2.1.

Approaching True PNR with an SMSPD

The SMSPD comprises a single meandering microstrip. As shown in Fig. 1(a), the meandering microstrip is divided into three distinct functional segments for the purposes of detecting, connecting, and bending according to their width, which are designated as A, B, and C, respectively, and color-coded accordingly. The detecting section A features a circular active area of $50\mu \mathrm{m}$ diameter and contains a series of parallel $1\text{-}\mu \mathrm{m}$-wide microstrips with 100-nm gaps, resulting in a high fill factor of 91%. In the bending section C, we utilized the L-shaped layout for the microstrips to maintain a high fill rate while mitigating the current-crowding effect^38,40, which incorporates an optimized 90 deg turn before the 180 deg turnaround. The 90 deg turn transitions the strip width from 1.05 to $1.5\mu \mathrm{m}$, while the 180 deg turnaround has a trip width of $1.5\mu \mathrm{m}$ and an interval of $2\mu \mathrm{m}$. The connecting wires in part B for linking sections A and C are $1.05\mu \mathrm{m}$ wide and are spaced 50 nm apart from each other. This design, where strips for connecting and bending are wider than those for detection, aims for decreasing the impact of defects to increase the device yield and minimizing the current crowding to enhance the switching current. Figure 1(b) illustrates the simulated current density distribution in an L-shaped bend using the RF module of COMSOL Multiphysics. It is evident that the current density is highest in detecting section A, while in other sections, such as points b through e, it is lower. Particularly, the current density in point e in the bending section is only 87% of that in point a in the detection section. To ensure stable operation without a shunt resistor, the detector was connected in series with $1.2\text{-}\mu \mathrm{m}$-wide meandering microstrips, resulting in a total inductance of $6\mu \mathrm{H}$. The detector was fabricated using a 7-nm-thick NbN film on a silica substrate and exhibited a switching current of $122\mu \mathrm{A}$ at the operation temperature of 2.2 K. Its basic performance (as shown in Figs. S1 and S2 in the Supplementary Material) is discussed in section 1 of Supplementary Material.

Fig. 1

SMSPD. (a) Scanning electron micrograph of the SMSPD. (b) Simulated current density distribution in three sections: detecting (A), connecting (B), and bending (C). Points a–e indicate the relative current density. This structure ensures a higher current density in the detecting section than in the other sections. (c) Rising section of typical output waveforms generated from different detections of one, two, and three photons.

We investigated the multiphoton response of the SMSPD by using an attenuated 1064 nm pulsed laser. In contrast to a previously reported cryogenic readout,²⁹ we directly amplified the photon-triggered electrical pulses using a room-temperature low-noise amplifier (LNA1800) and captured them with a high-speed oscilloscope. In Fig. 1(c), the rising-edge section of typical output waveforms exhibited distinct slope separations. To determine the rising-edge time $\mathrm{\Delta }{\tau }_{\text{rise}}$ of electrical pulses, we employed two fixed discrimination levels: a low level $V_L$ set to the maximum value of the system’s electrical noise, and a high level $V_H$ set to the minimum amplitude of the electrical pulses, as shown in Fig. 1(c). Owing to the large inductance and relatively small resistance, the output pulses have a long time in rising edges. The corresponding rising-edge time of 4.845, 3.447, and 2.667 ns indicates one-, two-, and three-photon events, respectively. Figure 2(a) exhibits the histograms of the rising-edge time (black dots) at two effective mean photon numbers per pulse $\tilde{\mu }$ of 2.5 and 5.1. Here, $\tilde{\mu }$ took into account the detector efficiency and coupling losses. Notably, we observed distinct separations of rising-edge time for up to 10 photons. We group the counting probability for $n\ge 11$, as significant overlap renders these photon events indistinguishable.

Fig. 2

Photon-number resolution in an SMSPD. (a) Histograms (dots) and Gaussian fitting (lines) of the rising-edge time of response pulses under pulsed laser illumination with an effective mean photon number $\tilde{\mu }$ at 2.5 and 5.1. Color areas represent the decomposed Gaussian functions. (b) Confusion matrix illustrating the probabilities of assigning $n$ detected photons to $m$ reported photons, where the diagonal terms $P_n^n$ represent the photon-number readout fidelity. (c) Photon count statistics reconstructed from the distributions of pulse rising-edge time at different effective mean photon numbers $\tilde{\mu }$ ranging from 0.05 to 5. The measured photon count statistics (color bars) align closely with the Poisson statistics of the coherent source (dashed lines).

