length of each attenuator was approximately 20 mm and therefore the corresponding delay of $\sim 100$ ps determined the timing resolution of VDL. The eight PC${}_{2\times 1}$ outputs are combined using an $8\times 1$ power combiner (PC${}_{8\times 1}$) into a single electrical line. The combined signal was properly amplified, then used as a start signal for a coincidence counter (HydraHarp 400, PicoQuant). Finally, the specification of our SSPDs are summarized in Table 1, which were individually measured using a CW laser. The maximum detection efficiencies (DE${}_{\mathrm {max}}$) were obtained at low CRs with sufficiently attenuated optical input, while the maximum CRs (CR${}_{\mathrm {max}}$) were measured for reference with stronger optical power.

 

Fig. 2. (a) Setup for the HSP experiment. The upper side shows 8-mux SSPDs for heralding, and the lower for $g^{(2)}$ measurement. (b) Typical temporal response of 8-mux SSPDs to pulsed coherent light. (c) Averaged waveform of (b).

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Table 1. Specification of SSPDs.

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The intensity correlation of the HSPs generated at 1535 nm was evaluated using another FBS${}_{1\times 2}$ followed by two other SSPDs placed in "$\displaystyle {g^{(2)}}$ measurement" in Fig. 2(a). A −20 dB attenuator (ATT) is inserted to avoid saturation of the two SSPDs for higher CRs. The experimental $g^{(2)}$ values of the heralded photons (subscripted with "H") can be calculated by $\displaystyle {g^{(2)}_{\mathrm {H}}=N_1 N_{123}/\left (N_{12} N_{13}\right )}$ [27] using time-tagged CRs. Here, $N_1$ denotes a single CR of the start signal. Two-fold ($N_{12}, N_{13}$) and three-fold ($N_{123}$) coincidence CRs were accumulated using a coincidence window of $\Delta t_{\mathrm {HSP}} = 720$ ps. This corresponds to the four standard deviations of the coincidence jitter between 8-mux SSPDs and SSPD9 or SSPD10, which was equally $\sim 420$ ps (FWHM). Meanwhile, the detection jitter of each SSPD spanned from 93 to 162 ps (FWHM) as listed in Table 1, and the coincidence jitter between an arbitrary pair of the SSPDs were $\sim 180$ ps (FWHM) on average. The relatively larger coincidence jitter when using 8-mux SSPDs was possibly caused by a residual timing delay remained unadjusted in each electrical line before PC$\displaystyle {{}_{8\times 1}}$. Although the timing resolution of VDL is smaller than the jitter of 8-mux SSPDs, a finer resolution may be required to reduce the residual timing delay. Streamed single CRs $\displaystyle {N^{\mathrm {st}}_j}$ of each channel were also recorded for, e.g., Klyshko efficiencies discussed later, where individual single counts unbounded with the start trigger are required. Here, the saturation of the detectors and the coincidence counter are assumed as linear losses which should not change $\displaystyle {g^{(2)}_{\mathrm {H}}}$. Additionally, since we only focus on $\displaystyle {g^{(2)}(\tau = 0)}$, we omit the time delay argument in this paper.

Figure 2(b) shows the typical temporal response of 8-mux SSPDs to a pulsed diode laser (pulse width $\sim 30$ ps, ALS) used as a test source. 10000 waveforms collected by an oscilloscope (Tektronix) are overlaid. The temporal resolution is 0.1 ns (10 G sample/sec). The waveforms are histogrammed into fifty voltage bins in the vertical axis. 8-mux SSPDs, working as a beamsplitter-based photon-number-resolving (PNR) detector [28] to the pulsed coherent light, can be used as a dead-time-reduced single-photon detector [29]. Because a single photon is routed to fire the only one of the detectors at one time, our configuration ideally works as an "OR" circuit for single photon input, which can multiplex detection signals for heralding without being temporally overlapped each other. The possible residual delay issued above, however, does not seem to be resolved by the maximum signal bandwidth of 1 GHz in our data. An averaged waveform of Fig. 2(b) is shown in Fig. 2(c). The FWHM of the negative polarity pulse was 1.65 ns, whose falling edge was used as triggering.

