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Yuta Kochi, [Sunao Kurimura](https://orcid.org/0000-0001-5220-1873), Junko Ishi-Hayase

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[Optimization of crystal length in a  pulse-pumped up-conversion single-photon detector for decoding femtosecond time-bin qubits](https://mdr.nims.go.jp/datasets/b7124cfd-9681-4141-8ffa-de8056000708)

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Optimization of crystal length in a pulse-pumped up-conversion single-photon detector for decoding femtosecond time-bin qubitsResearch Article Vol. 32, No. 26 / 16 Dec 2024 / Optics Express 47549Optimization of crystal length in apulse-pumped up-conversion single-photondetector for decoding femtosecond time-binqubitsYUTA KOCHI,1,2 SUNAO KURIMURA,3 AND JUNKOISHI-HAYASE1,2,*1School of Fundamental Science and Technology, Keio University, Kanagawa 223-8522, Japan2Center for Spintronics Research Network, Keio University, Kanagawa 223-8522, Japan3National Institute for Materials Science, Ibaraki 305-0047, Japan*hayase@appi.keio.ac.jpAbstract: In advancing ultrafast quantum communication and computing, it is crucial todevelop precise time-resolved measurement techniques for single-photon pulses. However,the measurement of photonic qubits, especially time-bin qubits, is limited by the temporalresolution of single-photon detectors, typically around tens of picoseconds. In this study, wedeveloped a pulse-pumped up-conversion single-photon detector (UCSPD) using periodicallypoled Mg-doped stoichiometric lithium tantalate (PPMg: SLT) crystals of varying lengths tooptimize femtosecond up-conversion. We evaluated the UCSPD’s efficiency and temporalresolution using a convolution model that accounts for group delay in nonlinear crystals. Ourresults demonstrate that the model calculations enable the accurate prediction of the crystallength dependence of temporal resolution and up-conversion efficiency without fitting parameters.The UCSPD achieved 415 fs resolution and 10.1 % efficiency with a 2 mm crystal, enablingsuccessful characterization of time-bin qubits with 800 fs pulse intervals.© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement1. IntroductionIn recent years, the development of quantum communication and quantum computing technologiesutilizing photons has garnered significant attention. A key component of these technologies isthe ability to accurately measure single-photon pulses with high temporal resolution. However,temporal resolution of conventional single-photon detectors is typically limited to 10∼100 psrange. Recently, Korzh et al. [1] achieved a breakthrough with superconducting nanowiresingle-photon detectors, demonstrating a sub-3 picosecond temporal resolution. Nevertheless,achieving ∼100 fs temporal resolution with single-photon detectors remains a challenge whilefemtosecond single-photon pulses have already been generated [2]. Consequently, the temporalspacing of photonic qubits, particularly time-bin qubits used in prior quantum communicationdemonstrations, has been constrained to the sub-nanosecond range [3–6]. Focusing on relativephase evaluation, Donohue et al. [7] demonstrated the determination of time-bin qubit phaseswith a pulse interval (∆t) of 2.2 ps. They decoded phase information from variations in frequencycomponents, however, the measurement accuracy was constrained by the monochromator’sresolution, which currently limited the pulse interval to approximately ∆t ∼ 1 ps. Severaldemonstrations of time-bin qubit decoding with picosecond-scale time intervals have beenachieved using an optical Kerr effect or by employing separate pulses with different polarizations[8–11]. However, due to the need to eventually detect the qubits with single-photon detectors, thetime intervals were typically stretched to the nanosecond range before decoding. This not onlycomplicates the experimental setup but also prevents full utilization of the advantages offered byultrafast communications.