How does photon-number resolution improve multiplexed SNSPD detector arrays?

Local resolution relaxes the quadratic detector scaling

A multiplexed array of on/off SNSPDs approximates photon-number resolution by spreading N photons across D pixels. Because each pixel only reports a click or no click, multi-photon collisions inside one pixel blur the readout, and the number of elements needed to keep that collision probability low grows quadratically with the photon number N. The paper asks whether that requirement can be relaxed if each individual element already carries some photon-number information of its own.

The authors generalize the existing binary formulation to detectors with an arbitrary number of resolvable states. An s-state detector reports s output classes: 0, 1, ..., s-2 resolved photons plus a final saturated class (s-1)+ that groups all occupancies at or above s-1 photons. The paper particularizes this to 2-, 3- and 4-state multiplexed systems, where a 3-state detector distinguishes 0-, 1- and 2-plus-photon events and a 4-state detector distinguishes 0-, 1-, 2- and 3-plus-photon events.

The headline claim is that increasing local photon-number resolution systematically improves both detector fidelity and Bayesian photon-number reconstruction, and that this substantially relaxes the quadratic scaling between detector number and resolvable photon number required by purely binary multiplexed arrays. That is a model result derived from the response tensor, not a measurement on a fabricated array.

The response tensor and binomial loss convolution

The core object is the probability response tensor P(K(s)|N,D,eta): the probability of observing a particular output vector K(s) when N photons arrive on a system of D s-state detectors with quantum efficiency eta. The output vector records how many detectors sense 0, 1, 2, ... (s-1)+ photons, and its components sum to D.

For the ideal lossless case the authors derive this tensor combinatorially using exponential generating functions, counting the number of ways k1 detectors receive one photon, k2 detectors receive two photons, and so on up to the saturated class. For s=2 the expression reduces exactly to the earlier lemma for distributing a Fock state of N photons evenly among D ideal on/off detectors.

Finite efficiency is handled separately through a binomial loss convolution: because the spatial splitting is uniform, a lossy system with N incident photons behaves like an ideal system injected with a reduced number R of effectively detected photons. Dark counts are assessed separately, and the authors note that the typical sub-hertz dark count rate in SNSPDs makes their effect negligible compared with collision-driven and resolution-driven contributions, so the numerical results use noiseless 4- and 8-detector systems.

Likelihood matrices approach the ideal diagonal

To compare readouts directly, the full tensor is projected onto a scalar estimator N_s = K(s) . v(s), where v(s) = (0, 1, ..., s-1). This marginalization yields a likelihood matrix P(s)(n_s|N,D,eta) over all output configurations that give the same estimator value. Because of saturation, N_s is a lower-bound estimator: generally N_s(K) <= N.

For D=8 and eta=0.92, the paper shows likelihood matrices with N varied up to 16. The white diagonal marks the ideal response where the readout equals the number of arriving photons. The 2-state response is strongly compressed because, as N grows, the number of detected photons increases at a reduced rate. The 3-state likelihood distributes along values of n3 closer to N, and the 4-state matrix is clearly more concentrated along the diagonal because collisions and losses remain limited.

The practical reading is that the projected readout becomes more informative about the number of incident photons as the state number increases, at fixed detector count and efficiency. This is a statement about the modeled likelihood, not about a specific fabricated device.

Fidelity gains and the 0.55 threshold

The paper derives a generalized N-photon detector fidelity from the response tensor and evaluates it numerically. Figure 2a reports fidelity as a function of photon number for different efficiencies using D=8. The 3-state system provides fidelities sensibly higher than the 2-state system up to N=12, and the lossless 2-state system is superior to the lossy 3-state one only up to 2 photons.

Figure 2b reports the number of detectors required to satisfy a minimum fidelity threshold of 0.55. For 2-state systems with limited efficiency the required number grows faster than e^N and becomes unpractical for 7 photons, where D exceeds 1000. For 3-state systems, up to 5 photons the requirement stays below ten detectors, with minor influence of losses when eta >= 0.92.

That comparison is the concrete relaxation of the earlier rule for on/off detectors, which held that high fidelity requires the number of resolving features to exceed N squared. The threshold of 0.55 and the D=8 setting are the conditions under which these numbers hold; they are not universal performance guarantees.