Figure 3 shows the result of the HSP experiment using 8-mux SSPDs. First, we compare the two $R_{\mathrm {h}}$ values: $N_1^{\mathrm {st}}$ directly obtained as the streamed value and $N_1$ calculated from time-tagged data. In Fig. 3(a), the streamed $R_{\mathrm {h}}$ (circle) with respect to coupled Pump1 were almost saturated about 77 Mcps, reaching about a half of the maximum countable rate (150 MHz) [30]. This saturation implies that the multiplexed detection signals were too closely arrived at the counter that it was not able to distinguish them. By contrast, the time-tagged $R_{\mathrm {h}}$ (square) saturated with CR approaching to 12.5 Mcps (the red dotted line shown for reference), which is related to the dead time in saving data [30]. The green dotted line is a fitted curve to the streamed $R_{\mathrm {h}}$ using $\displaystyle {f_{\mathrm {sat}}(p) = A(1-\exp (-p/P))}$ as a function of the coupled pump power $p$, with $A_{\mathrm {h}} = 75.8 \pm 0.4$ Mcps and $P_{\mathrm {h}}= 14.01 \pm 0.24$ mW. Hereafter, $R_{\mathrm {h}}$ stands for the streamed values. Each dead time of our single detector involved in 8-mux SSPDs, was distributed in the range of 33–111 ns, which we calculated as the inverse of CR${}{_{\mathrm {max}}}$ listed in Table 1. Hence, the dead time for 8-mux SSPDs was shortened to 13 ns ($\displaystyle {\sim 1/77}$ Mcps).

 

Fig. 3. Results of the HSP experiment. (a) Streamed and time-tagged $R_{\mathrm {h}}$ by 8-mux SSPDs. (b) Klyshko efficiency of 8-mux SSPDs. The dotted line is the linear fit to the lower CR region. (c) $g^{(2)}$ depending on $p$. (d) $g^{(2)}$ depending on the streamed $R_{\mathrm {h}}$. In (c) and (d), heralded and intrinsic $g^{(2)}$ are from raw and accidental-noise removed CRs, respectively. In (c), the dotted lines are the fitting results of $y_{\mathrm {H}}(p)$ (blue) and $y_{\mathrm {int}}(p)$ (red), while those in (d) are the parametric plots of ($x_{\mathrm {h}}(p)$, $y_{\mathrm {H}}(p))$ and ($x_{\mathrm {h}}(p)$, $y_{\mathrm {int}}(p))$. Error bars are omitted for ease of viewing in (d).

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Next, Klyshko efficiencies [31] for the overall setup can be evaluated using the following equations,

Figure 3(c) shows $g^{(2)}_{\mathrm {H}}$ values measured for the HSPs at 1535 nm without (cross) or with (diagonal cross) −20 dB attenuator, depending on the coupled pump power. Typical results of $g^{(2)}_{\mathrm {H}}$ and the corresponding $y_{\mathrm {int}}(p)$ are listed in Table 2. Here, $y_{\mathrm {int}}(p)$ was calculated from the fitted curve to $g^{(2)}_{\mathrm {int}}$ describing the intrinsic intensity correlation only for correlated pair generation explained later. It was confirmed from $g^{(2)}_{\mathrm {H}}<1/2$ that the presence of the single-photon state existed in the heralded photons in the $R_{\mathrm {h}}>20$ Mcps range. Moreover, $g^{(2)}_{\mathrm {H}}<1$ was observed at the even higher $R_{\mathrm {h}}$ over 50 Mcps. Thus the nonclassicality was successfully observed at $R_{\mathrm {h}}$ in the tens of Mcps range for the first time, to the best of our knowledge. The errors in $g^{(2)}_{\mathrm {H}}$ were evaluated based on the standard error propagation method assuming that the photon counting statistics obey the Poissonian distribution. The blue dotted line is a fitted curve by $f_{\mathrm {sat}}(p)$ to the entire area with $\displaystyle {A_{\mathrm {H}}=1}$ (fixed) and $y_{\mathrm {H}} = 7.04 \pm 0.18$ mW. Hence, it was revealed that $g^{(2)}_{\mathrm {H}}$ depending on $p$ can be traced phenomenologically by the standard saturation model with the single parameter.

