Detection jitter quantifies variance introduced by the detector in the determination of photon arrival time. It is a crucial performance parameter for systems using superconducting nanowire single photon detectors (SNSPDs). In this work, we have demonstrated that the detection timing jitter is limited in part by the spatial variation of the photon detection events along the length of the wire. We define this jitter source as a geometric jitter since it is related to the length and area of the SNSPD. To characterize the geometric jitter, we have constructed a differential cryogenic readout with less than 7 ps of an electronic jitter that can amplify the pulses generated from the two ends of an SNSPD. By differencing the measured arrival times of the two electrical pulses, we were able to partially cancel out the difference of the propagation times and thus reduce the uncertainty of the photon arrival time. We proved that the variation of the differential propagation time was a few ps for a 3 *μ*m × 3 *μ*m device, while it increased up to 50 ps for a 20 *μ*m × 20 *μ*m device. In a 20 *μ*m × 20 *μ*m large SNSPD, we achieved a 20% reduction in the overall detection timing jitter for detecting the telecom-wavelength photons by using the differential cryogenic readout.

Single-photon detectors are suitable for the applications that require single-photon signal sensitivity from visible to infrared wavelengths. The quantum key distribution,^{1} the time-of-flight laser ranging (LIDAR),^{2} the fluorescent imaging,^{3} and the space based optical communication^{4} all require single photon detectors. The superconducting nanowire single photon detectors (SNSPDs) excel relative to the other commercially available single-photon detectors,^{5} with efficiencies up to 96% at 1550 nm, response spectrum from 0.5 to 5 *μ*m, low dark count rates below 100 counts per second (cps), and a count rate above 100 × 10^{6} cps.^{6–9} These applications require precise time resolution, and thus low timing jitter. Prior work has demonstrated that the jitter of the SNSPD can be as low as 18 ps.^{11} However, a wide range of jitter values—from 18 ps up to hundreds of ps—have been reported for the SNSPDs of different sizes and different materials.^{7–9,11} This inconsistency is an indication that the mechanisms involved in the setting device jitter remain poorly understood.

The source of the measured timing jitter can be divided into two main parts: (i) an electronic readout component and (ii) an intrinsic component. The electronic components such as amplifiers, passive components, ground loops, and measurement equipment generate noise.^{10,11} The existence of this limit is inferred from the observation that a reduction of the signal-to-noise ratio (SNR) and slew rate of the SNSPD signal increases jitter: with an RMS noise of 4 mV, a reduction in the slew rate from 0.57 mV/ps to 0.15 mV/ps increased the electronic jitter from 6 ps to 24 ps.^{10} The origin of the intrinsic jitter remains unclear: O'Connor^{15} showed that a possible mechanism for an intrinsic jitter is variation in the hotspot lengths. In their work, the nanowire defects or constrictions distributed over the detector area were shown to produce different signal slopes and amplitudes, thus generating different SNSPD signals that increased the jitter by up to 50 ps in a 20 *μ*m × 20 *μ*m device. However, the limits of an intrinsic jitter in devices without intrinsic inhomogeneity remains a puzzle.

Our recent discovery that a superconductive nanowire acts like a transmission line^{12,13} has suggested a geometry-based approach to the understanding of the intrinsic jitter. In particular, due to a large kinetic inductivity of these wires, the signal velocity of such a transmission line is slower than the speed of light, while its characteristic impedance is much higher than 50 Ω. ^{14} For a typical SNSPD, the effective signal wavelength is thus significantly reduced, approaching the scale of the wire's physical length. Therefore, instead of describing the electrical pulse generation from the perspective of an SNSPD behaving like a lumped inductor, the device should be treated as a distributed circuit element, i.e., a transmission line. In the distributed-element scenario, after a photon is absorbed in the nanowire, two electrical pulses are generated, which then propagate towards either end of the nanowire, arriving at times *t*_{1} and *t*_{2}. Assuming the transmission line has a constant velocity *v*, *t*_{1} and t_{2} can be written as linear functions of the photon landing location *x*_{p} and arrival time *t*_{p}, which are $t1=tp+xpv$ and $t2=tp+D\u2212xpv$, where *D* is the total length of the nanowire. In a typical SNSPD with a meandered shape, photons are usually uniformly illuminated on the detector. Consequently, the spatial variation of the absorbed photon will contribute to the variation of *t*_{1} and *t*_{2}, introducing a geometric jitter *j*_{G} that is closely related to the size of an SNSPD. Although this fully distributed description oversimplifies the microwave behaviour of the SNSPD, it is qualitatively useful in understanding our results.

