Large-inductance superconducting microstrip photon detector enabling 10 photon-number resolution

Abstract. Efficient and precise photon-number-resolving detectors are essential for optical quantum information science. Despite this, very few detectors have been able to distinguish photon numbers with both high fidelity and a large dynamic range, all while maintaining high speed and high timing precision. Superconducting nanostrip-based detectors excel at counting single photons efficiently and rapidly, but face challenges in balancing dynamic range and fidelity. Here, we have pioneered the demonstration of 10 true photon-number resolution using a superconducting microstrip detector, with readout fidelity reaching an impressive 98% and 90% for 4-photon and 6-photon events, respectively. Furthermore, our proposed dual-channel timing setup drastically reduces the amount of data acquisition by 3 orders of magnitude, allowing for real-time photon-number readout. We then demonstrate the utility of our scheme by implementing a quantum random-number generator based on sampling the parity of a coherent state, which guarantees inherent unbiasedness, robustness against experimental imperfections and environmental noise, as well as invulnerability to eavesdropping. Our solution boasts high fidelity, a large dynamic range, and real-time characterization for photon-number resolution and simplicity with respect to device structure, fabrication, and readout, which may provide a promising avenue towards optical quantum information science.


Introduction
Single photons acting as qubits have emerged as a prominent approach to quantum information technology [1][2][3] .The ability to accurately discriminate the photon number cannot be overstated in various quantum systems, including linear optical quantum computation 4,5 , preparation 6 and characterization 7,8 of quantum light sources, quantum random number generation 9 , as well as quantum-enhanced communication 10 and detection 11 in general.In these schemes, information on quantum states is primarily monitored with photon-number-resolving detectors (PNRDs), whose resolving fidelity and dynamic range greatly affect the operating accuracy.The PNR fidelity is defined as the probability of accurately recording the number of incident photons 12 .It is simultaneously determined by detection efficiency, readout fidelity (the separating degree of neighboring photon numbers in readout features), and splitting fidelity particularly in the case of spatial-multiplexing arrays (the probability of all photons simultaneously activating different pixels).
Extensive efforts have been dedicated to the advancement of PNRDs 13,14 , yet their dynamic range and fidelity remain constrained by either significant readout overlap [15][16][17] or low detection efficiency 18 .Superconducting transition edge sensors (TESs), as the most successful PNRDs so far, can achieve high-fidelity PNR over 30 photons 9 .However, their operation at ultralow temperatures (~100 mK) poses a challenge, and the trade-off between PNR capability and recovery time restricts their count rate 14 , which is also undesirable for quantum information applications.
Superconducting nanostrip single-photon detectors (SNSPDs) 19 are a cutting-edge singlephoton detection technology, renowned for near-unity detection efficiency [20][21][22] , negligible dark counts 23,24 , gigahertz-level counting rate 25,26 , and timing jitter at the picosecond scale 27 .These exceptional attributes have catapulted themselves to significant applications within the field of 4 / 31 quantum information technology 28 .However, unlike TESs, SNSPDs possess an inherently weak PNR ability 29 due to the highly nonlinear signal amplification through hotspot resistance following a single photon detection 30 .Most existing solutions primarily aimed at implementing spatialmultiplexing arrays to distribute the optical wavepacket amongst multiple pixels 26,31,32 .In these quasi-PNR detectors, to achieve high fidelity, the pixels must significantly outnumber the incident photons to minimize the likelihood of multiple photons striking the same pixel.Even with perfect efficiency and readout distinguishability, a 100-pixel detector can only achieve fidelity of less than 90% at over-5-photon detection 33 .On the other hand, the presence of a localized hotspot resistor controlled by electrothermal feedback suggests that an SNSPD with an adequately long superconducting nanostrip is equivalent to a cascade of thousands of elements and n-photon simultaneously activating different elements should generate n non-overlapping hotspots 34 .Thus the rising-edge slope of the readout electrical pulses can convey the number of photons 29 , adhering to the electrical rule where the rising time of generated voltage approximately follows the time