#542524 https://doi.org/10.1364/OE.542524Journal © 2024 Received 25 Sep 2024; revised 25 Nov 2024; accepted 29 Nov 2024; published 13 Dec 2024https://orcid.org/0000-0002-4238-305Xhttps://doi.org/10.1364/OA_License_v2#VOR-OAhttps://crossmark.crossref.org/dialog/?doi=10.1364/OE.542524&amp;domain=pdf&amp;date_stamp=2024-12-13Research Article Vol. 32, No. 26 / 16 Dec 2024 / Optics Express 47550To enhance temporal resolution, time-gating methods that have utilized nonlinear effects withultrashort pulses have been developed. One such method was the optical Kerr gate, which hasdemonstrated measurements of single-photon pulse with a temporal resolution of 224 fs [12].However, this technique relies on a third-order nonlinear optical effect, making it challenging toincrease detection efficiency [13,14]. Another promising method is the frequency up-conversionsingle-photon detector (UCSPD). This detector has been designed to detect photons at telecomwavelengths using Si APDs, which offer higher efficiency compared to InGaAs APDs [15,16].Recent implementations of UCSPDs using continuous-wave (CW) lasers as pump light haveachieved up-conversion efficiencies exceeding 90 % [17–19]. In these systems, both the temporalresolution and dead time is influenced by the performance of the Si APDs. On the other hand,employing a femtosecond pulse laser as the pump light has enabled femtosecond order temporalresolution [20–24]. Kuzucu et al. [25,26] measured a temporal waveform correlation of photonpairs using a UCSPD with a 1-mm-long periodically poled Mg-doped stoichiometric lithiumtantalate (PPMg:SLT) crystal and a femtosecond pulse laser. In sum-frequency generation withinthe femtosecond regime, the process is significantly affected by group delay in the crystal, makingit challenging to increase up-conversion efficiency by simply extending the crystal length, as ispossible with CW pumping. In fact, using a longer crystal can degrade temporal resolution andsaturate the up-conversion efficiency due to the group delay. To date, however, no quantitativeevaluation of the efficiency or temporal resolution, nor an estimation of the optimal crystallength, has been conducted for pulse-pumped UCSPDs operating in the femtosecond regime.Recently, Allgaier et al. [27,28] achieved high temporal resolution and up-conversion efficiencyby employing a 27-mm-long periodically poled lithium niobate (PPLN) crystal in a waveguidethat satisfied group velocity matching (GVM) between the pump and signal beams. However,this crystal features a unique 4.4 µm periodically poled period, making it extremely difficult toobtain. In any case, there are no demonstrations on measuring femtosecond time-bin qubits withpulse intervals shorter than 1 ps.Fig. 1. (a) Previous method for evaluating time-bin qubits and (b) the current methodemploying frequency up-conversion techniques. SFG: sum-frequency generation; BPF:bandpass filter.In this study, a UCSPD specifically developed for measuring femtosecond range ultrashortpulse signals was employed (see Fig. 1(b)). To evaluate the temporal resolution and up-conversionefficiency of this detector, commercial PPMg:SLT crystals of varying lengths were utilized.By fine-tuning the crystal length and pump power, the efficiency and temporal resolution ofthe UCSPD were quantitatively assessed. Additionally, a convolution model was introducedto quantitatively evaluate the dependence of the detector’s performance on crystal length. Ourresults show that this model accurately predicts the crystal length dependence of sum-frequencyResearch Article Vol. 32, No. 26 / 16 Dec 2024 / Optics Express 47551waveforms using only constants, such as group delay and pulse width, without any fittingparameters. This model enabled us to estimate the optimal crystal length for femtosecondUCSPD across arbitrary wavelengths and pulse widths using only the group delay of nonlinearcrystals. As a result, the optimized UCSPD successfully evaluated the relative phase of pseudofemtosecond time-bin qubits with the shortest pulse interval reported thus far, ∆t = 800 fs.2. Performance of UCSPD2.1. Experimental setupThe UCSPD was developed to overcome the performance limitations inherent in conventionalsingle-photon detectors. This device employs a frequency up-conversion technique, wheresignal photons are converted into visible photons through sum-frequency generation (SFG) uponintroduction into the detector. Utilizing a femtosecond pulse laser as the pump light, SFG occursexclusively when there is temporal overlap between the signal photons and the pump pulse. Bydetecting the frequency component of SFG using commercial single-photon detectors, such asSi APDs, femtosecond order temporal resolution is achieved through time-gating of the signalwaveforms.Figure 2 illustrates the schematic setup of the UCSPD. An 820 nm femtosecond Ti:Sapphirelaser (Coherent Mira-XP, pulse width ∆p: 200 fs, repetition frequency: 76.3 MHz) was utilizedto pump the nonlinear crystals, paired with a 1520 nm pulse from an optical parametric oscillator(Coherent Mira OPO-X, pulse width ∆s: 240 fs) serving as the signal light. For frequency up-conversion, 1 mol.