Detector fidelity is not reconstruction fidelity

The paper distinguishes two metrics that are easy to conflate. N-photon detection fidelity asks whether the detector output matches the true photon number. Reconstruction fidelity asks how much information about the true photon number is gained from the detector output, and it matters especially when the photon-number estimate informs a further decision.

The authors make the difference concrete: a saturated outcome that matches the true photon number does not degrade detector fidelity, while the posterior it produces remains spread over several photon numbers and the reconstruction fidelity is reduced. Any protocol acting on a photon-number threshold is only reliable when, given the observed outcome, the residual probability mass over that threshold is sufficiently small, and reconstruction fidelity quantifies exactly that probability.

The Bayesian reconstruction inverts the response tensor to infer the likelihood of N given an observed output vector, using a prior distribution. The paper also defines an ensemble reconstruction fidelity that averages the local fidelity over both detector outcomes and the prior distribution, and notes that the prior appears implicitly in the Bayesian term and explicitly as the weighting over source statistics.

Poisson-averaged reconstruction rewards high efficiency

For the reconstruction comparison the authors assign the number of incoming photons N and distribute them randomly among the D detectors using a multinomial distribution; for equiprobable detectors each detector receives a photon with probability 1/D. They also show that output vectors of lower-state systems can be retrieved from a higher-state output vector through the hierarchical relations, for example K(4)=[2,1,2,1] giving K(3)=[2,1,3] and K(2)=[2,4] for a Poissonian source with mu=8 and D=6.

Expressed as a function of detector efficiency, the Poisson-averaged reconstruction fidelities are smaller than 0.6 for eta < 0.9, but increase largely for greater eta, especially for D=8. The authors note this is particularly advantageous for multiplexed multi-state SNSPD devices because SNSPDs can easily exceed eta=0.9. Smaller prior mean mu gives greater efficiency for the same reasons, yet 4-state detectors with mu=8 outperform 2-state detectors with mu=4.

These are model outcomes under stated priors and detector counts. They describe how reconstruction fidelity scales with efficiency and state number in the framework, not a measured reconstruction on a physical array.

Where these results apply and where they stop

The framework is general in the number of resolvable states, but the numerical results are particularized to 2-, 3- and 4-state multiplexed systems with 4 or 8 detectors, noiseless operation, and efficiencies including eta=1 and eta=0.92. The dark-count treatment is placed in an appendix and is not part of the main numerical results, on the grounds that sub-hertz SNSPD dark counts are negligible relative to collision and resolution effects.

The dark-count appendix also shows that the coincidence window tau strongly changes the per-window dark-count probability epsilon: with a 1 Hz rate, tau around 300 ps gives epsilon near 3E-10, while tau around 100 microseconds gives epsilon near 1E-4. The authors assume tau in the resistive-domain lifetime range of 50 to 800 ps, which is the condition under which their negligible-dark-count conclusion holds.

The paper is a preprint and its claims are analytical and numerical. It does not report a fabricated multi-state array, so the fidelity and reconstruction numbers should be read as value-of-information assessments of the modeled architectures rather than demonstrated device performance.

The priority is reconstruction fidelity, not just detector fidelity

The paper's closing argument is that much emphasis in characterizing photon-number-resolving detectors is traditionally placed on N-photon detection fidelity, but reconstruction fidelity answers a different and often more decision-relevant question: how much information about the true photon number is actually gained from the output. A saturated outcome can match the true photon number and leave detector fidelity intact while the posterior stays spread and reconstruction fidelity drops.

For readers weighing multi-state SNSPD arrays, the operative comparison is therefore between 2-, 3- and 4-state detectors at fixed D and eta, judged on both metrics. The 0.55 fidelity threshold and the D>1000 requirement for 2-state systems at 7 photons mark where binary multiplexing becomes impractical, while 3-state systems stay below ten detectors up to 5 photons with minor loss influence when eta >= 0.92.

The most useful takeaway is scenario-specific: when a photon-number estimate feeds a later threshold decision, such as assessing whether a heralded single-photon source carries hidden multi-photon contamination, reconstruction fidelity is the metric that quantifies the residual risk, and it is the metric that local photon-number resolution improves.

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