Two-photon comb with wavelength conversion and 20-km distribution for quantum communication

Quantum computing and quantum communication, have been greatly developed in recent years and expected to contribute to quantum internet technologies, including cloud quantum computing and unconditionally secure communication. However, long-distance quantum communication is challenging mainly because of optical fiber losses; quantum repeaters are indispensable for fiber-based transmission because unknown quantum states cannot be amplified with certainty. In this study, we demonstrate a versatile entanglement source in the telecom band for fiber-based quantum internet, which has a narrow linewidth of sub-MHz range, entanglement fidelity of more than 95%, and Bell-state generation even with frequency multimode. Furthermore, after a total distribution length of 20-km in fiber, two-photon correlation is observed with an easily identifiable normalized correlation coefficient, despite the limited bandwidth of the wavelength converter. The presented implementation promises an efficient method for entanglement distribution that is compatible with quantum memory and frequency-multiplexed long-distance quantum communication applications.

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The solution to overcoming the distance limitations is to use quantum repeaters, which mainly consist of Bell-state operators with [14] or without [15] quantum memory (QM). The maximum distance of quantum key distribution (QKD) without using quantum repeaters was estimated to be approximately 550 km [16]; the maximum distance achieved experimentally to date is 421 km [17].
Aiming beyond secure communication of classical information by QKD, for instance, to combine quantum nodes such as quantum simulators and/or quantum computers, QM is required [18] to assist the repeater and preserve quantum information with arbitrary operation timing for multi-partite processing.
Based on the above discussion, QM-compatible versatile entanglement source (VES) is ardently desired to approach long-distance fiber-based quantum internet. The VES should emit telecom wavelength photons (~1.5 μm) to minimize fiber loss, have a sufficiently narrow linewidth for many types of QM ( ≪ 10 MHz ), and achieve high-fidelity quantum entanglement. Each of these specifications, however, is very difficult to realize that there have been only a few reports [19] about VES to date: It is still challenging to obtain all of high photon-count rate, narrow linewidth and high entanglement fidelity in the telecom regime. In a previous study, cavity-enhanced spontaneous parametric down conversion (SPDC) photon source was demonstrated for many applications [20]; for example, demonstrating theoretical/proof-of-principle models [21]- [26], highly bright single modes [27]- [31], and narrow linewidths [32]- [35]. On the other hand, for compatibility between telecom and QM absorption line, wavelength conversion (WC) is demonstrated using a laser [36] [37] and single photons [38] [39], for application to nitrogen-vacancy centers, rare-earth-doped crystals, and rubidium gas, among others. WC still has difficulty about conversion efficiency and noise photon to realize photonic interface.
In this study, we demonstrate a VES with wavelength conversion after 10-km fiber transmission.
We utilize the two-photon comb (TPC) technique, which realizes a large number of frequency multimodes and entangled photon pairs with a narrow sub-MHz linewidth even in a telecom band.
The WC is based on sum-frequency generation (SFG) and target Pr 3+ :YSO QM. We successfully realize two-photon 10-km transmission in an optical fiber or overall 20-km distribution and subsequent WC, resulting in a clear observation of comb structure of wavelength-converted TPC (WC-TPC), which is most suitable for use in frequency multiplexed quantum communication. This technical development can be applied not only to quantum information science but also to experimental optics.

TPC specifications
The TPC used herein consists of degenerate 1514-nm SPDC crystals and a surrounding cavity.