% MgO-doped stoichiometric lithium tantalate (PPMg:SLT) bulk crystals wereemployed. These crystals offer considerable benefits, including low absorption and dispersioncoupled with high nonlinearity of 10 pm/V [29]. Their minimal linear and nonlinear absorbances[30] contribute to thermally stable operation even under high-power green light [31]. Notably,due to its short-band edge below 280 nm [32], the refractive index dispersion of PPMg:SLT inthe long-wavelength range is substantially lower than that of lithium niobate (LN) and potassiumtitanyl phosphate (KTP). This characteristic effectively minimizes pulse broadening and reducesgroup velocity mismatch [25] between signal and pump pulses. PPMg:SLT crystals of 1, 2,and 3 mm lengths with periodically poled periods of 8.55 µm were selected, optimal for type-0phase-matched SFG (820 nm + 1520 nm → 533 nm) at a crystal temperature of 85◦C. Thewavelength that satisfies the phase matching condition can be adjusted through the temperaturecontrol of the crystal. When the pump wavelength is fixed at 820 nm, the signal wavelengthsatisfies the phase matching condition within the range of approximately 1510 to 1560 nm byvarying the temperature from 20 to 200◦C. The pump and signal pulses were focused throughseparate lenses to prevent chromatic-aberration-induced focal shifts [33]. The focal length ofthe pump beam lens fp was 200 mm, and that of the signal beam lens fs was 150 mm. Thebeam spot sizes were set to 80 µm for the signal and 100 µm for the pump. The beam diameterswere set larger than the optimal values estimated based on the crystal length, according toBoyd et al.’s theory [34] since focusing to a smaller spot size could introduce instability due tooptical path fluctuations and unexpected noise, such as supercontinuum generation. Moreover,by using this looser focus, we were able to effectively compare the crystal length dependenceof the UCSPD performance independently of other factors. The up-converted photons weresubsequently detected using a Si APD (Hamamatsu C13001-01). The quantum efficiency of usedSi APD was 41 % at 532 nm, and the dark count rate was 100 cps.2.2. Results and discussionThe initial assessment of temporal resolution and up-conversion efficiency was performed bymeasuring a single pulse at the single-photon level (average photon number: 0.1/pulse) usingthe UCSPD, with the pump power set to 300 mW. The lengths of the PPMg:SLT crystals wereResearch Article Vol. 32, No. 26 / 16 Dec 2024 / Optics Express 47552Fig. 2. Schematic of the UCSPD system. BPF: band-pass filter; DM: dichroic mirror;MMF: multimode fiber; Si APD: silicon avalanche photodiode.varied, and the positions of the lenses were readjusted to optimize the focal points of the pumpand signal light, thereby maximizing the up-conversion efficiency. Figure 3(a) shows the countrate of the up-converted photons as a function of the time delay between the signal and pumppulses. It was observed that the peak count rate increased with the crystal length from 1 to 2mm but reached saturation with a 3 mm crystal due to the reduced temporal overlap between thepump and signal pulses, which was caused by group delay in the crystal.Fig. 3. (a) Measured waveforms of single-photon level pulses obtained with the UCSPDare shown. Solid lines represent the calculated waveforms using our convolution model. (b)Graph showing temporal resolution (black) and up-conversion efficiency (red) as functionsof crystal length. Square dots represent experimental data, while solid lines are theoreticalpredictions based on Eq. (4). The hollow dots at a crystal length of 0 mm represent estimatedvalues, with an up-conversion efficiency of 0 % and a temporal resolution of 200 fs (same asthe pump pulse width).To estimate the temporal resolution, the temporal resolution function T(t) is defined as follows:T(t) ≡ (P ∗ G)(t)G(t) ={︄1 −τgL2 ≤ t ≤ τgL20 t<−τgL2 , τgL2 <t,(1)where P(t) represents the temporal waveform of the pump pulse, and G(t) is a rectangular functionwith a width equal to the group delay τgL (with τg being 204.3 fs/mm [35]) between the pumpResearch Article Vol. 32, No. 26 / 16 Dec 2024 / Optics Express 47553and signal pulses. This model assumes that the pump electric field is sufficiently weak, such thatthe SFG intensity is proportional to the pump power. The waveforms measured by the UCSPD,Sout(t), can be expressed as:Sout(t) = (Sin ∗ T)(t). (2)Assuming Gaussian temporal waveforms for both the pump and signal pulses,P(t) ∝ exp⎡⎢⎢⎢⎢⎣−(︄2√ln 2∆pt)︄2⎤⎥⎥⎥⎥⎦Sin(t) ∝ exp⎡⎢⎢⎢⎢⎣−(︄2√ln 2∆st)︄2⎤⎥⎥⎥⎥⎦ ,(3)waveform Sout(t) is computed as follows:Sout(t) = (Sin ∗ T)(t)∝∫ τgL/2−τgL/2exp[︄−4 ln 2∆2p + ∆2s(t − τ)2]︄dτ=∫ τgL/2−t−τgL/2−texp[︄−4 ln 2∆2p + ∆2sτ′2]︄dτ′∝ erf[︄√︄ln 2∆2p + ∆2s(︁τgL − 2t)︁ ]︄− erf[︄√︄ln 2∆2p + ∆2s(︁−τgL − 2t)︁ ]︄.(4)This convolution model illustrates that as the crystal length (L) increases, the correspondinggroup delay (τgL) also increases, causing the measured waveform Sout(t) to approximate arectangular shape. Temporal resolution was defined as the FWHM of T(t). Analyzing thedata presented in Fig. 3(a), the temporal resolutions of the UCSPD were estimated to be 255,415, and 591 fs for crystal lengths of 1, 2, and 3 mm, respectively, as indicated by the squaredots in Fig. 3(b). The pulse shapes of solid lines in Fig. 3(a) show good agreement with theexperimental results across all crystal lengths, despite the calculation using Eq. (4) involvingonly the group delay value of 204.3 fs/mm and the pulse widths of pump and signal pulses. Tocompare the experimental and calculated waveforms, we only adjusted the amplitude of thecalculated waveforms for the 1-mm-long crystal. Although the relative amplitudes between the1, 2, and 3 mm crystals were kept unchanged, the calculated waveforms closely replicates theexperimental results. In this model, it was assumed that there were no changes in the beamdiameters of the signal or pump, nor any pulse broadening (chirping) due to group velocitydispersion. This approximation is valid because, in femtosecond range UCSPD, increasing thecrystal length reduces pulse overlap due to group delay, and only millimeter-scale crystals canbe practically used. Consequently, changes in beam diameter have a negligible effect, and thecalculated results closely match experimental values, validating this assumption. However, whenthe crystal length reaches around 3 mm, changes in beam diameter can no longer be ignored. Inthe experiments, the spot size of the signal beam was set smaller than that of the pump beam toensure full overlap. Nonetheless, due to the shorter wavelength of the pump light, it was observedthat for crystal lengths exceeding 2 mm, the signal beam diameter becomes partially larger thanthe pump beam, reducing spatial overlap. In practice, for the 3-mm-long crystal, the calculatedwaveform displayed a larger amplitude than observed experimentally.Figure 3(b) depicts the dependence of the temporal resolution and up-conversion efficiency oncrystal length. These results demonstrate that the model outlined in Eq. (4) adequately explains theResearch Article Vol. 32, No. 26 / 16 Dec 2024 / Optics Express 47554experimental results using only constants such as the group delay of the crystals and pulse widths,with no additional fittings required. This suggests that it will enable the selection of the appropriatecrystal length according to the required temporal resolution and up-conversion efficiency foreach experiment in the future femtosecond UCSPD designs. Regarding up-conversion efficiency,the calculated value for the 3-mm-long crystal exceeds the experimental result due to variationsin beam diameter. Therefore, further analysis is needed for longer crystal lengths, and futureresearch should focus on calculations that account for changes in beam diameter.The subsequent investigation focused on the dependence of up-conversion efficiency (ηUC)and the detection limit of the UCSPD on pump power. As shown in Fig. 4(a), the up-conversionefficiency increases proportionally with the pump power of up to 400 mW and the efficiencybegun to saturate beyond this power. In our experiments, including those conducted thereafter, weused a pump