Table 2. Typical values of $R_{\mathrm {h}}$, $g^{(2)}_{\mathrm {H}}$, and $y_{\mathrm {int}}(p)$ depending on $p$ for HSPs.

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It is notable here that the observed $g^{(2)}_{\mathrm {H}}$ should be larger than that of the intrinsic one, because uncorrelated coincidence counts mixed to the intrinsic counting data often degrade $g^{(2)}_{\mathrm {H}}$. More specifically, an accidental coincidence happened by photons generated with adjacent excitation pulses is not distinguishable, because the single temporal bin ($1/f_{\mathrm {rep}}=20$ ps) is much shorter than $\Delta t_{\mathrm {HSP}}$, in which 36 temporal bins are included. To estimate the accidental-noise-free $g_{\mathrm {int}}^{(2)}$ corresponding to the intrinsic intensity correlation only for correlated pair generation, we introduce the following analysis. First we define the noise-to-signal ratio as $\displaystyle { \chi \equiv \alpha n_{\mathrm {noise}} / \left ( n_{\mathrm {H}} - \alpha n_{\mathrm {noise}} \right ) }$, where $\alpha = \displaystyle { \left ( m_{\mathrm {pulse}}-1 \right ) / m_{\mathrm {pulse}}}$ and $\displaystyle {m_{\mathrm {pulse}} = f_{\mathrm {rep}} \Delta t_{\mathrm {HSP}}}$. $n_{\mathrm {H}}$ is heralded photon counts measured within the coincidence window centered at the peak position, while $n_{\mathrm {noise}}$ denotes accidental coincidence counts taken with a coincidence window placed $20\,\mu$s away from the peak with the same temporal width as $\Delta t_{\mathrm {HSP}}$. The number of intrinsic coincidence counts is thus estimated by subtracting the noise counts from $n_{\mathrm {H}}$. When the light pulse for $g^{(2)}_{\mathrm {H}}$ is assumed to be a mixture of statistically independent, intrinsically correlated and noisy uncorrelated photons, $g^{(2)}_{\mathrm {H}}$ can be described as $\displaystyle {g^{(2)}_{\mathrm {H}} = \left ( g^{(2)}_{\mathrm {int}} + \chi ^2 g^{(2)}_{\mathrm {n}} + 2\chi \right )/\left (1+\chi \right )^2}$ [32]. Here, $g^{(2)}_{\mathrm {n}}$ values are the intensity correlation of noise photons and were observed as unity within certain errors. $\chi$ is measured by one of the detectors for the $g^{(2)}$ measurement. Hence, $g^{(2)}_{\mathrm {int}}$ can be estimated using the experimental parameters:

Table 3. Fitting results for $R_{\mathrm {h}}$, $g^{(2)}_{\mathrm {H}}$, and $g^{(2)}_{\mathrm {int}}$ for HSPs.