In order to assign a photon number $m$ for a detection event with pulse rising-edge time $\mathrm{\Delta }{\tau }_{\text{rise}}$, a series of dividing thresholds ${\tau }_{k,1\le k\le 11}$ need to be determined. This allows us to assign the event to a photon number $m$, if ${\tau }_m<\mathrm{\Delta }{\tau }_{\text{rise}}<{\tau }_{m+1}$. To begin, we fitted the histograms of rising-edge time with Gaussian models [color area in Fig. 2(a)]. Subsequently, we normalized each distribution $G_n(t)$ associated with the photon number $n$. The dividing thresholds were determined at the position of the intersection between two adjacent Gaussian distributions. Note that as the photon numbers increase, the center distance decreases, and the overlap between adjacent distributions increases—thus the certainty of assigning a photon number decays. To access the accuracy of the photon-number assignments using the rising-edge time of output pulses, we calculate the probability distribution of assigning photon number $m$ to the detection event when $n$ photons detected $P_n^m$, which is described as⁴¹

Figure 2(b) illustrates the confusion matrix of photon-number assignment, providing insights into the readout fidelity of $n$ photons, represented by $P_n^n$. In the ideal case, the confusion matrix would be an identity matrix, with all diagonal terms equal to 1 and all nondiagonal terms equal to 0. Our readout quality approaches the ideal case in the low photon-number regime. For example, the readout fidelity achieves a parts-per-hundred-billion precision of $P_1^1=0.99999999993$ for photon number $n=1$ and a parts-per-million precision of $P_2^2=0.999998$ for $n=2$. For higher photon numbers, the readout fidelity gradually decreases. It remains above 0.98 for $\le 4\text{photons}$ but significantly decays to below 0.90 for $\ge 7\text{photons}$.

After establishing the dividing thresholds ${\tau }_{k,1\le k\le 11}$, the photon events in different threshold-to-threshold regions can be counted to reconstruct the photon statistics $Q(n)$. When the photon number ranges from 1 to 10, each hotspot has a length between 10 and $3\mu \mathrm{m}$, estimated from electrothermal simulation. Since it is far smaller than the total length ($\sim 2000\mu \mathrm{m}$) of the microstrip, the probability of hotspot overlap is negligible. Therefore, as shown in Fig. 2(c), the photon statistics $Q(n)$ directly followed the Poisson distribution $Q(n)={\tilde{\mu }}^n{e}^{-\tilde{\mu }}/n!$ without requiring a conditional probability typically necessary for most array-based quasi-PNR detectors,³¹ which indicates that the SMSPD infinitely approaches the true PNRD. The effective mean photon number per pulse $\tilde{\mu }$ is estimated by measuring the photon clicking probability ${\eta }_{\text{click}}=R_{\text{click}}/f_{\mathrm{rep}}$ and solving the equation $1-e^{-\tilde{\mu }}={\eta }_{\text{click}}$, where $R_{\text{click}}$ is the photon clicking rate and $f_{\mathrm{rep}}$ is the repetition rate of pulse laser.

2.2.

Influence Mechanism of the Photon-Number Readout Capability

To investigate the influence mechanism of the PNR capability in SNSPDs/SMSPDs, we designed a comparison vector including five detectors with different widths and inductance: $100\mathrm{nm}-1\mu \mathrm{H}$, $100\mathrm{nm}-3\mu \mathrm{H}$, $100\mathrm{nm}-5\mu \mathrm{H}$, $300\mathrm{nm}-5\mu \mathrm{H}$, and $500\mathrm{nm}-5\mu \mathrm{H}$. The SNSPDs with nanostrip widths of 100, 300, and 500 nm covered different circular active areas of 20, 30, and $50\mu \mathrm{m}$ diameter with a filling rate of 33%, 50%, and 50%. The target inductance was achieved by connecting in series with 1.5 times wider meandering nanostrips than the photon-sensitivity nanostrips. The nanostrips acting as inductance were located approximately $50\mu \mathrm{m}$ away from the photon-sensitivity area to avoid photon absorption.