Further, SPDC occurred in type-0 periodically poled lithium niobite (PPLN) with temperature stability on the order of a few millikelvins. As shown in Fig. 1, we used a mutually orthogonal arrangement to generate two-photon polarization entanglement |HH⟩ + |VV⟩, where and are the probability amplitudes including relative phase [40]. This type of entanglement generation method has three advantages: high brightness or photon-generation rate compared with other phase matching types based on overlapped SPDC cones [40], low dependence on optics alignment, and path/phase compensation caused by birefringence in the crystal. However, this type of entanglement generation reduces the finesse of the surrounding cavity because of optical losses (e.g., via absorption, scattering, and reflection) in the crystal, which is a serious problem related to the absorption line of QM. For example, when two 1-cm-long PPLN crystals are placed inside a typical 0.5-m cavity, the theoretical linewidth of the cavity is ~5.3 MHz with a typical PPLN loss of 0.06 dB/cm [41] in the impedancematched case. Photons with such linewidth will be coupled to a 10-MHz-linewidth QM with an efficiency of ~70% (we assume Lorentzian photon profile and squarish QM absorption line).
Furthermore, because this value can only be obtained in a perfectly aligned case, careful cavity alignment is always required, which requires an additional complicated maintenance procedure. To overcome this disadvantage, we developed a ~2.5-m-long bow-tie cavity with a free spectral range (FSR) of ~120 MHz, which enabled a sub-MHz linewidth along with an increase in the number of frequency modes. All frequency modes are entangled in each photon pair and have good compatibility with atomic-frequency-comb QM, which enables time-and frequency-multiplexed quantum communication [42]. Cavity locking was achieved using the Pound-Drever-Hall technique to target the absorption line of QM and stabilize the relative phase of entanglement.
To evaluate the quality of the generated two photons, we measured two-photon statistics with a Hanbury Brown-Twiss-type setup, which consists of a 50:50 beam splitter, two superconducting single photon detector (SSPDs), and a time-correlated single-photon counting module (TCSPC). We utilized SSPD and silicon avalanche photodiode (SiAPD) depending on whether the wavelength of photon was telecom or visible. The SSPDs have efficiencies of approximately 85% at telecom wavelength; Fig. 2 (a) shows the raw data of two-photon correlation with 10-μW pump power. This type of Glauber's correlation function, which is similar to a comb structure, indicates frequency multimodality as in the Fourier relation. The measured time interval, corresponding to the time taken for a round trip in the cavity, was 8.6 ns, and the FSR was 116 MHz. In general, the ratio between the time interval and the pulse width of each peak gives the number of frequency modes or comb range: if some fiber dispersion exists, the number of modes can be evaluated because the pulse width becomes substantially larger than the timing jitter and thus the timing jitter becomes ignorable. Then, we performed 10-km fiber transmission and evaluated the width of the frequency range by measuring the dispersion. We observed ~2 ns broadening per one channel of TCSPC, and we estimated that TPC had a comb range of approximately 1-2 THz from the fiber dispersion of 15 ps·nm −1 ·km −1 (please see The exponential envelope with a long coherence time implies a narrow cavity linewidth Δ that is affected by the loss over the round trip. By approximating the envelope as −2 (Δ ) , we calculated as 0.95 MHz (or, according to Ref. [25], the degenerate photon linewidth will be 0.64 times this value, i.e., ~0.61 MHz). Whereas this is the case with an output mirror of 99% reflectivity, an output mirror of 95% reflectivity resulted in a broader cavity linewidth of 1.35 MHz and a higher brightness because the escape efficiency was approximately 2 to 3 times higher.
Photonic-state tomography was performed using the mirror of 95% reflectivity to obtain more precise counts in a shorter acquisition time; Fig. 2 (b) and (c) show the reconstructed density matrix of the absolute value after applying the maximum-likelihood method to 16 measurements [43]. The maximal fidelity to an arbitrary pure state was 96.1% and the concurrence was 93.0%, which were the highest values in the multimode regime (please see Supplementary note 6). Through 16 similar measurements, we further obtained a Clauser-Horne-Shimony-Holt parameter [44] S of 2.47 (> 2 implies nonlocality). Subsequently, we placed zero-order half-wave plates on the path after the beam splitter to adjust the relative phase of the entangled state and form a Bell state. Four Bell states were achieved (Fig. 3) by changing the angle of the horizontal plane, i.e., the yaw angle and slow axis in the vertical plane. The fidelity to |Φ + ⟩ = |HH⟩ + |VV⟩, |Φ − ⟩ = |HH⟩ − |VV⟩, |Ψ + ⟩ = |HV⟩ + |VH⟩, and |Ψ − ⟩ = |HV⟩ − |VH⟩ was 90.0%, 90.2%, 89.4%, and 88.1%, respectively (we omitted the coefficients 1/√2 ). These results show that this technique is effective, even for frequencymultiplexed entanglement.