constant
, where represents the total kinetic inductance of the detector and the L k / hs () L k  hs corresponds to the total resistance of all n hotspots.Although some modified readouts have () been exploited to augment the PNR capability by using a wideband cryogenic amplifier 29 or by integrating an impedance-matching taper 35 , the PNR dynamic range still hovers around 3~4 and the PNR fidelity is somewhat constrained in terms of readout distinguishability.
Recently, superconducting microstrip single-photon detectors (SMSPDs) [36][37][38] have shown saturated sensitivity to near-infrared single photons and have demonstrated over 90% system detection efficiency using low-energy-gap WSi 38 or ion-irradiated NbN 37 .SNSPDs and SMSPDs operate in a similar manner with respect to photon-number resolution.Their primary differentiation 5 / 31 lies in the line width, granting SMSPDs several advantages, including improved signal-to-noise ratio of response pulses, polarization insensitivity, and an increased active area.Here, we have presented, for the first time, compelling evidence of a large-dynamic-range and high-fidelity photon number readout in an SMSPD.Even without the need for cryogenic amplifiers, a large-inductance SMSPD can resolve up to 10 photons, achieving a readout fidelity of 98% and 90% for 4-photon and 6-photon events respectively.Furthermore, we demonstrated that the PNR capability of a conventional SNSPD or SMSPD can be enhanced by stretching the rising edges of photon-triggered electrical pulses, which encode the photon number information.By increasing the total inductance or strip width, the stretched rising edges become more distinguishable in terms of reading out the number of photon-induced hotspots and thus more sensitive to photon numbers.For reading out the photon number in real time, we implemented a dual-channel timing setup to measure the risingedge time.The combination of high fidelity, large dynamic range, and real-time measurement opens up possibilities for a variety of cutting-edge photon-number-resolving applications.For example, we illustrate the practicality of our system for quantum cryptography applications by creating a quantum random-number generator (QRNG) based on sampling the photon statistics of a coherent state.This method is fundamentally unbiased, robust to experimental imperfections and environmental noise, and invulnerable to eavesdropping 9,39 .This article is structured into four parts: (1) first introducing the SMSPD's architecture and photon-number resolution, (2) then illustrating the influence mechanism of the photon number readout capability, (3) next proposing the real-time readout setup, (4) and finally applying the PNR system to a QRND.6 / 31

Approaching true PNR with an SMSPD
The SMSPD comprises a single meandering microstrip.As shown in Fig. 1a, 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 μm diameter and contains a series of parallel 1-μm-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° turn before the 180° turnaround.The 90° turn transitions the strip width from 1.05 μm to 1.5 μm, while the 180° turnaround has a trip width of 1.5 μm and an interval of 2 μm.The connecting wires in part B for linking sections A and C are 1.05 μm wide and space 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 1b illustrates the simulated current density distribution in a 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-μm-wide meandering microstrips, resulting in a total inductance of 6 μH.The detector was fabricated using a 7-nm-thick NbN film on a silica substrate and exhibited a switching current of 122 μA at the operation temperature of 2.2 K. Its basic performance is discussed in Section 1 of Supplementary Information.We investigated the multi-photon response of the SMSPD by using an attenuated 1064 nm pulsed laser.In contrast to 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. 1c, the rising-edge section of typical output waveforms exhibited distinct slope separations.To determine the rising-edge time of ∆ rise electrical pulses, we employed two fixed discrimination levels: a low level set to the maximum  L value of the system's electrical noise, and the high level set to the minimum amplitude of the  H electrical pulses, as shown in Fig. 1c.Owning to the large inductance and relatively small resistance, 8 / 31 the output pulses have a long time in rising edges.The corresponding rising-edge time of 4.845 ns, 3.447 ns, and 2.667 ns indicate 1-, 2-, and 3-photon events, respectively.Figure 2a exhibits the histograms of the rising-edge time (black dots) at two effective mean photon numbers per pulse  of 2.5 and 5.1.Here, 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 ≥10, as significant overlap renders these photon events indistinguishable.In order to assign a photon number m for a detection event with pulse rising-edge time , ∆ rise a series of dividing thresholds need to be determined.This allows us to assign the event to a photon number m, if .To begin, we fitted the histograms of rising-edge  m < ∆ rise <  m + 1 time with Gaussian models (color area in Fig. 2a).Subsequently, we normalized each distribution associated with the photon number n.The dividing thresholds were determined at the position  n () 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 photonnumber 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 ,    which is described as 41 Figure 2b illustrates the confusion matrix of photon number assignment, providing insights into the readout fidelity of n photons, represented by .In the ideal case, the confusion matrix would be  n n 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 =0.99999999993 for photon  1 1 number n=1 and a parts-per-million precision of =0.999998 for n=2.For higher photon numbers,  2 10 / 31 the readout fidelity gradually decreases.It remains above 0.98 for ≤ 4 photons but significantly decays to below 0.90 for ≥ 7 photons.
After establishing the dividing thresholds , the photon events in different  k, 1 ≤ k ≤ 11 threshold-to-threshold regions can be counted to reconstruct the photon statistics .When the () photon number ranges from 1 to 10, each hotspot has a length between 10 μm and 3 μm estimated from electrothermal simulation.Since it is far smaller than the total length (~2000 μm) of the microstrip, the probability of hotspot overlap is negligible.Therefore, as shown in Fig. 2c