power of 300 mW, and it was found that our simple model is sufficiently applicableat this intensity. For higher pump powers, such as in SFG under high-intensity pumping, it will benecessary to consider the effects of reverse processes and the saturation of conversion efficiency.Concurrently, Fig. 4(b) illustrates that the noise count rate rises exponentially with increasingup-conversion efficiency. Considering that the dark count rate of the Si APD was approximately100 cps, this noise is primarily attributable to photons originating from the PPMg:SLT crystals.The predominant source of noise photons is the spontaneous parametric down-conversion (SPDC)process, which is reconverted by the pump light, as Raman scattering is minimal in crystals of afew millimeters in length [36]. Previous studies have suggested that SPDC noise can be reducedby employing a longer pump wavelength of approximately 1.9 µm [37–39].Fig. 4. (a) Up-conversion efficiency plotted against pump power for different crystal lengths:black for 1 mm, red for 2 mm, green for 3 mm. (b) Variation in noise count rates. (c) Averagenumber of photons per pulse at the detection threshold.Research Article Vol. 32, No. 26 / 16 Dec 2024 / Optics Express 47555The detection limit is defined as the number of photons resulting in a count rate three timesthe standard deviation of the noise count rate (σNCR):Average photon number of the detection limit =3σNCRηtot. (5)Here, ηtot = ηUC × ηother represents the total UCSPD system efficiency, which includes theefficiency ηother considering other losses such as the Si APD detection efficiency of 41 %, fibercoupling efficiency of 85 %, and filter transmittance of 94 %. Fig. 4(c) illustrates that the detectionlimit remains relatively stable for pump powers exceeding 200 mW, registering minimum valuesof 8.6, 3.3, and 3.1 ×10−5 photons per pulse for crystal lengths of 1, 2, and 3 mm, respectively, ata pump power of 300 mW. The internal up-conversion efficiencies ηUC were calculated to be6.1, 10.1, and 10.2 % for the 1, 2, and 3 mm crystals, respectively (Table 1). In this experiment,we did not optimize the beam diameter for the above-mentioned reasons. However, if the beamdiameter can be optimized for each crystal length, further efficiency improvements expectedto be possible. Furthermore, we did not use a transform-limited pulse for the pump light ofthe UCSPD. Compared to a laser with the same pulse width, a transform-limited pulse has anarrower spectrum, which increases the frequency components of the pump light within thecrystal’s bandwidth, and is expected to result in higher conversion efficiency. When compared toother detectors, the UCSPD achieves notably lower noise levels than InGaAs detectors at roomtemperature (300 K) and delivers high temporal resolution on the femtosecond scale (Table 2).Table 1. Dependence of UCSPD performance on crystal length.Crystal Temporal Up-conversion NoiseLength Resolution Efficiency Count Rate1 mm 255 fs 6.1 % 1910 cps2 mm 415 fs 10.1 % 800 cps3 mm 591 fs 10.2 % 700 cpsTable 2. Performance comparison of various single-photon detectors at telecom wavelength.Typical values for the SNSPD are based on SCONTEL TCOPRS-CCR-TW-60.Single-photon Operating Temporal Quantum Noise DeadDetector Temperature Resolution Efficiency Count Rate TimeInGaAs/InP [16] 223 K 170 ps 27.5 % 1200 cps 100 nsSNSPD (Typ.) 2.1 K 57 ps 80 % 10 cps 10 nsSNSPD [1] 0.9 K 4.3 ps - - -optical Kerr gate [12] 300 K 0.2 ps - - -UCSPD (CW) [19] 300 K 400 ps 40.2 % 200 cps 100 nsUCSPD (pulse) [27,28] 300 K 0.3 ps 16.9 % - -UCSPD (this work) 300 K 0.4 ps 3.3 % 700 cps -Recent studies have shown that GVM can be effectively mitigated using type-II quasi-phasematching in PPLN crystals across the wavelength range of 780–1560 nm [40,41]. Allgaier’sresearch [27,28] reached high up-conversion efficiency and temporal resolution by addressingthis phase matching in a PPLN crystal with a periodically poled period of 4.4 µm withina 27 mm optical waveguide. Although effective, this setup presents significant challenges,particularly because obtaining the crystal required for this design is extremely difficult. Incontrast, our UCSPD achieves femtosecond order temporal resolution and practical detectionefficiency using commercially available crystals, offering a more accessible and equally robustResearch