Wavelength-converted two-photon comb
The principle of WC is sum frequency generation (SFG) in a PPLN waveguide with a strong auxiliary laser. Our target wavelength was ~606 nm, which is the center of the Pr 3+ :YSO absorption line 3 H4(0)-1 D2(0) [45]. In the whole experiment, we used 1514-nm photons and a 1010-nm laser, which satisfy the following two conditions: their sum frequency must be 606 nm and they can be stabilized by molecular gases of acetylene and iodine, respectively. Our setup aimed at conversion of a time-bin state which is suitable for long-distance fiber transmission and would achieve proper quantum frequency conversion because the temporal/linewidth profile of input photons would be preserved. Fig. 4 (a) shows WC-TPC with a SPDC pump power of 10 mW and SFG laser power of 50 mW. The WC-TPC has two important parameters: noise floor level and signal-to-noise ratio (SNR).
The noise floor level is influenced by many factors, including the presence of a residual laser, which can induce other nonlinear processes such as SPDC and Raman scattering [36] [37], dark counts in the detector, and stray light (related results are presented in the Supplementary note 5).
Two-photon correlation was characterized by the normalized second-order signal-idler correlation coefficient g , (2) (0) , which is defined as the ratio of the highest count to the average noise count, similar to SNR ( Fig. 4 (b)). The blue circles connected by a line represent g , (2) (0) before WC (1514nm two-photon data without WC), and the orange squares represent the values after WC-TPC (the log scale one is in Supplementary note 2). Some scattering was observed in the values after WC-TPC because the timing jitter increased by one order in the visible range compared with the telecom range.
The variation in g , (2) (0) clearly showed that the low SNR in the signal from the two photons in the telecom range can be increased by putting the two photons through the wavelength converter (see Discussion for details). When we tried to demonstrate only single-photon WC, which corresponds to a correlation between the telecom photon and the visible-range photon, we did not observe a clear correlation function, despite extensive adjustment of the SPDC/SFG pump power and spectral filters; we observed g , (2) (0) ≈ 1, i.e., almost the noise floor (data not shown).

Wavelength conversion after fiber transmission
For achieving memory-assisted quantum communication, TPC and WC were combined in an attempt to achieve long-distance communication, assuming that each wavelength has a distinct advantage: telecom qubits can be transmitted across long distances through a fiber with an attenuation of 0.2 dB/km, whereas the visible-range qubits can interact with highly efficient QM. It is very important to ensure that the whole system is applicable to fiber-based quantum communication because other problems such as the degree of polarization mixing and modulation by wavelength dispersion exist. can be reduced at telecom wavelengths, and our measured transmittance value was approximately 62% at 1514 nm. The second factor can be corrected only when the environmental conditions remain almost unchanged. The third factor is very small because the wavelength converter has a bandwidth of ~0.03 nm. The fiber dispersion of 15 ps·nm −1 ·km −1 therefore induces a pulse spreading of 4.5 ps which is much smaller than the timing jitter. For further increase in g , (2) (0) with or without a long fiber, the most effective approach is to develop a filter with a suitable frequency range to remove WC noise from the WC-TPC spectra; this technique is discussed in the following section.

