Single-photon detectors are indispensable tools in quantum-photonic systems as well as in classical applications. Quantum-key distribution [1], linear optical quantum computation [2], photonic boson samplings [3–6], quantum metrology [7], Moon-to-Earth optical communications [8], and lidar [9] all need high-performance single-photon detectors. In particular, these detectors are often configured to measure coincidence of photons, for example, in Hong–Ou–Mandel [10,11] and Hanbury–Brown–Twiss interferometers [12,13]. Measuring N-fold coincidence typically requires N sets of single-photon detectors as well as the associated bias and readout electronics [11,13]; these resources could become prohibitive if N goes large and if the detectors need to work at cryogenic temperatures. While individual superconducting nanowire single-photon detectors (SNSPDs) [14] have achieved unprecedented matrices of performance [15–20], it remains a challenge to scale single elements to large arrays to measure N-fold coincidences, due, mainly, to the limitation of the cooling-power budget on the readout electronics accessing the detectors. Several multiplexing schemes, in time [10], frequency [21,22], amplitude [23–25], space [26,27], and row-column [28], have been proposed and demonstrated to simplify the otherwise formidably complicated readout circuits. However, it is still elusive whether it is possible to scale N to a large number without resorting to a large number of electronic components.

In this paper, we propose a concept and a device architecture to achieve an n-fold coincidence photon counter, in which n can be set from 1 to N by controlling the bias condition. The device is composed of N photosensitive superconducting nanowires and one wider nanowire functioning as a current reservoir [29]. With its bias and readout circuitry as simple as one individual SNSPD, from the external point of view, the detector outputs a voltage pulse if and only if n photons are detected. We therefore name the detectors of this type superconducting nanowire multi-photon detectors (SNMPDs).

In Fig. 1, we contrast the commonly used technology for N-fold coincidence photon counting with the proposed one using an SNMPD. As shown in Fig. 1(a), conventionally, in order to count N-fold coincidence, N individual single-photon detectors as well as the associated electronics are needed. After the N detectors detect the N photons, the multi-channel, time-correlated pulse counter counts the coincidences. In contrast, in this work, we propose to use a single SNMPD to count N photons. For one N-fold coincidence, the SNMPD directly outputs one voltage pulse. Thereby, we anticipate that the SNMPD could dramatically simplify the circuitry of the N-fold coincidence counting.

Figure 2 presents the device architecture of an SNMPD and its operating principle as an N-photon detector. As shown in Fig. 2(a), the N photosensitive nanowires (N1,N2,…,NN) and the current reservoir (R) are all connected in parallel and then are connected with a choke inductor (with inductance of Lc) in series. We assume the N photosensitive superconducting nanowires to be identical, although in principle they can be designed to be different. The set of parameters to characterize each of them includes kinetic inductance Lk, switching current Isw, and initial bias current Ib(0). Accordingly, the kinetic inductance, switching current, and initial bias current of the current reservoir can be noted as Lk/α, βIsw, and αIb(0), respectively. The initial allocation of the total bias current Ib to each branch is inversely proportional to the kinetic inductance of this branch, which explains the coefficient α in αIb(0).

Figures 2(b)–2(d) present the operating principle. Figure 2(b) shows the initial bias condition before any photon detection event occurs. As shown in Fig. 2(c), once the first photon is detected by the first photosensitive nanowire N1, which then temporarily switches to a high-impedance state, the current gets redistributed—the bias levels of the remaining N−1 photosensitive nanowires and the current reservoir all increase but do not exceed their corresponding switching currents. Similar procedures are repeated for the second, the third, …, and the (N−1)^th photons. When they are detected by the second, the third, …, and the (N−1)^th photosensitive nanowires, the bias level in the current reservoir increases in a staircase pattern. As shown in Fig. 2(d), when the N^th photon is detected by the N^th photosensitive nanowire NN, the redistribution of the current into the reservoir causes its switching to the high-impedance state, as the bias current exceeds its switching current βIsw. Most of the bias current in the reservoir then backflows to the N photosensitive channels, causing all of them to switch to the high-impedance states. Therefore, the N^th photon functions as the “last straw” to switch the entire device. The bias current then flows through the load impedance Z0, outputting a voltage pulse. The current subsequently restores in the photosensitive nanowires and the current reservoir after their superconductivity recovers. Note that this operating mode has no requirement on the sequence that the N photons fire the N photosensitive nanowires or on the time interval between two subsequent firings.