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 5 detectors with different widths and inductance: 100 nm-1 μH, 100 nm-3 μH, 100 nm-5 μH, 300 nm-5 μH, 500 nm-5 μH.The SNSPDs with nanostrip widths of 100 nm, 300 nm, and 500 nm covered different circular active areas of 20 μm, 30 μm, and 50 μm 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 11 / 31 nanostrips.The nanostrips acting as inductance were located approximately 50 μm away from the photon-sensitivity area to avoid photon absorption.Figure 3a-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 (Fig. 3a-c), the detector is only capable of distinguishing 2 photons.Traditional wisdom argues that a larger inductance imposes two disadvantages on SNSPDs including degrading the signal-to-noise ratio 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 Fig. 3a-c, with increasing the inductance, the overlaps between 1-and 2-photon events get smaller.Upon maintaining the inductance at 5 μH and increasing the strip width , the PNR  capability improves significantly.Therefore, the 100-nm-wide detector can only achieve over 90% readout fidelity for assigning between single-photon and multi-photon events, while the 500-nmwide and 1-μm-wide detectors can respectively resolve 4 and 6 photons with 90% readout fidelity.
Note that in the subgraph of Fig. 3a, 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 Supplementary Information).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.
We conclude that the phenomenon described above was caused by the limited bandwidth of the readout electronics and the signal-to-noise ratio (SNR) of response pulses.A universal amplifier for SNSPDs (such as LNA1800 here) typically has a 3 dB bandwidth of ~ 1 GHz, which restricts the rise time (10%-90% amplitude) to (~300 ps).Consequently, when the intrinsic response  lim pulses have a fast rising-edge time approaching or smaller than , their rising edges will be  int  lim 13 / 31 broadened after passing through the amplifier, following .This addition of rising time  2 int +  2 lim prevents us from accurately acquiring the real rising time and also brings the pulse rising time of multi-photon events closer.
According to Eq. ( 4) and ( 5), increasing the total inductance of SNSPDs directly increases  k the intrinsic rising-edge time on the scale of , which subsequently weakens the bandwidth  int  k limit of readout electronics.We further fitted these histograms of rising-edge time with Gaussian models (color area in Fig. 3a-e) to respectively extract the mean and standard deviation  mu  std of rising-edge time at each photon number n.By fitting these mean versus photon number  mu  with Power functions, they scale approximatively as theoretical except for the 100 nm-1 μH  -0.5 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 of response pulses, which  int similarly scale approximately as .Additionally, a wider superconducting strip increases the bias  current linearly and improves the SNR, and reduces the jitter of rising-edge time.Figure 3f  detectors.We also define the relative SNR as to evaluate the 2( mu,i + 1 - mu,i ) ( std,i + 1 +  std,i ) quantitative impact on photon number readout capability.When the strip width is fixed at 100 nm and inductance increases, despite the and both increase, the relative SNR still becomes  mu  std larger, which implies a better readout fidelity (Fig. 3g).Increasing the strip width increases the  mu but reduces , as a result of which the relative SNR was significantly enhanced.In summary,  std larger inductances and wider strips can enhance the photon number readout capability by stretching 14 / 31 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.

Real-time readout setup
For many PNR applications, it is crucial to utilize the photon number information as immediate feedback [4][5][6]10,11,42 . However,the traditional approach of waveform discrimination using a digitizer necessitates considerable data acquisition on the order of 1 GB/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 MB/s and thus enable real-time readout by using a dual-channel constant-threshold timing tagger.As illustrated in Fig. 4a, the response signal after the low-noise amplifier is divided into two equal pulses by a power splitter.Then the pair of pulses are measured by two time-todigital converters (TDCs) at two different voltage thresholds.One TDC measures the timestamp  L at the low discrimination level while the other one obtains the high-level ( ) timestamp .
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 here does not participate in discriminating photon number, which is necessary in PNR-enhanced communication 10 and LiDAR 11 .and 1σ data).c, The of rising-edge time for two readout setups.The results using the 2-TDC  τ 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 (1σ data).
To validate the accuracy of the dual-channel TDCs method, we compared the histograms of time difference generated from this setup with that obtained directly from an oscilloscope  H - L without a splitter.As shown in Fig. 4b, the 2-TDC method can distinguish up to 8 photons, which was basically consistent with the results of the oscilloscope method (9 photons).However, the readout fidelity of the 2-TDC method is always lower than that of the oscilloscope method (Fig. ).The slight decrease in PNR capability results from the increased the jitter of rising-edge time (Fig. 4c), which is attributed to two factors: the decreased SNR of electrical pulses and the additional timing jitter of the TDCs.We first evaluated the SNR of pulses before and after the power  TDC splitter.Before the splitter the standard deviation of thermal noise, the standard deviation  th  amp and mean of pulse amplitudes were 1.9 mV, 3.75 mV, and 354.6 mV, while after the splitter  amp these values were 1.9 mV, 2.71 mV, 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 only decayed a little from 94.56 to 90.85.This observation indicates that  amp  amp 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 of 34 ps, which increased  TDC the measurement uncertainty of rising-edge time by .Therefore, the 2-TDC setup is an  2 τ +  2 TDC 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. 4b).If the regions of certainty are confined within of each peak, ± 1 2  τ 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 8 photon numbers, as depicted in Fig. 4b and d.In summary, this post selection can reduce the binning error rates by less than 1/4 but at the cost of a 62% loss in detection efficiency.17 / 31