Discussion
The TPC has a wide spectral range of ~1 THz, but a very narrow linewidth of ~1 MHz, and it shows polarization entanglement with a fidelity of ~90% to an arbitrary Bell state. In addition, it has a very long time interval and ensures high-speed modulation or time-bin state generation. A narrow linewidth is guaranteed by the long cavity length (or short FSR), even with low finesse. Therefore, the highest finesse in the current setup is not required because the linewidth will remain narrower than the absorption line of QM, even if the finesse reduces slightly due to either a shift in alignment or something else; in our case, the photon with 1-MHz linewidth has a redundancy and will couple to 4.6-MHz window of Pr 3+ :YSO QM even if finesse becomes less than a half of the present value.
Accordingly, TPC is free of frequent cavity alignment, resulting in good compatibility with the troublesome Bell-state generation and connection to QM. At smaller finesse, it has been reported that the intensity of side (or additional) clusters of the main frequency comb increases [28]; however, because of their spectral separation, the side clusters can be removed using a spectral filter or can be used to increase the number of modes without filtering. The cavity condition is affected by the temperature and convection of air, which can be compensated for by ensuring that the temperature stability of the crystal is of the order of several millikelvins over a period of few days. The condition of the crystal is the main factor preventing high fidelity because many frequency and time modes exist.
The relative phase and coherence will clearly deteriorate if the crystal position is changed. To approach the best condition, we used a polyethylene-terephthalate board to eliminate thermal interaction between the crystal holder and the positioning stage, and a wind shield to reduce the changes in air conditions and cavity length. For further improvement, an external compensation crystal, similar to that used for single-pass SPDC [46], and separative adjustment of the signal/idler using a conjoint double cavity [47] could be included.
To ensure good compatibility between QM and TPC, the QM requires an atomic frequency comb with a tailored absorption line composed of a finely arranged comb-like structure in the wide inhomogeneous-broadening of the rare-earth-ion ensemble (see Supplementary note 7 Based on the tomographic results, we consider that the dispersion or wavelength-dependent retardance of the waveplates affects the tomographic reconstruction. To realize desired states, additional waveplates unrelated to the tomography are utilized: one for |Φ ± ⟩, and two for |Ψ ± ⟩.
Although estimating the magnitude of the effect is difficult due to complex system, the obtained fidelity does appear lower when this is taken into account.
The increase in g , Although our WC crystal has a limited bandwidth of ~25 GHz that is restricted by the phasematching condition, WC-TPC can achieve an adequate SNR despite attenuation by a dissonance between the bandwidths of WC and TPC. We demonstrated that the two-photon rate can be increased by controlling the SFG pump power owing to the correlation-filtering effect, which enables detection of photon pairs even if an uncorrelated photon exists within the coherence time of a pair. A trade-off exists between conversion efficiency and noise count depending on the auxiliary laser power. A lower noise count at an SPDC pump power < 1 mW is appropriate despite the decrease in conversion efficiency. To achieve an optimal conversion efficiency and noise count, a dedicated filter focusing on the TPC spectrum is required. We propose a filter containing Pr 3+ :YSO with a high dopant concentration. Using rare-earth ions as a dynamical bandpass filter is not a new idea [50][51]; however, Pr 3+ :YSO with a high dopant concentration of ~1% has not been studied well, and is expected to absorb a broad noise spectrum. Although we consider that QM absorption also plays a role in decreasing the noise from outside the two-photon spectra, it degrades the atomic frequency comb because ions may transfer from other hyperfine states.
After developing a special filter, WC can be customized as a polarization-insensitive type [39], which is difficult to achieve with the same efficiency as that of a single-polarization type. Polarizationinsensitive WC is beneficial in the case of randomly rotating polarization when the conversion efficiency is more than half that of single-polarization WC. Irrespective of the type, the flying qubits should be in a time-bin state that is resistant to polarization changes along the transmission path [52].
Our single-polarization type WC can be directly applied to real quantum communication by using time-bin states because of the advantages of relative simplicity and higher efficiency: the polarization of time-bin photons will become unpolarized along with long-fiber transmission, and the condition will become the same as in this experiment, except for the quantum state.
