Machine learning inspired photon number resolution in superconducting nanowire single-photon detectors

TL;DR

This work addresses the absence of a systematic framework for photon-number resolution in SNSPDs by applying PCA to full detector traces and showing that the essential information resides in the first principal component, which corresponds to the derivative of the mean response $d\overline{V_1(t)}/dt$. A time-shift model links PCA projections to photon-number dependent delays, enabling efficient photon-number discrimination using full trace data and modest hardware ($5$ GSa/s, $3$ GHz bandwidth). The authors introduce a Bhattacharyya-coefficient-based confidence metric to benchmark resolvability, fit photon-number peak distributions with exponentially-modified Gaussians, and demonstrate robust performance across datasets, including an open dataset, with potential FPGA implementation for real-time classification and feed-forward in quantum photonic systems. The work thus provides a scalable, hardware-friendly framework for real-time photon counting and comparative benchmarking of SNSPD-based photon-number resolution capabilities, guiding hardware improvements mainly through jitter reduction.

Abstract

Photon-number resolved detection with superconducting nanowire single-photon detectors (SNSPDs) attracts increasing interest, but lacks a systematic framework for interpreting and benchmarking this capability. In this work, we combine principal component analysis (PCA) with a new readout technique to explore the photon-number resolving capabilities of SNSPDs and find that the information of the photon number is contained in a single principal component which approximates the time derivative of the average response trace. We introduce a new confidence metric based on the Bhattacharyya coefficient to quantify the photon-number-resolving capabilities of a detector system and show that this metric can be used to compare different systems. Our analysis and interpretation of the principal components imply that photon-number resolution in SNSPDs can be achieved with moderate hardware requirements in terms of both sample rate (5 GSample/sec) and analog bandwidth (3 GHz) and could be implemented in an FPGA, giving a highly scalable solution for real-time photon counting.

Figures (6)

• Figure 1: Schematic overview of the experimental setup showing the laser, attenuators and readout electronics.
• Figure 2: Principal component analysis of detector pulses. (a) Time trace of the output signal of the SNSPD. (b) Example of the projection of the dataset at $\mu=1.77$ to the first two principal components. (c) Comparison of the first principal component for $\mu=0.003$ (blue), $\mu=1.77$ (orange) and the average of the derivative of the output at $\mu=0.003$ (green dashed line). (d) Projection of datasets of various mean photon numbers to the mean derivative of the $\mu=0.003$ dataset. The results for projecting the data onto the first principal component of the $\mu=0.003$ dataset are indistinguishable from the projection on the average derivative.
• Figure 3: Count statistics as a function of photon number. (a) Measured histogram (black) and fitted exponentially modified gaussian distribution (blue) and gaussian distribution (orange) for $\mu=1.77$. The residuals are plotted in green. (b) Relative probability for the different photon numbers. The probabilities for 1 and 2 photons are directly derived from the Gaussian fit. The 3+ photon contribution is obtained by normalizing the residuals. The solid lines give the expected zero-truncated Poisson distribution.
• Figure 4: Analysis of the dataset of Schapeler et al. (a) Projection on the first two principal components for the original dataset and (b) a filtered and downsampled version of the dataset. The dashed and dotted lines indicate the projection and orthogonal axis used for assigning photon numbers to traces, respectively. (c) Histogram of the data projected onto the average derivative (blue) and onto the effective principal component (orange, dashed). The latter is linearly scaled and shifted to align with the derivative projection, allowing direct comparison.
• Figure 5: Scree plot showing the explained variance of the first 10 principal components in the principal component analysis for $\mu=1.77$. The inset shows the cumulative explained variance as a function of the number of principal components.