In this work, we have identified, studied, and provided experimental evidence for *j*_{G} in an SNSPD by implementing a differential cryogenic readout to determine the absolute and the relative arrival times of output pulses at the two ends of the nanowire. As shown in Fig. 1(a), we pre-amplified these two pulses by means of the two HEMT-based cryogenic amplifiers, increasing the signal-to-noise ratio (SNR) to reduce the electrical jitter *j*_{E} below 7 ps. The amplifier had a high-input impedance and operated at 4.2 K while immersed in liquid helium (see Fig. 1(a)). The SNSPD was placed into a symmetric electrical network on a custom-printed circuit board (PCB), which was made from a 1.2-mm-thick FR4 laminate sheet. The SNSPD was biased with a DC current supplied by a voltage source with a serial resistor, while its output pulses were AC coupled to two HEMTs.^{18} In this way, positive and negative pulses arriving at each end of the SNSPD were separately amplified (Fig. 1(b)). The electrical noise was lowered by using multiple high-pass filters that are designed to have a low cut-off frequency around 200 MHz. The filter cut-off was defined by the input, output capacitors, and by the RC network placed at the source terminals of the HEMTs. Furthermore, the input stage of the amplifier comprised of a resistor and an inductor; the SNSPD was shunted to the ground by a 50 Ω resistor *R*_{shunt} (that provided a low-impedance path, to avoid latching events) in series with a 10 *μ*H conical broadband inductor *L*_{shunt} (self-resonant frequency up to 40 GHz) that was used to boost the high-frequency component of the SNSPD signal. The conical inductor provided a high-impedance block for the rising edge of the SNSPD signal: in the frequency range between 200 MHz and 1.5 GHz (consistent with the bandwidth of the SNSPD signals), the input impedance of the each HEMT was higher than 600 Ω. The output pulses were read out at the room temperature, amplified by a single stage 20 MHz–4-GHz-bandwidth amplifier, and analysed with a 6 GHz oscilloscope. The gain of the differential stage was evaluated comparing its output to a classic one without amplifier,^{7–12}obtaining a gain of 18 dB (see Fig. 1(b)). Fig. 1(b) shows the output signals (Ch_{1} and Ch_{2}) and the corresponding arrival times of the differential readout for a signal event from a 3 *μ*m × 3 *μ*m device of a nanowire meandered in a square geometry, biased at 24 *μ*A. A clear delay was observed between Ch_{1} and Ch_{2}. This delay varied due to the variation in an event position along the length of the wire. We measured an output signal with an 800 mV amplitude, a rise time of 500 ps, and a slew rate of ∼1.5 mV/ps. The RMS voltage output noise was 4.5 mV, which guaranteed a maximum electronic RMS jitter of 3 ps (FWHM jitter of 7.3 ps) per channel.^{10}