Figures 3(a)–3(e) depict the histograms of rising-edge time of response pulses from the comparison vector. In general, detectors with larger inductance and wider strips exhibit improved photon-number resolution. Based on 100-nm wide strips [Figs. 3(a)–3(c)], the detector is only capable of distinguishing two photons. Traditional wisdom argues that a larger inductance imposes two disadvantages on SNSPDs, including degrading the SNR of rising edges and increasing the recovery time. However, the stretched rising edges are beneficial for photon-number discrimination due to more distinguished rising edges among different-photon-number events. As shown in Figs. 3(a)–3(c), by increasing the inductance, the overlaps between one- and two-photon events get smaller. Upon maintaining the inductance at $5\mu \mathrm{H}$ and increasing the strip width $w$, the PNR capability improves significantly. Therefore, the 100-nm wide detector can only achieve over 90% readout fidelity for assigning between single-photon and multiphoton events, while the 500-nm-wide and $1\text{-}\mu \mathrm{m}$-wide detectors can, respectively, resolve four and six photons with 90% readout fidelity. Note that in the subgraph of Fig. 3(a), the shoulder in the histogram of rising time exceeding 600 ps arises from the counting events at bends of meandering nanostrips, where the hotspot resistance is smaller than that of straight sections (as illustrated in Fig. S3 in the Supplementary Material). If ensuring full coupling of the incident photon onto the straight nanostrips, this phenomenon can also aid in eliminating intrinsic dark counts at the bends of meandering nanostrips.

Fig. 3

Photon-number readout capability versus inductance and width. (a)–(e) Histograms and Gaussian fitting of the rising-edge time of response pulses generated from detectors with varying inductance and width. Black dots, measurement data; blue lines, Gaussian fitting results; color areas, decomposed Gaussian functions. The rising-edge time shows a power function with an exponent of 0.5 in relation to the photon number. Green diamonds, extracted mean of rising-edge time; orange dashed lines, fitted power functions. (f) The $\overline{{\tau }_m}$ and $\overline{{\tau }_{\mathrm{std}}}$ of rising-edge time and relative SNR at different detector conditions. (g) Photon-number readout fidelity of detectors with different inductance and width, which are extracted from (a)–(e).

We conclude that the phenomenon described above was caused by the limited bandwidth of the readout electronics and the SNR of response pulses. A universal amplifier for SNSPDs (such as LNA1800 here) typically has a 3 dB bandwidth of $\sim 1\mathrm{GHz}$, which restricts the rise time (10%–90% amplitude) to ${\tau }_{\lim }$ ($\sim 300\mathrm{ps}$). Consequently, when the intrinsic response pulses have a fast rising-edge time ${\tau }_{\mathrm{int}}$ approaching or smaller than ${\tau }_{\lim }$, their rising edges will be broadened after passing through the amplifier, following $\sqrt{{\tau }_{\mathrm{int}}^2+{\tau }_{\lim }^2}$. This addition of rising time prevents us from accurately acquiring the real rising time and also brings the pulse rising time of multiphoton events closer.

According to Eqs. (5) and (6), increasing the total inductance $L_{\mathrm{k}}$ of SNSPDs directly increases the intrinsic rising-edge time ${\tau }_{\mathrm{int}}$ on the scale of $\sqrt{L_{\mathrm{k}}}$, which subsequently weakens the bandwidth limit of readout electronics. We further fitted these histograms of rising-edge time with Gaussian models [color area in Figs. 3(a)–3(e)] to, respectively, extract the mean ${\tau }_{\mathrm{mu}}$ and standard deviation ${\tau }_{\mathrm{std}}$ of rising-edge time at each photon number $n$. By fitting these mean ${\tau }_{\mathrm{mu}}$ versus photon number $n$ with power functions, they scale approximatively as theoretical $n^{-0.5}$, except for the $100\mathrm{nm}-1\mu \mathrm{H}$ detector, whose exponent is $-0.39$. These fitting results further support the above analysis, indicating that readout bandwidth weakens the photon-number readout capability. Increasing the width leads to a smaller hotspot resistance and hence a longer rising-edge time ${\tau }_{\mathrm{int}}$ of response pulses, which similarly scale approximately as $\sqrt{w}$. Additionally, a wider superconducting strip increases the bias current linearly and improves the SNR, and reduces the jitter of rising-edge time. Figure 3(f) presents the average ${\tau }_{\mathrm{mu}}$ and ${\tau }_{\mathrm{std}}$ (referred to as $\overline{{\tau }_{\mathrm{mu}}}$ and $\overline{{\tau }_{\mathrm{std}}}$) of different superconducting strip detectors. We also define the relative SNR as $2({\tau }_{\mathrm{mu},\mathrm{i}+1}-{\tau }_{\mathrm{mu},\mathrm{i}})/({\tau }_{\mathrm{std},\mathrm{i}+1}+{\tau }_{\mathrm{std},\mathrm{i}})$ to evaluate the quantitative impact on photon-number readout capability. When the strip width is fixed at 100 nm and inductance increases, although the $\overline{{\tau }_{\mathrm{mu}}}$ and $\overline{{\tau }_{\mathrm{std}}}$ both increase, the relative SNR still becomes larger, which implies a better readout fidelity [Fig. 3(g)]. Increasing the strip width increases the $\overline{{\tau }_{\mathrm{mu}}}$ but reduces $\overline{{\tau }_{\mathrm{std}}}$, as a result of which the relative SNR was significantly enhanced. In summary, larger inductances and wider strips can enhance the photon-number readout capability by stretching the rising edges to break the bandwidth limitation of readout electronics and by enhancing the SNR of readout pulses to reduce the rising-time jitter.