In summary, we demonstrated VES with telecom TPC and wavelength conversion of two photons with a total fiber distribution length of 20 km. This technique renders a very narrow linewidth in the telecom range and high fidelity even with a long coherence time and multi-frequency modes, thereby creating prospects for quantum applications such as quantum networks combined with WC and the correlation-filtering effect. For further improving the TPC, a new method for separating two degenerate photons is desired to increase the distribution rate and investigate photon statistics. In the case of WC, a specialized filter is required to increase the SNR and establish a good relationship with QM. In addition, Bell state measurements for multimodes are preferable for frequency-multiplexing quantum communication applications. Our future plan is to convert polarization basis to time-bin basis and to add the loss of basis-conversion system to the WC-TPC result, which will enable us to estimate the whole quantum-communication system performance.

Two-photon comb
Two photons were generated by the SPDC process in two PPLN crystals (manufactured by Jinan Institute of Quantum Technology) having the dimensions of 0.5 × 3 × 10 mm and 0.5 × 0.5 × 10 mm , arranged orthogonally in a bow-tie cavity. The small dimensions allowed the crystals to precisely align with the laser path. The temperatures of the crystals and their holders were stabilized to be within ~1 mK. An SPDC pump laser of 757 nm was obtained by second harmonic generation (SHG) of a 1514-nm external-cavity diode laser (Sacher, TEC420-1530-1000), whose wavelength was stabilized using acetylene molecules. The pump laser was focused to a waist size of ~25 μm using lenses and a plano-concave mirror to achieve strong parametric interaction [53].
The optical cavity was stabilized using the Pound-Drever-Hall technique for this laser with an optical chopper with a duty cycle of 1/3. The two photons were separated using a 50:50 laser-line beam splitter, following which they entered a tomographic setup consisting of a zero-order 1514-nm quarter-wave plate, half-wave plate, and vertical-transmittance polarizer, similar to that in Ref. [43].
To generate a Bell state, two additional half-wave plates were placed in the path of one photon: one plate was used as a phase shifter by aligning the yaw angle, and the other was used as a bit flipper with a slow axis of 45°. The measurement time for 1 basis was 15 s, which was sufficient to converge the correlation function with a relatively strong pump power of 100 μW. The total testing time was ~10 min, including a rest time of 20 s. In almost all our experiments, the SSPDs were superconducting single-photon detectors with a detection efficiency of ~85%, which was the maximum value for our setup. However, a detection efficiency of ~60% was used in the experiments yielding the results shown in Fig. 4 (b) (blue dots) because our SiAPDs had an efficiency of 60% for visible wavelengths (SPCM-AQRH-14-FC). The maximal input power was 10 mW, resulting in a detected count rate approaching the limit of ~10 7 . We utilized HydraHarp 400 as a TCSPC module, whose resolution is 32 ps for that shown in Fig. 2 (a), and 16 ps and for the others. The 32 ps resolution was used because it could record longer interval times, which was required for measuring the coincidences with an adequate margin.

Wavelength converter
By removing the beam splitter and the tomographic setup, telecom photons were coupled to a polarization-maintaining fiber to be transported to the wavelength converter. An output collimator, consisting of a triplet lens, was placed immediately after the fiber for producing a high-quality singlemode Gaussian beam of diameter ~3 mm. The PPLN for the wavelength converter was of the waveguide type with an area of 9.9×11 μm and length of 48 mm (NTT electronics) with a type-0 phase matching condition (all three interacting lights are vertically polarized). The auxiliary laser for SFG has a wavelength of 1010 nm (TOPTICA, TA pro), where the emitted SHG light was stabilized using iodine molecules. Telecom photons and an auxiliary laser were coupled to this waveguide with efficiencies of ~60%. The external conversion efficiency was calculated by multiplying the coupling efficiency and the internal conversion efficiency. The external conversion efficiency was ~60%, whereas the internal counterpart was ~96% (details are presented in the Supplementary note 4). To obtain higher efficiency, we examined a lot of experimental conditions like changes in the combination of lenses, careful observation of waveguides by using camera, and optimization of noise filters. We found the best setup and realized the wavelength conversion of both two-photons. Two dichroic mirrors and one bandpass filter were installed as filters to remove WC noise, because using a spectrometer or numerous filters can cause optical loss, decreasing the two-photon coincidence count.
