Practical considerations for assignment of photon numbers with SNSPDs

TL;DR

This work investigates practical factors in assigning photon numbers with SNSPDs by examining how optical pulse shape and duration influence arrival-time histograms used for PNR. Using a programmable pulse shaper, the authors compare Gaussian-filtered versus band-filtered pulses and demonstrate that shorter, Gaussian-shaped pulses yield cleaner $n$-photon separation in arrival times, while longer pulses degrade discrimination. They advance the analysis by adopting exponentially-modified Gaussian (EMG) distributions to model arrival-time histograms, showing superior accuracy over Gaussian fits and enabling more reliable estimation of misidentification and loss; they also reconstruct the detector's POVMs via quantum detector tomography, revealing intrinsically sharp PNR features. Overall, the study provides actionable guidance on pulse shaping, histogram modeling, and POVM reconstruction to optimize SNSPD-based PNR, with implications for heralded photon sources and quantum information protocols.

Abstract

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.

Figures (9)

• Figure 1: Experimental setup used to investigate the influence of multiple experimental parameters of the input light pulses on the photon-number resolution of SNSPDs. The laser pulses from a pulsed laser pass through two electro-optic modulators for high extinction ratio pulse-picking, a programmable optical processor (waveshaper) to manipulate the temporal shape of the pulse and variable optical attenuators to set the mean photon number per pulse. For more detail refer to the main text in Sec. \ref{['sec:setup']}. QWP: Quarter-wave plate; HWP: half-wave plate; EOM: electro-optic modulator; PBS: polarizing beam splitter.
• Figure 2: Arrival-time histogram of different waveshaper bandwidth settings, for a mean photon number per pulse of $\bar{n}=3$ and center wavelength of $\lambda=1550~\mathrm{nm}$. The temporal pulse widths of $60~\mathrm{ps}$ (red), $25~\mathrm{ps}$ (light blue) and $2.9~\mathrm{ps}$ (orange) correspond to bandwidths of $0.01~\mathrm{nm}$, $0.14~\mathrm{nm}$ and $2.66~\mathrm{nm}$, respectively. The mean photon number for the bandwidth of $0.01~\mathrm{nm}$ is 2.5, due to limited optical power.
• Figure 3: (a) Sum of the arrival-time histograms (black line) with fits (green dashed line and the colored lines) based on EMG distributions (for an ensemble of arrival-time histograms for different mean photon numbers), according to the model from Ref. sidorova2025jitter with $\chi^2_\mathrm{EMG}=0.02$. (b) Sum of the arrival-time histograms (black line) with a fit (red dashed line) based on a sum of Gaussian distributions weighted according to Poissonian statistics (Eq. \ref{['eqn:gaussianSum']}) with $\chi^2_\mathrm{Gauss}=0.06$. Vertical black dashed lines separate the photon-number regions, based on the intersection points between neighboring distributions. Individual underlying Gaussian distributions for photon-number contributions up to $n=20$ are shown with different colored lines.
• Figure 4: (a) Overlap matrix of every EMG distribution with every photon-number region, bounded by the separation lines, i.e., intersection points of the underlying distributions (from Fig. \ref{['fig:gaussfit']}(a)). (b) Probability of missing $p_{\mathrm{missing,}n}$ and misidentifying $p_{\mathrm{misidentified,}n}$ an $n$-photon event as a function of photon number $n$ for both fitting methods (based on Gaussian and EMG distributions). For the Gaussian model, the probabilities are underestimated due to inaccuracies in the fit of the exponential tail of the underlying distributions (see Fig. \ref{['fig:gaussfit']}(b)).
• Figure 5: (a) Outcome matrix $\boldsymbol{P}$ that results from applying the separation lines from Fig. \ref{['fig:gaussfit']}(a) to the recorded data, by creating histograms with variable bin sizes for the possible detector outcomes. The largest outcome is $n'=3$, given by the resolvability criterion Eq. \ref{['eqn:res_fwhm']}. (b) Reconstructed POVM elements for a smoothing parameter of $10^{-6}$. The inset shows the $\Pi_{1,1}=p(1|1)$ POVM element as a function of the smoothing parameter. Over-smoothing is manifested in a reduction of the $p(1|1)$ probability for a larger smoothing parameter.