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Finally, we analyzed $g^{(2)}$ depending on $R_{\mathrm {h}}$. Because both parameters were saturated, we used parametric plotting for the fitted curves. Namely, the blue and red dotted lines in Fig. 3(d) were drawn as $\displaystyle {\left (x_{\mathrm {h}}(p),\, y_{\mathrm {H}}(p)\right )}$ and $\displaystyle {\left (x_{\mathrm {h}}(p),\, y_{\mathrm {int}}(p)\right )}$, where

4. EPP experiment using 3-mux detectors

In this section, we demonstrate that our scheme is also applicable to the EPP experiment. In Fig. 4, Alice and Bob receive the polarization-EPPs at 1535 nm and 1565 nm, respectively, which were sent from the photon source (Fig. 1(a)). The experimental state is described as $\displaystyle {|\varphi _{\mathrm {e}} \rangle \sim |\varphi ^{-}_{\mathrm {ab}} \rangle = \left ( | H_{\mathrm {a}} H_{\mathrm {b}}\rangle - |V_{\mathrm {a}} V_{\mathrm {b}}\rangle \right ) / \sqrt {2} }$, and was evaluated via quantum state tomography (QST) for polarization qubits [33] using a set of QWP (quarter-wave plate), HWP, and PBS on both sides. For this experiment, we selected the four SSPDs showing both high DEs and CRs from Table 1. As a consequence, the single detector (assigned to SSPD1 here) with the highest DE was chosen for Alice to generate the start signal, while the second highest three were multiplexed (3-mux SSPDs) for Bob. Unlike the HSP experiment, we separately analyzed the detection signals from 3-mux SSPDs channel by channel.

 

Fig. 4. Setup for the EPP experiment. Electrical lines after SSPDs are omitted for simplicity.

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Figure 5 shows the results for the polarization-EPP experiment. In Fig. 5(a), we show single and coincidence CRs depending on the sum of coupled Pump1 and Pump2: single CRs of SSPD1 ($S_{\mathrm {a}}^{\mathrm {st}} = N_{1}^{\mathrm {st}}$, and time-tagged one for reference), those of 3-mux SSPDs $\left (S_{\mathrm {b}}^{\mathrm {st}} = \sum _{j=2}^4 N_{j}^{\mathrm {st}}\right )$, their coincidence CRs (time-tagged $C_{\mathrm {ab}}=\sum _{j=2}^4 N_{1j}$ and $C_{\mathrm {ab},\varphi _{\mathrm {e}}}$). Here, ${S_{\mathrm {a}}^{\mathrm {st}}}$, ${S_{\mathrm {b}}^{\mathrm {st}}}$, and $C_{\mathrm {ab}}$ were measured with $\displaystyle {|HH\rangle }$ basis, to be used to estimate Klyshko efficiencies later. $C_{\mathrm {ab},\varphi _{\mathrm {e}}}$ is a sum of the coincidence CRs for the polarization-EPPs measured with the four polarization bases of $\displaystyle {|HH\rangle }$, $\displaystyle {|HV\rangle }$, $\displaystyle {|VH\rangle }$, and $\displaystyle {|VV\rangle }$. The coincidence window used here to accumulate $N_{1j}$ was $\Delta t_{\mathrm {EPP}} = 270$ ps in common, which is the four standard deviations of the mean coincidence jitter between SSPD1 and SSPD$j$ ($j>1$). The maximum CRs of $S_{\mathrm {a}}^{\mathrm {st}}$, $S_{\mathrm {b}}^{\mathrm {st}}$, and $C_{\mathrm {ab},\varphi _{\mathrm {e}}}$ were 10.6, 12.3, and 1.6 Mcps, respectively, with the maximum pump power $\sim 1.3$ mW. The blue, green, and brown dotted lines are fitted curves using $\displaystyle {f_{\mathrm {sat}}(p)}$ with the parameters listed in Table 4. Next, we calculated Klyshko efficiencies shown in Fig. 5(b) using $\displaystyle {C_{\mathrm {ab}}/S_{\mathrm {b(a)}}^{\mathrm {st}} \approx T_{\mathrm {a(b)}} = T^{\mathrm {com}}_{\mathrm {a(b)}}\,\overline {\eta }_{\mathrm {a(b)}}}$, which are only valid for the lower CR region as similarly before. A linear fitting yielded $\displaystyle {T_{\mathrm {a}} = 12.6\,\%}$ and $\displaystyle { T_{\mathrm {b}} = 12.3\,\%}$ (shown as the purple and green dotted lines, respectively). We also obtained $\displaystyle {T^{\mathrm {com}}_{\mathrm {a}} = 16.6\,\%}$ and $\displaystyle {T^{\mathrm {com}}_{\mathrm {b}} = 18.7\,\%}$ calculated from the mean detection efficiencies of $\displaystyle {\overline {\eta }_{\mathrm {a}} = \eta _{\mathrm {a}} = 75.7\,\%}$ and $\displaystyle {\overline {\eta }_{\mathrm {b}} = 65.9\,\%}$. Hence, the appropriate multiplexing of SSPDs can outperform the best one in CR without sacrificing the overall detection efficiency. Additionally, $T^{\mathrm {com}}_{\mathrm {a(b)}}$ was resolved in the free-space propagation and the fiber-coupling efficiencies. We obtained the same free-space propagation efficiency of $\sim 69$ % using backward-propagating CW light emitted from the fiber input of Alice or Bob to just before PPLN-WG2. Therefore, the fiber-coupling efficiencies of the forward-propagating correlated photons were estimated to be $\displaystyle {T^{\mathrm {fc}}_{\mathrm {a(b)}} \sim 24\,\% \, (27\,\%)}$. However, $> 90$ % of the backward-propagating CW light was coupled to PPLN-WG2. The discrepancy yet to be identified would possibly be caused by, e.g., multi-spatial-mode excitation in a waveguide photon source as was implied in Ref. [34].