We measured the pulse arrival times for both pulses to extract *t*_{1} and *t*_{2} after removing fixed delays from the connections and the amplifiers. As shown in Fig. 1(c) (histograms of *t*_{1} and *t*_{2}) for the measurements of a 10 *μ*m × 10 *μ*m SNSPD, if we only use *t*_{1} or *t*_{2} to determine *t*_{p}, which is similar to readout of a conventional SNSPD, the overall detection timing jitter *j*_{t1} = *j*_{t2} = 35 ps is defined as the FWHM of the distribution of *t*_{1} or *t*_{2}. This value includes not only the intrinsic jitter but also our hypothesized geometric jitters *j*_{G1} or *j*_{G2} due to the distribution of photon landing locations along the wire length. For each photon-detection event, the difference of arrival times $\Delta t=t2\u2212t1=(D\u22122xp)v$ is independent of the photon arrival time *t*_{p}, and only depends on the photon landing location *x*_{p}. Therefore, the variation of Δ*t*, which we will call the differential jitter *j*_{Δ}, is used as a metric to evaluate the contribution of the geometric jitter to the overall jitter. Referring to the previous expression, *j*_{G} can be evaluated as $j\Delta 2$. The sum of *t*_{1} and *t*_{2,} $\Sigma t=t2+t1=2tp+Dv$ is a function of *t*_{p} and independent of the photon landing locations *x*_{p}, from which *t*_{p} can be determined by calculating $tp=(\Sigma t\u2212Dv\u2009)2$, where $Dv$ is assumed to be constant. As shown in Fig. 1(c) (histogram of ($\Sigma t2$)), the variation of $\Sigma t2$ gives the detection sum timing jitter *j*_{Σ}, which is 29 ps, representing a 12% reduction relative to the *j*_{t1} or *j*_{t2}. A clear delay was observed between *t*_{1} and *t*_{2} for the hotspot nucleation event shown. This delay varied randomly from event to event, which we interpret as resulting from the variation in the position of the event along the length of the wire.

With the metrics defined above, we implemented the differential readout on the NbN SNSPDs of various geometries. The devices were fabricated from ∼4-nm-thick NbN films on the SiN_{x}-on-Si substrates. The standard meander structure was defined using electron-beam lithography, followed by reactive-ion etching. The nanowire width was 100 nm, the fill factor was 50%, and the detector area varied from 3 *μ*m × 3 *μ*m up to 20 *μ*m × 20 *μ*m. Assuming a uniform distribution in the hotspot stimulation from the flood laser illumination on the active area, we measured the *t*_{1} and *t*_{2} for each photon detection event and thus derived *j*_{Δ} and *j*_{Σ.}

Fig. 2(a) shows that increasing the dimension of the SNSPD, at the same biasing current, increased the *j*_{Δ}$\u2009$ as well. For larger SNSPDs, photons were spread over a larger area. Because it takes time for a pulse generated at one end to reach the other end, a larger area resulted in a higher Δ*t*. We compared the detection jitters *j*_{t1} and *j*_{t2}, each measured from a single output, to the *j*_{Σ} determined by using both outputs. The jitters were acquired with a LeCroy 6 GHz oscilloscope. The number of photon events acquired was more than 50 000 and the measurement time was 1 h each. The laser power was set to a single photon level. As shown in Fig. 2(c), as the size of the SNSPD was varied from 3 *μ*m × 3 *μ*m to 20 *μ*m × 20 *μ*m, jitters *j*_{t1} and *j*_{t2} increased from 22 ps to 45 ps. In comparison, the increase of *j*_{Σ} over the same range was smaller, resulting in a reduction of 17% in the variance of the determination of the photon arrival time by using *j*_{Σ} relative to using *j*_{t1} (or *j*_{t2}) for a 20 *μ*m × 20 *μ*m SNSPD. As a result, we can use the differential readout to reduce the single-photon detection jitter without changing the design of an SNSPD, and this reduction grows as the device area grows. We noticed that the jitter reduction was higher in 10 *μ*m × 10 *μ*m instead of 15 *μ*m × 15 *μ*m. This was explained by the biasing ratio: the 15 *μ*m × 15 *μ*m showed higher defects, low critical current (Fig. 2(c)), than 10 *μ*m × 10 *μ*m, leading in higher hot-spot jitter,^{15} which cannot be compensated in j_{Σ}.

For sufficient small detector areas, the geometric effect ought to be negligible, and so *j*_{Δ} should approach the intrinsic jitter of the hot-spot generation process, while *j*_{Σ} should approach *j*_{t1} or *j*_{t2.} To test this hypothesis, we fabricated the SNSPDs with a minimal detection area, named the constricted SNSPD. For each detector, there was only a 1-*μ*m-long and a 100-nm-wide nanowire region (shown in red and blue) for sensing photons. Because photon-detection events are limited to the 1-*μ*m-long nanowire region, the geometric jitter from this small portion of the device *j*_{ΔS} should approach the limit set by the combined effect of the variance in the intrinsic hot-spot generation process and electrical noise (as expected from Ref. 12, the electrical jitter increases with the overall inductance of the nanowire, including both narrower and wider regions). We considered the inductance variation decreasing the constricted SNSPD biasing current (i.e., changing signal slopes): at higher signal slope (representing the 3 *μ*m × 3 *μ*m), we achieved a jitter, *j*_{ΔS}, lower than 12 ps instead for the 20 *μ*m × 20 *μ*m *j*_{ΔS} was 18 ps.