2.3.

Real-time Readout Setup

For many PNR applications, it is crucial to utilize the photon-number information as immediate feedback.^{4–6,10,11,42} However, the traditional approach of waveform discrimination using a digitizer necessitates considerable data acquisition on the order of $1\mathrm{GB}/\mathrm{s}$, coupled with extensive postprocessing, thereby impeding real-time access to photon-number information. Benefiting from the projection mechanism from photon number to rising-edge time, we can reduce the data stream to the order of $1\mathrm{MB}/\mathrm{s}$ and thus enable real-time readout by using a dual-channel constant-threshold timing tagger. As illustrated in Fig. 4(a), the response signal after the low-noise amplifier is divided into two equal pulses by a power splitter. Then the pair of pulses is measured by two time-to-digital converters (TDCs) at two different voltage thresholds. One TDC measures the time stamp $t_{\text{L}}$ at the low discrimination level $V_{\text{L}}$, while the other one obtains the high-level ($V_{\text{H}}$) time stamp $t_{\text{H}}$. Compared to time-correlated jitter measurement,⁶ our method is not affected by intrinsic timing jitter, although it does reduce the SNR of pulses. More importantly, the photon arrival time represented as $(t_{\text{L}}+t_{\text{H}})/2$ here does not participate in discriminating the photon number, which is necessary in PNR-enhanced communication¹⁰ and lidar.¹¹

Fig. 4

Real-time readout and binning error reduction. (a) Equivalent circuit diagram of the setup. The response pulse through the power splitter is divided into two equal pulses, which then enter two TDCs. One TDC measures the high-level ($V_{\text{H}}$) time stamp $t_{\text{H}}$, while the other TDC measures the low-level ($V_{\text{L}}$) time stamp $t_{\text{L}}$. (b) Histograms and Gaussian fitting of the rising-edge time of response pulses measured by an oscilloscope without a splitter or the 2-TDC setup (all data and $1\sigma$ data). (c) The ${\sigma }_{\tau }$ of rising-edge time for two readout setups. The results using the 2-TDC setup are slightly inferior to those obtained using an oscilloscope, which is due to the additional timing jitter of the TDCs. (d) The photon-number binning error for three readout methods including oscilloscope, 2-TDC (all data), and 2-TDC ($1\sigma$ data).

To validate the accuracy of the dual-channel TDCs method, we compared the histograms of time difference $t_{\text{H}}-t_{\text{L}}$ generated from this setup with that obtained directly from an oscilloscope without a splitter. As shown in Fig. 4(b), the 2-TDC method can distinguish up to ten photons, which was basically consistent with the results of the oscilloscope method. However, the readout fidelity $P_n^n$ of the 2-TDC method is always lower than that of the oscilloscope method [Fig. 4(c)]. The slight decrease in PNR capability results from the increased jitter of rising-edge time [Fig. 4(c)], which is attributed to two factors: the decreased SNR of electrical pulses and the additional timing jitter ${\sigma }_{\mathrm{TDC}}$ of the TDCs. We first evaluated the SNR of pulses before and after the power splitter. Before the splitter, the standard deviation ${\sigma }_{\mathrm{th}}$ of thermal noise, the standard deviation ${\sigma }_{\mathrm{amp}}$, and the mean $V_{\mathrm{amp}}$ of pulse amplitudes were 1.9, 3.75, and 354.6 mV, while after the splitter, these values were 1.9, 2.71, and 246.2 mV. The thermal noise remained constant, but amplitude noise (including thermal noise and amplitude fluctuation) decreased with amplitude; thus the pulse SNR $V_{\mathrm{amp}}/{\sigma }_{\mathrm{amp}}$ only decayed a little, from 94.56 to 90.85. This observation indicates that splitting the electronic pulses does not significantly compromise the SNR. In addition to the decreased SNR of readout pulses, the TDC’s timing jitter is the primary cause of the weakened PNR capability. The TDC in our experiment has an RMS timing jitter ${\sigma }_{\mathrm{TDC}}$ of 34 ps, which increased the measurement uncertainty of rising-edge time by $\sqrt{{\sigma }_{\tau }^2+{\sigma }_{\mathrm{TDC}}^2}$. Therefore, the 2-TDC setup is an effective readout method that minimally affects the SNR but only requires low-jitter TDCs.