 

Fig. 5. Results of the polarization-EPP experiment. (a) Single and coincidence CRs. (b) Klyshko efficiencies. (c) Fidelity, purity, and EoF of the reconstructed state. The dotted lines in (a) and (b) are fitted curves, while those in (c) outlining the trend.

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Table 4. Fitting results of $ {S}_{\mathrm {a}}^{\mathrm {st}}$, $ {S}_{\mathrm {b}}^{\mathrm {st}}$, and $ {C}_{\mathrm {ab},\varphi _{\mathrm {e}}}$ for EPPs.

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Finally, we implemented QST of the generated EPPs and reconstructed their density matrices $\hat {\rho }_{\mathrm {e}}=|\varphi _{\mathrm {e}}\rangle \langle \varphi _{\mathrm {e}}|$ via diluted maximum likelihood estimation (MLE) [35]. Experimental results of fidelity ($F_{\mathrm {raw}}$ and $F_{\mathrm {max}}$), purity $\left ( \mathrm {Tr}\left [\hat {\rho }_{\mathrm {e}}{}^2\right ] \right )$, and entanglement of formation (EoF) [36], are shown in Fig. 5(c). $F_{\mathrm {raw}}$ were calculated from the raw data, while $F_{\mathrm {max}}$ were for maximized results using $\displaystyle { F_{\mathrm {max}} = \mathrm {max}_{-\pi \leq \theta \leq \pi }\langle \varphi _{\theta }^{}| \hat {\rho }_{\mathrm {e}}|\varphi _{\theta }^{}\rangle }$ [23]. Consequently, $\displaystyle {F_{\mathrm {raw}}}$ almost matched with $\displaystyle {F_{\mathrm {max}}}$, where $\displaystyle {\theta _{\mathrm {max}}\sim - 0.94\,\pi }$ rad on average and its discrepancy from $\theta _{\mathrm {e}}$ for the experimental states were small as was indicated in Fig. 5(c). At the maximum $\displaystyle {C_{\mathrm {ab},\varphi _{\mathrm {e}}}}$, the values of $F_{\mathrm {raw}}$, $F_{\mathrm {max}}$, purity, and EoF were $0.881 \pm (0.254 \times 10^{-3})$, $0.889 \pm (0.240 \times 10^{-3})$, $0.822 \pm (0.390\times 10^{-3})$, and $0.718 \pm (0.642 \times 10^{-3})$, respectively. The negligibly small errors are standard deviations of 1000 density matrices numerically obtained with diluted MLE using simulated photon counting data, which were generated via Poissonian random number generator from the experimental counts. Note, the CRs in the weaker pump region were more susceptible to the background, likely yielding the larger fluctuation of the figures of merit compared to those in the stronger pump region in Fig. 5(c). Furthermore, these figures of merit should be limited by the imperfections in BPFs (the bandwidth discrepancy and the side lobes), or the back reflection at the waveguide facets which can be mitigated by the special AR coatings [24]. These results shown in this section indicate that the quality of the entangled state was preserved at the high CR, and thus demonstrate the applicability of our scheme to polarization-EPP experiments.