By comparing the difference between *j*_{Δ} in the typical SNSPDs *j*_{ΔN} and *j*_{ΔS} in the 1-*μ*m-long devices, as shown in Fig. 2(b), we could evaluate how much the geometric jitter *j*_{G} contributes to the jitter of a typical device after removing the contribution from electrical noise. We assumed that *j*_{ΔN} and *j*_{ΔS} add in quadrature, so $jG=j\Delta N2\u2212j\Delta S22$.

Fig. 3 shows the design of a comparison experiment, designed to characterize the maximum propagation time that can result when a photon hits at either edge of the nanowire. In this experiment, we used an on-chip multiplexing circuit to operate the two SNSPDs (named L and R), which were designed to have limited photon-sensing area close to the edge of the meander shape shown in Fig. 3. Each of the SNSPDs had a 1-*μ*m-long nanowire region placed in opposite corners, with the rest of the wire comprised of a 200-nm-wide nanowire. The devices had a total area of 20 *μ*m × 20 *μ*m of which only the 100 nm × 1 *μ*m length was active. The arms were connected in parallel but oriented in opposite directions. The two SNPSDs were selectively biased by means of the superconductive switches—nanocryotrons (nTrons)^{16,17}—one placed in series with each side of the device (L and R) along the meander. When the nTron gate was biased over its critical current, the nTron channel became resistive and blocked the bias current on one side of the device. Fig. 3(b) shows the distribution of the Δ*t* acquired in the two different configurations of Fig. 3(a): (i) arm L ON, where the photon detection location was far from OUT_{1}; (ii) arm R ON, where the photon detection location was close to OUT_{1}. As shown in Fig. 3(b), the two resulting distributions of *t*_{2}–*t*_{1} had almost the same FWHM but had a spacing of Δ*t* = 42 ps between the distribution centres. Thus, Δ*t*/2 corresponds to the propagation delay for a signal to travel the entire length of the nanowire in this geometry (noting of course that the velocity of signal propagation along the 200 nm wide region will not be exactly the same as that for the 100-nm-wide region in typical devices).

In conclusion, we developed: (i) a differential cryogenic readout circuit demonstrating less than 7 ps of readout jitter by using a high impedance input stage and (ii) a post-processing compensation method improving the detection timing jitter of an SNSPD, reducing the jitter by 17% for a 20 *μ*m × 20 *μ*m device, and thus achieving 38 ps of detection timing jitter. We also demonstrated the existence of geometric jitter in an SNSPD and characterized the dependence of geometric jitter on the SNSPD size. We showed that this effect contributes with 20 ps to the jitter of a 20 *μ*m × 20 *μ*m device. This work also implies that the observed system jitter should depend on how light is coupled to a detector (whether focussed on a single spot or distributed evenly across the full device).

This work has shown that some improvement in jitter is available by a simple change in a readout method; we believe that in future work, a comprehensive treatment of the full microwave environment and device characteristic could result in a substantial modification in the existing model of the SNSPD operation. The SNSPD architectures can now be envisioned in which, for example, an improved microwave biasing, readout, and signal amplification of the device are considered. We believe that any future efforts to drive the SNSPD jitter and reset performance to higher levels will require consideration of the microwave characteristic of the nanowire geometry.

This research was supported by the National Science Foundation (NSF) grant under Contact No. ECCS1-509486 and the Air Force Office of Scientific Research (AFOSR) grant under Contract No. FA9550-14-1-0052. Niccolò Calandri would like to thank his financial support from the Roberto Rocca project when he was a visiting student in MIT. Di Zhu is supported by National Science Scholarship from A*STAR, Singapore. Andrew Dane was supported by NASA Space Technology Research Fellowship, Grant No. NNX14AL48H. Emily Toomey is acknowledged for helpful comments on the manuscript. Adam McCaughan is acknowledged for helpful discussions.