If a partial relinquishment of counts is deemed acceptable, the accuracy of assigning photon numbers can be significantly improved by re-establishing the dividing regions near the centers of each decomposed Gaussian function. Only events falling within this region are included in the statistics [see Fig. 4(b)]. If the regions of certainty are confined within $\pm \frac{1}{2}{\sigma }_{\tau }$ of each peak, approximately 62% of data is discarded. However, this results in a decrease in the binning error rates from 24% to 6% or lower for the first eight photon numbers, as depicted in Figs. 4(b) and 4(d). In summary, this postselection can reduce the binning error rates by $<1/4$ but at the cost of a 62% loss in detection efficiency.

2.4.

Unbiased Quantum Random-Number Generation

Due to the significant improvement in the dynamic range and readout speed of the SMSPD-based PNRD, they can now be directly employed in QRNGs. Random numbers play a critical role in science and technology, with applications ranging from simulation to cryptography. QRNGs leverage the inherent randomness in quantum mechanics to generate perfect sources of entropy for random numbers.⁴³ Classical or quantum light serves as a convenient and affordable source of quantum randomness. QRNGs based on homodyne measurement of random vacuum fluctuations can easily achieve high bit rates up to Gbit/s,^44,45 but these methods also suffer from nonuniform randomness with bias. Photon-counting methods that harness the intrinsic randomness of photon-number statistics are inherently unbiased.^39,42,43 However, the main challenges lie in achieving high PNR capability and detector speed. In this work, we have successfully implemented a QRNG by sampling the parity of the Poisson distribution of a coherent state using an SMSPD. This approach is resilient against various experimental imperfections (such as photon loss, detector inefficiency, phase and amplitude fluctuations of the laser), environmental noise contamination, and potential eavesdropping.^9,39

To generate random numbers, we simply transform photon-number detections into binary outputs using the $\pm \frac{1}{2}{\sigma }_{\mathrm{n}}$ method, as depicted in Fig. 5(a). In this conversion, odd photon-number events are assigned an outcome of “1,” while even ones are assigned “0.” According to Gerry’s theory,³⁹ the expectation of parity is given as

Fig. 5

Generation and testing of quantum random numbers. (a) Operating principle of the QRNG. The graph on the right shows a sequence consisting of 50 rising-edge times, along with the random numbers at an effective mean photon number per pulse $\tilde{\mu }$ of 5.1. (b) Results of the NIST randomness tests on $1000\times {10}^6$ binary bit strings. The confidence interval, represented by the dashed blue lines, ranges from 0.98 to 1. It is calculated using a normal distribution as an approximation to the binomial distribution.

4.1.

Fabrication

Here, 7-nm-thick NbN film was deposited on a silicon substrate with a 268-nm-thick thermal oxide layer using reactive DC magnetron sputtering. The NbN film had a critical temperature of $T_{\mathrm{c}}=7.6\mathrm{K}$. Then, the NbN film was patterned into a meandered nanowire structure using 100 kV electron-beam lithography with a 70-nm-thick positive-tone resist (ZEP520A) and reactively etched in ${\mathrm{CF}}_4$ plasma at a pressure of 4 Pa and RF power of 50 W.

4.2.

Numerical Calculation

To analyze the effects of inductance and width on rising-edge time, we utilize the electrothermal feedback model. The lumped equivalent electrical model of the SNSPD consists of a hotspot-number-dependent resistor in series with a kinetic inductor. After the initiation of a detection event, the dynamics of the 1D electrothermal system are governed by the interaction between the SNSPD and the readout circuit, which can be mathematically described as⁵³

In Sec. 2 of the Supplementary Material, more accurate results are obtained through a finite-element simulation of the electrothermal process.^55–57