To operate the device in the N-photon regime outlined above, we need to bias it in the correct bias conditions. These conditions can be summarized as the following. (i) After N−1 photosensitive nanowires have been fired by incident photons, the N^th photosensitive nanowire has not been switched to the normal state by current redistribution. (ii) After N−1 photosensitive nanowires have been fired by incident photons, the current reservoir has not been switched to the normal state by current redistribution. (iii) After N photosensitive nanowires have all been fired by incident photons, the current reservoir will be switched to the normal state by current redistribution. These three conditions set constraints on the initial bias current Ib(0); specifically, conditions (i) and (ii) set the upper bounds, whereas (iii) sets the lower one. Appendix A details the derivations of these constraints. Two assumptions are important: one assumes that the inductance of the choke inductor is so large that each redistribution of the current does not cause leaky current to Z0; another assumes that the j^th photon-detection event on the j^th photosensitive nanowire, total amount of current ηIj(j−1), flows out from the photosensitive nanowire, where η is a constant and Ij(j−1) is the bias current in the j^th nanowire prior to the j^th detection event. The first assumption can be realized experimentally to ensure the leaky current is negligibly small; the second assumption of η to be a constant is an approximation, which has been justified by comparing the values of η as a function of Ib(0)/Isw obtained by electrothermal simulations [30]. The justification is detailed in Appendix A. If we further design the device such that α=β=N, the bias condition for the detector to work in the N-photon regime is (2N−12N+η−1)N<Ib(0)Isw<(2N−12N+η−1)N−1.(1)Specifically, electrothermal simulation shows that η is approximately 0.8. The bias condition Ib(0)/Isw for each N is calculated analytically and is presented in Fig. 2(e). The dashed lines present the bias regimes for the device to work as an N-photon detector.

We present a simplest example, where α=β=N=2, to quantitatively illustrate the electrothermal dynamics of the SNMPD. The bias condition is Ib(0)/Isw=0.75, fulfilling the requirement set by Eq. (1). The values of the parameters used in the electrothermal simulation are presented in Appendix B. Assume that at t=0, the first photon is absorbed by the first photosensitive nanowire at the location x=0, where x is the direction along the nanowire. Figure 3(a) shows the electrothermal evolution of the device after the first photon is absorbed. The photon generates a resistive region around x=0, but as the bias current Ib(0) is relatively low, the resistive region soon shrinks and disappears. The net effect is that the first photosensitive nanowire stabilizes in the superconducting state with an even decreased bias current I1(1), and the bias levels in the second photosensitive nanowire and the current reservoir become elevated to I2(1) and IR(1), respectively, due to the redistribution of the current. However, as the bias currents are still less than their switching currents, both the second photosensitive nanowire and the current reservoir stay superconductive. Assume that the second incident photon is absorbed by the second photosensitive nanowire at t=10ns and the location is noted as x=0. Figure 3(b) presents the subsequent electrothermal evolution of the device. The second photon triggers a complete electrothermal evolution in the second photosensitive nanowire, in which the bias level has already been elevated by the previous current redistribution; furthermore, the redistribution of the current this time fires the current reservoir. The current in the reservoir backflows to the first and the second photosensitive nanowires, firing the first photosensitive nanowire even though I1(1) is low. At this moment, the two photosensitive nanowires and the current reservoir all switch to high-impedance states, and the current flows through the load impedance Z0, generating an output voltage pulse. Afterwards, the absorbed photon energy dissipates, the “hotspots” shrink and disappear, and the system recovers to its initial state. Therefore, it is the synergic effect of the first and the second photons that fires the entire detector—the first photon prepares for the avalanche by increasing the bias levels in the second photosensitive nanowire and the current reservoir, and the second photon initiates the avalanche. The current dynamics I1(t), I2(t), IR(t), and Iout(t) are presented in Fig. 3(c). We also present a video (Visualization 1) to visualize this complete electrothermal evolution.

We now investigate how the device functions if the bias conditions given by Eq. (1) and shown in Fig. 2(e) are not satisfied. Apparently, if Ib(0)Isw<(2N−12N+η−1)N, even N photons cannot fire the entire device to signal an output. On the other hand, if Ib(0)Isw>(2N−12N+η−1)N−1, we hypothesize that fewer photons are sufficient to fire the entire device so that it works as an n-photon detector, where n<N. Analysis on these more general cases shows the bias conditions for an n-photon detector can be uniformly expressed as (2N−12N+η−1)n<Ib(0)Isw<(2N−12N+η−1)n−1,(2)where n=1,2,…,N. Figure 4 presents the bias conditions Ib(0)/Isw for different N and n in two different ways: Fig. 4(a)shows the bias regions with the maximum photon numbers for each N (n=N), with the second largest photon numbers for each N (n=N−1), and so on. Figure 4(b) shows the bias regions for a given photon number n, for example, n=1,2,…. We note that the data sets presented in Figs. 4(a) and 4(b) (the dots in these two subfigures) are identical; the difference is the lines for guiding the eyes and guiding the understanding of these data.

The role of the current reservoir is pivotal in our SNMPD, and it makes the device distinguished from SNSPDs and superconducting nanowire avalanche photodetectors (SNAPs). An SNSPD itself can work as a multi-photon detector [31,32], but in a low bias regime, and therefore the detection efficiency is low. So is that of an SNAP [33]. In contrast, the current reservoir allows us to engineer the bias condition, in particular, to boost the multi-photon regime to the high-bias region. Figure 5 presents the comparison of the bias conditions for multi-photon detection for an N-SNAP and SNMPDs with two configurations. Compared with the N-SNAP, the bias region for multi-photon detection is elevated in the SNMPD with α=N and can further be elevated if we flexibly engineer the current reservoir; for example, let α=2N. See Eq. (A13) in Appendix A. Therefore, multi-photon detection with relatively high detection efficiency is the major advantage of SNMPDs integrated with current reservoirs.