5. Conclusion

In summary, we have demonstrated the ultra-high-rate nonclassical light source using the 50 GHz mode-locked pump pulses via SHG of the interleaved EO comb, the multiplexed SSPDs with CR reached about 77 Mcps, and the photon source using the PPLN waveguide placed in the Sagnac-loop interferometer. With 8-mux SSPDs, we obtained $g^{(2)}_{\mathrm {H}} < 1/2$ indicating the presence of the single-photon state in the heralded photons up to the $R_{\mathrm {h}}$ over 20 Mcps. The nonclassicality of the photons was observed with $g^{(2)}_{\mathrm {H}} < 1$ beyond the $R_{\mathrm {h}}$ of more than 50 Mcps. We have also obtained 1.6 Mcps coincidence CR for the polarization-EPPs by 3-mux SSPDs, whose single CR outperformed that by the best one, with the overall detection efficiency maintained comparable.

For the HSP experiment, a coincidence detector with a short timing jitter could realize $g^{(2)}_{\mathrm {H}}$ approaching to $g^{(2)}_{\mathrm {int}}$. For example, the $32.3$ ps coincidence jitter (FWHM) [37] would realize almost accidental-noise-free detection for $f_{\mathrm {rep}} = 50$ GHz with $m_{\mathrm {pulse}}\sim 2.8$. However, $y_{\mathrm {int}}(p)$ were still in the order of 1e-1 in the highest CR, implying other noises in the strongly pumped region. One may stem from multi-photon generation, which should be mitigated by eliminating $n>1$ detection events from $n>0$ ones. Though we did not see so much $n>1$ events (therefore so much improvement in the conditional $g^{(2)}_{\mathrm {H}}$) due to the limited overall efficiency $T_{\mathrm {h}}$, utilizing the PNR capability of our scheme, which can be specified with, e.g., detector tomography [38], would decrease $g^{(2)}_{\mathrm {H}}$ when $T_{\mathrm {h}}$ becomes higher. Another possible noise would be uncorrelated photons excited in multiple spectral distributions, i.e., Schmidt modes in such a broadband SPDC source [39]. In fact, the gradually increasing deviation from the single-mode model was observed in the heralded $g^{(2)}$ [14]. To detour the effect, the most promising way is to apply our system to a spectrally pure HSP source. An improvement of $T_{\mathrm {h}}$ should also mitigate both noises by allowing the weaker pump. In aiming at higher $R_{\mathrm {h}}$, spatially beamsplit detection by arrayed SSPDs [40,41] especially realizing a high CR with small number of pixels [42] is promising for the further speedup as far as a coincidence counter follows. As for the optical setup, $f_{\mathrm {rep}}$ could be reachable to a 100 GHz range, if our EO comb, which can generate a flat bandwidth up to $> 650$ GHz ($> 26\, f_{\mathrm {mod}}$) with $f_{\mathrm {rep}}$ multiplicable by waveshaping, is combined with a HSP source with broader bandwidth (e.g., [43]). Finally, the following improvement possibly enables $\displaystyle {C_{\mathrm {ab},\varphi _{\mathrm {e}}}}$ over 10 Mcps for the polarization-EPP experiment: to increase $T_{\mathrm {a(b)}}$ to the level that has been reported in [24] and to multiplex SSPDs having higher DEs as was demonstrated in Ref. [44].