Article Vol. 32, No. 26 / 16 Dec 2024 / Optics Express 47556alternative. Moreover, our simple convolution model accurately represented the experimentalresults using only the group delay value and the pulse widths of the pump and signal pulses. Thismodel provides the optimal crystal length for femtosecond regime up-conversion across variousnonlinear crystals, wavelengths, and pulse widths.3. Evaluation of femtosecond time-bin qubits3.1. Experimental setupFigure 5 depicts the experimental setup utilized for generating and assessing pseudo femtosecondtime-bin qubits. This arrangement comprises two unbalanced Michelson interferometers, eachconfigured with identical time delays (∆t), which establish the pulse intervals for the time-binqubits. Each interferometer is equipped with a polarization beam splitter and two quarter-waveplates to minimize light loss upon reflection. Additionally, a cover glass was positioned in onearm of each interferometer to facilitate the adjustment of the relative phase by varying the angleof the glass plate.Fig. 5. Experimental setup for generating and evaluating femtosecond time-bin qubits. CG:cover glass; ND: neutral density filter; PBS: polarization beam splitter; λ/2: half-wave plate;λ/4: quarter-wave plate.For this experiment, light pulses at the single-photon level (average photon number: 0.1/pulse)from an optical parametric oscillator were employed as the light source. The time delay ∆t was setto 800 fs, significantly exceeding both the pulse width and the temporal resolution of the UCSPD.As a single-photon traverses the interferometer, it is transformed into a time-bin qubit encoded as1√2(|e⟩ + eiφ |l⟩), where |e⟩ denotes the photon traveling through the short arm, and |l⟩ signifiesthe photon passing through the long arm. As these time-bin qubits reenter the interferometer,each photon state evolves into |e⟩ → 1√2(|ee⟩ + eiφ′|el⟩) and |l⟩ → 1√2(|el⟩ + eiφ′|ll⟩), resulting inthe output state |ψout⟩ of the interferometer:|ψout⟩ ∝ |ee⟩ + (eiφ + eiφ′)|el⟩ + |ll⟩. (6)Research Article Vol. 32, No. 26 / 16 Dec 2024 / Optics Express 47557The ratios of the probability amplitudes Pee, Pel, and Pll (corresponding to photon states |ee⟩,|el⟩, and |ll⟩, respectively) are analyzed to provide insights into the system dynamics:Pee : Pel : Pll = 1 : 4 cos2(︃ϕ − ϕ′2)︃: 1. (7)Therefore, by time-gating the central peak Pel using the UCSPD, the relative phase differenceϕ − ϕ′ of the time-bin qubits can be evaluated effectively. We used a 2-mm-long PPMg:SLTcrystal and 300 mW pump power for the UCSPD.3.2. Results and discussionFigure 6(a) shows the measured temporal waveforms of pseudo femtosecond time-bin qubits atvarious relative phases. It is evident that the waveforms of time-bin qubits with an interval ofonly 800 fs have been clearly measured using our UCSPD. The solid blue line represents theestimated original waveforms, reconstructed through deconvolution using the temporal resolutionfunction T(t). This demonstrates that our convolution model for the UCSPD enables accuratereconstruction of the original signal waveforms prior to measurement. Notably, the interferencewaveforms of time-bin qubits with pulse intervals as short as 800 fs are clearly observablewhen the pump pulse delay is set to 0 fs. Under these detection conditions, the detection limitof our UCSPD was 3.3 × 10−5/time-bin, which is sufficient to measure single-photon pulses.This underscores the capability of our technique to achieve high precision in time-resolvedmeasurement and reconstruction of waveforms in the femtosecond range. For measurementsrequiring even higher temporal resolution, a 1-mm-long crystal can be used. In this case, althoughthe up-conversion efficiency decreases to 6.1% and the average photon number detection limitreduces to 8.6× 10−5/time-bin, this adjustment enables pulse intervals to be shortened to as closeas 540 fs, thereby significantly enhancing the temporal resolution capabilities.Fig. 6. (a) Temporal waveforms of time-bin qubits, with black and red dots representingexperimental data and the solid blue line depicting the original waveforms reconstructed bydeconvolution using the temporal resolution function T(t). (b) Variation in the count of thecentral peak at τ = 0 fs in (a), with solid lines representing sine curve fittings to the data.Black: ϕ′ = 0; Red: ϕ′ = π. Blue squares represent background noiseSubsequently, with the delay time maintained at 0 fs, the variation in interference intensityPel as a function of the relative phase ϕ − ϕ′ was measured. Figure 6(b) illustrates how thecount varies with the relative phase, demonstrating visibilities of 98.2 ± 0.09% for ϕ′ = 0 and98.1 ± 0.09% for ϕ′ = π. Hence, we have successfully optimized and evaluated the performanceof the femtosecond UCSPD, achieving high-visibility decoding of time-bin qubits with an 800 fspulse interval, which was previously challenging to measure.Research Article Vol. 32, No. 26 / 16 Dec 2024 / Optics Express 475584. ConclusionIn this study, a pulse-pumped up-conversion single-photon detector (UCSPD) was developedusing commercially available bulk PPMg:SLT crystals. To quantitatively assess the temporalresolution and up-conversion efficiency of the UCSPD, we introduced a convolution modelto estimate the optimal crystal length. Using this model, we accurately simulated the crystallength dependence of the temporal resolution and up-conversion efficiency of the UCSPD usingonly the group delay value and the pulse widths of the pump and signal pulses. This suggeststhat it will enable the selection of the appropriate crystal length according to the requiredtemporal resolution and efficiency for each experiment using various types of nonlinear crystals,wavelengths, and pulse widths in the future design of femtosecond UCSPD. In our UCSPD, wedetermined that the optimal crystal length for our setup was 2 mm. With this crystal length,the UCSPD demonstrated a temporal resolution of 415 fs, significantly surpassing conventionalsingle-photon detectors, and achieved an up-conversion efficiency of 10.1%. Additionally, thisstudy successfully evaluated single-photon-level femtosecond time-bin qubits with a pulse interval∆t of 800 fs, the shortest interval reported to date. This methodology promises a substantialenhancement in the information density of quantum communication—potentially increasing it bythree orders of magnitude—and in accelerating communication speeds. Future work will focuson further exploring the characteristics of time resolution and output waveforms identified in thisstudy. This includes optimizing beam diameter conditions and conducting detailed theoreticalanalyses using propagation equations for the frequency mixing of ultrashort pulses. Moreover,this method requires sweeping the stage to measure a wide range of waveforms. However, byintegrating it with time-to-space conversion [42], time-to-frequency conversion [43], and thedual-comb-based Asynchronous Optical Sampling (ASOPS) technique [24] will enable rapidwaveform acquisition over a wide time range without the need to move the stage. Subsequentresearch will also involve evaluating time-bin qubits with nonclassical light sources and exploringtheir potential for storage and retrieval in broadband quantum memory systems utilizing quantumdots [44,45], thereby advancing the field of quantum communication technology.Funding. Japan Society for the Promotion of Science (Grant-in-Aid for JSPS Fellows No. JP23KJ1916); CoreResearch for Evolutional Science and Technology (JST CREST No. JPMJCR24A5).Acknowledgments. This work was supported by Center for Spintronics Research Network and the Program forthe Advancement of Next Generation Research Projects at Keio University. The authors extend their gratitude to Prof. R.Shimizu for insightful discussions and technical support.Disclosures. The authors declare no conflict of interest.Data availability. Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.References1. B. Korzh, Q.-Y. Zhao, J. P. Allmaras, et al., “Demonstration of sub-3 ps temporal resolution with a superconductingnanowire single-photon detector,” Nat. Photonics 14(4), 250–255 (2020).2. E. Y. Zhu, Z. Tang, L. Qian, et al., “Poled-fiber source of broadband polarization-entangled photon pairs,” Opt. Lett.38(21), 4397–4400 (2013).3. A. Martin, F. Kaiser, A. Vernier, et al., “Cross time-bin photonic entanglement for quantum key distribution,” Phys.Rev. A 87(2), 020301 (2013).4. E. Saglamyurek, J. Jin, V. B. Verma, et al., “Quantum storage of entangled telecom-wavelength photons in anerbium-doped optical fibre,” Nat. Photonics 9(2), 83–87 (2015).5. 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