Unbiased quantum random-number generation
Due to the significant improvement in the dynamic range and readout speed of the SMSPD, 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.QRNDs based on homodyne measurement of random vacuum fluctuations can easily achieve high bit rates up to Gbps 44,45 , but these methods also suffer from uniform 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 method, as depicted in Fig. 5a.In this conversion, odd photon-number ± 1 2  n 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 by where represents the mean photon number of the coherent state.Consequently, the inherent bias  diminishes exponentially as the intensity of the coherent state increases.For instance, when ,  = 5.1 the theoretical expectation is .We subjected the 10 9 random bits generated by  -10.2 ≈ 3.7 × 10 -5 our protocol to a battery of tests from the NIST suite of randomness tests 46 .The extensive experimental data were divided into 1000 separate smaller streams of 10 6 bits each.With a significance level of α = 0.01, all proportions passing a particular test lie within the confidence interval of 0.98 to 1, as shown in Fig. 5b.These results validate the randomness of our measurements across all considered tests.However, due to finite sampling and binning errors, the measured parity in our experiment only reached , which leads to the "Frequency" test reaching the upper 9.7 × 10 -4 bound of 1.In the future, by leveraging a low-noise cryogenic readout to enhance the SNR of readout pulses, we can increase photon-number resolution and decrease the binning error rate, thus reducing the residual bias.In our experiment, the pulsed laser was operated at a repetition rate of 500 kHz to ensure full recovery of the SMSPD.Although slower than a conventional SNSPD, this rate is still 1-2 orders of magnitude faster than a TES 14 .In the future, by connecting a resistor in series, the counting rate can be increased to tens of MHz 47,48 .

Conclusion
In summary, we have first demonstrated that a large-inductance SMSPD is capable of resolving photon number up to 10, concurrently achieving high readout fidelity of over 98% and 90% for photon number up to 4 and 6 respectively.This is primarily attributed to the positive influence of large kinetic inductance and wide microstrips on the rising-edge time of readout pulses, enabling enhanced photon-number resolution.By incorporating the proposed 2-TDC method, we have further 20 / 31 achieved real-time photon-number readout, which outperforms traditional data acquisition methods by three orders of magnitude in terms of efficiency.Compared to other SNSPD-based PNR techniques, our solution stands out for its high fidelity, large dynamic range, and real-time characterization.Additionally, it boasts a simplified device structure, ease of fabrication, and streamlined readout processes.Moreover, we have demonstrated the utility of our PNR system in creating an unbiased and robust QRNG by sampling the photon-number statistics of a coherent state.
Beyond QRNG, as the detection efficiency of SMSPDs continues to increase, we envision its broad applicability in a variety of cutting-edge technologies, such as photonic quantum computing 49 , Boson sampling 50 , and quantum metrology 51 .

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  c = 7.6 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 CF 4 plasma at a pressure of 4 Pa and RF power of 50 W.

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 21 / 31 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 as 52  In Section 2 of Supplementary Information, more accurate results are obtained through a finiteelement simulation of the electrothermal process.

Fig. 4 |
Fig. 4 | Real-time readout and binning error reduction.a, Equivalent circuit diagram of the

Fig. 5 |
Fig. 5 | Generation and testing of quantum random numbers.a, Operating principle of the

( 2 )
and (3) easier to solve.The multiplication of it with the resistance of unit length  sq / quantifies the growing rate of resistance52 .By solving above coupled equations (2) and (3), we can approximately estimate the maximum value of total hotspot resistance and the corresponding risingedge time,  hs () =  hs,max ∝  k  sq  NS   ,#(4)  rise () ≈  k  hs () ∝  k   sq  NS  .#(5) 53  d = ( b - d ) L .#(3) and current through the SNSPD respectively, and represents the load impedance of the readout  L circuit.The normal-superconducting boundary propagates at a velocity of .Although this  NS expansion rateis dependent on time53, here we treat it to a constant value to make equations  NS