Practical considerations for assignment of photon numbers with SNSPDs

Superconducting nanowire single-photon detectors (SNSPDs) can enable photon-number resolution (PNR) based on accurate measurements of the detector’s response time to few-photon optical pulses. In this work we investigate the impact of the optical pulse shape and duration on the accuracy of this method. We find that Gaussian temporal pulse shapes yield cleaner arrival-time histograms, and thus more accurate PNR, compared to bandpass-filtered pulses of equal bandwidth. For low system jitter and an optical pulse duration comparable to the other jitter contributions, photon numbers can be discriminated in our system with a commercial SNSPD. At 60 ps optical pulse duration, photon-number discrimination is significantly reduced. Furthermore, we highlight the importance of using the correct arrival-time histogram model when analyzing photon-number assignment. Using exponentially-modified Gaussian (EMG) distributions, instead of the commonly used Gaussian distributions, we can more accurately determine photon-number misidentification probabilities. Finally, we reconstruct the positive operator-valued measures (POVMs) of the detector, revealing sharp features which indicate the intrinsic PNR capabilities.

💡 Research Summary

This paper investigates the intrinsic photon‑number‑resolution (PNR) capabilities of a commercial superconducting nanowire single‑photon detector (SNSPD) by analysing how the shape and duration of the incident optical pulse affect the accuracy of photon‑number assignment. The authors use a 1550 nm pulsed laser whose repetition rate is reduced to 9.5 kHz with an electro‑optic modulator (EOM). A programmable optical processor (WaveShaper 4000B) shapes the spectrum of the pulses to generate three Gaussian temporal profiles with full‑width‑half‑maximum (FWHM) durations of 2.9 ps, 25 ps and 60 ps, corresponding to spectral bandwidths of 2.66 nm, 0.14 nm and 0.01 nm respectively. Two variable optical attenuators set the mean photon number per pulse (⟨n⟩) from the single‑photon regime up to ~68 000 photons. The SNSPD (Single Quantum) operates at saturated detection efficiency with a measured jitter of 19 ps (FWHM). A time‑tagger records the timestamps of the laser trigger (derived from a fast photodiode) and the rising edge of the SNSPD electrical pulse, producing arrival‑time histograms for each ⟨n⟩.

The central finding is that the optical pulse duration contributes directly to the overall system jitter and therefore determines the separability of the photon‑number peaks in the arrival‑time histograms. With the shortest 2.9 ps pulses, the one‑photon and two‑photon peaks are clearly separated, enabling reliable photon‑number discrimination even with a single, commercially available SNSPD. As the pulse duration is increased to 25 ps, the peaks begin to broaden, and at 60 ps the overlap becomes severe, dramatically reducing discrimination performance. This demonstrates that for high‑speed PNR the optical pulse must be comparable to or shorter than the intrinsic jitter contributions of the detector and readout electronics.

To extract photon‑number probabilities from the histograms, the authors compare two fitting models. The conventional approach uses a sum of Gaussian distributions weighted by Poissonian input statistics. The more sophisticated model employs exponentially‑modified Gaussian (EMG) distributions, which incorporate an exponential tail arising from the detector’s electrical response and thermal relaxation. EMG fits are performed on the first nine input states (⟨n⟩ = 1–9) and include a small shift of the peak positions with increasing ⟨n⟩, ensuring independence from the mean photon number. The EMG model yields a χ² of 0.02, accurately reproducing the experimental tails over four orders of magnitude, whereas the Gaussian model gives χ² = 0.06 and fails to capture the tail, leading to underestimation of error probabilities.

Using the fitted distributions, the authors construct an overlap matrix that quantifies how much each photon‑number distribution spills into the decision regions defined by the intersection points of neighboring peaks. From this matrix they compute type‑I (false‑positive) and type‑II (false‑negative) error probabilities, p_misidentified,n and p_missing,n. The EMG‑based analysis predicts a misidentification probability for single‑photon events of 0.14 % (≈1 in 700), whereas the Gaussian model unrealistically reports only 0.003 % (≈1 in 30 000). For higher photon numbers the discrepancy grows, confirming that the exponential tail significantly contributes to misclassification. The authors also discuss a trade‑off: narrowing the decision windows reduces misidentification but introduces “loss” where events fall outside any window; sacrificing about 6 % of one‑photon events can lower the misidentification rate to 0.01 %.

Finally, the paper reconstructs the positive‑operator‑valued measures (POVMs) of the detector by combining the EMG fit parameters with the known Poissonian input distribution. The reconstructed POVM elements display sharp features up to at least 20 photons, with a pronounced drop in response beyond five photons, reflecting the formation of multiple resistive hotspots and the resulting non‑linear increase in nanowire resistance. This reconstruction provides a quantitative quantum‑measurement description of the SNSPD’s intrinsic PNR capability.

In summary, the work demonstrates that (i) the optical pulse duration must be comparable to or shorter than the total system jitter to preserve photon‑number discrimination; (ii) arrival‑time histograms should be modelled with EMG distributions to obtain realistic error estimates; and (iii) POVM reconstruction validates the intrinsic multi‑photon sensitivity of SNSPDs. These insights furnish practical guidelines for deploying SNSPDs in high‑speed quantum‑information protocols that require accurate, low‑noise photon‑number resolution.