In the process of obtaining WC-TPC, one-photon wavelength-converted coincidences were highly noisy that only the noise floor or an extremely low g , (2) (0) was observed because of an excessively high photon rate at the telecom detector (~10 Mcts/s) due to pump-power-dependent uncorrelated photons.   This follows Ref. [43] and demands only one waveplate rotation between each measurement, shortening the total measurement time. Supplementary Figure 1 shows coincidence counts for the Bell states |Ψ + ⟩ and |Ψ − ⟩, which are realized by installing 45° slow-axis half-wave plate without yaw angle and 0° slow-axis half-wave plate with a yaw angle. The apparent difference appears in the counts for n10 (|DD⟩) and n16 (|RL⟩): By changing the yaw angle of only one half-wave plate, these counts visibly increase and decrease. Because of this advantageous nature, we could achieve high fidelity and arbitral relative-phase adjustment. The counts for |Ψ − ⟩ is a slightly low because of the misalignment of the optical path due to larger yaw angle.
Tomographic reconstruction and maximum likelihood method are achieved based on Ref. [2], which is an available program in Git Hub; thus, there is no room for our conveniences or mistakes in tomography experiment. Furthermore, to depict the density matrix, QuTiP library in python is used [3]. In this whole processing, fidelity is defined as ℱ = ⟨ | | ⟩ , which is a common definition (however, this is not square-root fidelity [4]). For further higher fidelity, improvement of the basin-like structure, which appears small in n16 count (left side of Supplementary Figure 1), is required. This is caused by polarization rotating along cavity-round-trip phase shift by passing through crystals or reflection on dielectric mirror with a few angles; it can be reduced by finely adjusting the crystal position and decreasing the reflection angle.
Our main effort aiming at high fidelity is to reduce decoherence effect as much as possible. We construct a bow-tie cavity with a small reflection angle (approximately 2-3 degrees) to decrease senkrecht and parallel polarization dependence. This angle is limited by finite width of PPLN crystal; the minimum magnitude is proportional to arctangent of the half of the short arm of bow-tie configuration and the width of PPLN. Long cavity can make reflection angle smaller than ordinal-size cavity, resulting in decoherence. Also, a PPLN crystal has birefringence, and we adjust the temperature of two PPLNs with a few milli-kelvin stabilities to reduce total birefringence. Furthermore, the mirrors after the cavity for coupling to the PMF are all removed, and two photons straightly couple into the PMF. These are only a few examples to realize high fidelity: dephasing mechanism is very complex in reality. We demonstrated that g , (2) (0) of WC-TPC can overcome that of original TPC (Fig. 4 (b)). In this supplementary note, we focus on original TPC or telecom two-photon correlation before WC. (SiAPD) have almost the same efficiency of 60% because of regulation of the applied current, the timing jitters differ greatly: ~40 ps for SSPD and ~300 ps for SiAPD.

Supplementary note 3: Estimation of bandwidth of TPC spectrum
In general, spectrometer or optical grating is widely used to analyze a spectrum. However, these "bandpass method" has a little difficulty of high optical loss. Further, when measuring both two photons, two frequency-correlated setups are required because two photons satisfy energy conservation. Then, we suggest another new method to estimate two-photon spectrum with lower loss and relatively easy setup.