To show the scalability of N, we present our design of a superconducting nanowire eight-photon detector. As the electrothermal simulation becomes cumbersome when N goes large, we used the SPICE-based model [34] as a simpler but equally effective tool to model the current dynamics. Each of the eight photosensitive superconducting nanowires is identical to the counterpart in the two-photon case, whereas the width of the current reservoir is increased to 400 nm and its kinetic inductance is decreased to 10 nH. The values of the parameters used in the SPICE simulation are presented in Appendix B. Figures 6(a)–6(c) show the current dynamics in the photosensitive nanowires, the current reservoir, and the output voltage pulse, respectively. Similar to what we have seen in the two-photon detector, the device outputs a voltage pulse if and only if the eight photons are respectively detected by the eight photosensitive nanowires. The limitation on N is as presented in Fig. 2(e): when N goes large, the appropriate bias regions shrink.

By elevating the initial bias current, the eight-photon detector (N=8) can work in the n-photon regime, where n<8. For example, if we choose n=6, the bias condition is Ib(0)/Isw=0.756, satisfying the requirement set by Eq. (2). Figure 7 presents the current dynamics of the eight-photon detector working in the six-photon regime.

Another advantage of SNMPDs is their simplicity. Their one-port bias and readout circuitry is as simple as the circuitry of one individual SNSPD. Furthermore, when an SNMPD works as an N-fold coincidence counter, we can directly count the coincidences by counting the output voltage pulse, eliminating the need of external multi-channel pulse counters.

In addition to the scalability and the simplicity, it is also possible to configure an SNMPD to a photon-number-resolving detector. Very recently, researchers discovered that an SNSPD actually has intrinsic photon-number-resolving capability, but it is limited to the few-photon regime [35–37]. An interesting and distinct feature of SNMPDs is that prior to the “final fire” by the N^th photon, the information of the arrivals of the N−1 photons is essentially stored in the format of supercurrent in the current reservoir. For example, as shown in Fig. 6(b), the quantized step of the current increase per each photon is on the order of 3 μA, which can be readout by, for example, on-chip integrating yTron devices [38]. The successful readout of these steps of the current changes would also enable photon-number-resolving capability.

An alternative operating mode to resolving the photon number requires repetitive measurement. Assume that the incident light is in the m-photon state and the m photons are separated in m spatial modes. To determine m, we can repeatedly measure the light by configuring an SNMPD to be an N-photon detector, (N−1)-photon detector, …, and m-photon detector that would only then output voltage pulses.

We evaluate the performance characteristics of the SNMPDs. The detection efficiency is determined by the product of the detection efficiencies of those N photosensitive nanowires. The operating speed is set by Ltotal/R, where Ltotal is the total kinetic inductance, dominated by the inductance of the choke inductor Lc, and R is the 50-Ω load impedance. We expect that the accidentals, due to the false counts of the photosensitive nanowires, are negligibly low. The SNMPDs do not provide relative timing information among the N photons unless the current reservoir is configured as the current-sensing mode introduced above so that the arrival time of each photon can be extracted.

If the detection scheme was extended to the photon-number-resolving scheme, the probability that an SNMPD with N sections of photosensitive nanowires correctly measures the photon number in a weak optical pulse actually containing n photons could be calculated by [39] PDEN(n|n)=N!Nn(N−n)!DEn, where DE is the detection efficiency of each photosensitive superconducting nanowire. For simplicity, the detection efficiency of the N sections of photosensitive nanowires is assumed to be identical. The probability of error is therefore 1- PDEN(n|n). These calculations also assume, as in Ref. [39], that the weak optical pulse is originally in one spatial mode and is then split to couple with these N photosensitive nanowires.

The SNMPDs do not have additional constraints on materials, optical structures, or fabrication processes, compared with SNSPDs. They allow both polycrystalline [14,40] and amorphous [15] materials commonly used for SNSPDs and allow both meander [40], spiral [41], and fractal [42,43] geometries for free-space coupling and hairpin structures for waveguide integration [44]. Integrating with waveguides naturally separates the photosensitive nanowires in space, and the waveguide integration may be convenient for coupling these N photons that are already separated in different spatial modes, similar to what the authors of Refs. [27,45] presented. A schematic drawing of a waveguide-integrated SNMPD is presented in Appendix C. There is essentially no change in the fabrication processes used for making SNSPDs, except for that the device layout may become larger.

In conclusion, we have introduced the concept of SNMPDs. They are composed of N photosensitive superconducting nanowires and one current reservoir, and they can be configured to n-fold coincidence counters, where n≤N, by controlling the bias conditions. We have detailed the operating principle and have discussed their advantages, including simplicity in readout electronic circuitry and scalability. It is also possible to configure SNMPDs to photon-number-resolving detectors. The fabrication process for realizing SNMPDs is almost identical to that for SNSPDs. We anticipate realizing them experimentally as the next step and finding applications in measurement of high-order temporal correlations of light [13] and characterization of multi-photon entanglement [46].