We utilize 10 km single-mode fiber and observe dispersion of TPC. In general, optical fiber has birefringence and the optical path length could change depending on the fiber condition. When only the influence of dispersion is required, a correlation of two photons dispersed in the same fiber is suitable under the approximation that the fiber remains almost unchanged within the coherence time. , where c is a coefficient corresponding to peak counts; represents full width at half maximum (FWHM) of one frequency peak, i.e., linewidth; N is related to the number of timing combs;

Supplementary
denotes the time interval expressed as a reciprocal of free spectral range (FSR); is the FWHM of one timing peak which is approximated by a Gaussian-shape peak. Although a Voigt function should be used for the strict estimation of the spectrum, we used a Gaussian function to reduce complexity.
The red line corresponds to this function; although it appears smaller than raw data, the insets show that it has some variance around the peak counts. From this figure, despite passing through the 10 km fiber, frequency linewidth is almost the same as that of ~1. 35 MHz that corresponds to 95% reflectivity of cavity output coupler. is 2.0 ± 0.2 ns ns from (a) and 4.0 ± 0.3 ns from (b) among 10 peaks. The timing jitter of our measurement system is ~0.1 ns, and has ignorable influence on these values. The dispersion value around 1514 nm can be calculated using the dispersion equation as ~+15 ps·nm -1 ·km -1 ; then, we estimate the TPC frequency bandwidth of ~13 nm.
Such a very long timing can be compensated using a special fiber having an opposite sign of dispersion value. If this compensation can be considered as one of the re-modulations of wave packet, it can recover the quantum coherence as well [6]. Moreover, using band-pass effects such as WC, we can eliminate this broadening (Fig. 4 (c) in main text).  The main characteristics of a wavelength converter are the quantum efficiency of conversion and the counts of noise. Quantum efficiency is calculated as follows [7]:

Supplementary
where , , , , , and represent the number of photons, power of light, frequency of light, length of wavelength-conversion crystal, normalized power efficiency in the low-gain limit, and pump power, respectively. We used a 48 mm PPLN waveguide with a cross section of 9.9 μm × 11.5 μm to realize higher efficiency than bulk.
Supplementary Figure 4 (a) shows a quantum efficiency . We used telecom laser to directly measure the external efficiency, which is the conversion efficiency, including waveguide-coupling efficiency and transmission loss, but removed the influence of the auxiliary laser. Further, we obtained the telecom-laser coupling rate by turning the WC pump laser off, to estimate the internal efficiency.
The maximum external efficiency we could measure was 56.0%; however, it appears to increase slightly at higher pump power range. The coupling rates at 1514 nm and 1010 nm are 59.4% and 60.5%, respectively; then, we can estimate the internal efficiency. Our crystal is designed to have an efficiency of nearly 1, and the measured maximum value was 94.3%.
Noise count is obtained as shown in Supplementary Figure 4 (b). In this measurement, only WC pump laser is used, and the filters consist of two dichroic mirrors and one bandpass filter. The dotted curve represents quadratic-function fitting and the measured data set are almost positioned on fitting.
This indicates that noise is mainly caused by second-order nonlinear optical process in this setup. Then, we analyze the component noise spectrum using a spectrometer. The result indicates one peak, which is centered on 606 nm and a FWHM of ~ 25 GHz (not shown). We then try to adjust the number of filters or optical depth by adopting other bandpass filters or filtering crystals.
In Supplementary Figure 5, we defined the wavelength-conversion function * as the ratio of external efficiency and noise rate at kilo counts per second. In this process, the approximated function and the quadratic noise function are used. This function has no meaning in absolute value because of strange unit (% divided by kilo counts), but indicates the pump-dependent signal-to-noise ratio.
This calculation is similar to reference [8]. The maximum point is positioned at 7.97 mW pump power in our condition of dark counts of ~0.32 kHz; on using lower dark counts, this point shifts toward the left. Although this pump power was the best to obtain high SNR, we chose a higher power of 50 mW to obtain shorter experimental time for accurate comparison of g s,i (2) (0) (Supplementary Figure 2 (b)). As one of the indices, noise equivalent power (NEP) is often adopted [9], and the optimum pump power of that shows 46.8 mW. The actual comparison of the two-photon coincidences is shown in the next supplementary note.