Photon-Number-Resolving Architectures

• Photon-number-resolving architectures are detection systems that convert incident photon numbers into measurable signals using techniques like calorimetry, spatial partitioning, and timing analysis.
• They employ diverse mechanisms—such as integrated TES, segmented nanowires, and multiplexed ON–OFF detectors—to achieve high-resolution photon counting with efficiencies often exceeding 90%.
• Benchmarking frameworks and advanced signal-processing methods are integral for optimizing these systems, driving applications in quantum state engineering and integrated photonics.

Photon-number-resolving architectures are detector and readout configurations that produce outcomes correlated with the integer photon number of an incident optical field, rather than only a binary click/no-click response. Across contemporary implementations, number resolution is obtained through several distinct mechanisms: calorimetric energy measurement in transition-edge sensors (TESs), spatial partitioning across multiple superconducting nanowire elements, temporal or spatial multiplexing of ON–OFF detectors, distributed absorption in waveguides or standing-wave structures, and inference from analog pulse amplitude, timing, or waveform shape. These approaches span integrated waveguide TES systems with $79\% \pm 2\%$ single-pass efficiency and $88\% \pm 3\%$ double-pass efficiency, segmented nanowire devices with $\sim 98\%$ system detection efficiency and photon-number resolution up to 32, and single conventional SNSPDs whose pulse timing alone separates $N=1\ldots 5$ photon events (Calkins et al., 2013, Ding et al., 3 Apr 2025, Sauer et al., 2023).

1. Architectural principles and benchmarking frameworks

The central architectural question in photon-number resolution is where the number information is physically encoded. In calorimetric detectors, it is encoded in deposited energy. In parallel or segmented nanowire devices, it is encoded in the number of resistive elements that switch and hence in output pulse amplitude. In multiplexed detectors, it is encoded in click patterns across bins or pixels. In timing- and waveform-based approaches, it is encoded in analog observables such as leading-edge advance, trailing-edge shift, or a low-dimensional projection of the full electrical trace.

Benchmarking has correspondingly diversified. A common formalism models the detector as a discrete POVM $\{\Pi_n\}$, with matrix elements

$$P_{n,m}=\langle m|\Pi_n|m\rangle=\Pr(\text{output}=n\mid \text{input}=m),$$

and defines the event fidelity of exactly $n$ photons as $F(n)=P_{n,n}$ (Ding et al., 3 Apr 2025). In detector-tomographic studies of multiplexed architectures, the outcome operators are further analyzed through purity and mutual information, showing that more multiplexing outcomes increase photon-number-resolving power only when the splitting between those outcomes is also optimized (Schapeler et al., 2022). For waveform-domain classification, a different benchmark has been introduced via the Bhattacharyya coefficient $B_{n,m}$, with adjacent-class confidence

$C_{|n\rangle\to|n+1\rangle}=1-B_{n,n+1},$

so that overlap between neighboring photon-number histograms becomes a direct architecture-level figure of merit (Kuijf et al., 17 Nov 2025).

These metrics are not interchangeable. System detection efficiency, dark-count rate, count rate, timing jitter, event fidelity, purity, and confidence characterize different failure modes. The literature therefore treats photon-number resolution not as a single scalar property but as a coupled optical, electrical, and inferential architecture problem.

2. Calorimetric architectures: transition-edge sensors and integrated TES circuits

TES-based architectures remain the canonical intrinsic PNR platform because the detector directly measures absorbed energy. In the integrated waveguide TES architecture at telecom wavelength, the calorimeter is a $88\% \pm 3\%$0 tungsten square, $88\% \pm 3\%$1 thick, with superconducting transition at $88\% \pm 3\%$2. It is voltage-biased in the middle of its superconducting transition for high $88\% \pm 3\%$3 sensitivity, and its current pulses are read out by a low-noise SQUID chain, with pulse height proportional to deposited photon energy. Extended absorbers consist of two $88\% \pm 3\%$4-long fins, each with a tungsten layer and a gold spine; the coupled electrothermal model predicts sub-100 ps thermalization times and clear separation of $88\% \pm 3\%$5 photon peaks up to at least 4 photons (Calkins et al., 2013).

The same work embeds these TESs into UV-written silica-on-silicon waveguides. The platform uses a silicon wafer, $88\% \pm 3\%$6 thermal-oxide under-cladding, and a $88\% \pm 3\%$7 photosensitive Ge-doped silica core with no overcladding, allowing evanescent coupling to surface detectors. TM-like modes were selected for highest absorption, with simulated evanescent absorption coefficients $88\% \pm 3\%$8 and $88\% \pm 3\%$9. Exploiting mode transmission failure, three TESs were multiplexed in series so that

$\sim 98\%$0

with measured single-pass efficiencies $\sim 98\%$1, $\sim 98\%$2, and $\sim 98\%$3 for the three detectors, yielding $\sim 98\%$4 combined efficiency in a single pass. With a terminal Bragg reflector, the waveguide-grating-TES chain is traversed twice and the measured combined efficiency becomes $\sim 98\%$5. Classical Bragg-grating measurements agreed with TES-measured absorption, and the paper reports no significant unexplained loss due to scattering or reflections (Calkins et al., 2013).

A distinct TES architecture addresses the readout bottleneck rather than the absorber geometry. A real-time FPGA-based processor for TES pulses digitizes at 14 bit and 250 MS/s, performs digital low-pass filtering, differentiation, rise finding, and hardware extraction of pulse area, height, length, and slope, then streams event packets over Ethernet. This system resolves photon number up to $\sim 98\%$6 in real time, achieves discrimination at the parts-per-billion level for low photon numbers, reduces data rate by $\sim 98\%$7, and completes FPGA-side measurement in $\sim 98\%$8 after pulse end (Morais et al., 2020).

Signal-processing architecture materially changes effective TES resolution. For unsupervised classification of TES traces, previous methods such as signal area and PCA resolve up to 16 photons with confidence above $\sim 98\%$9, while nonlinear dimensionality-reduction methods resolve up to 21 at the same confidence threshold. In the reported synthetic-uniform dataset, the maximum photon number with $N=1\ldots 5$0 rises from 12 for pulse area and 16 for PCA to 21 for two-dimensional t-SNE or UMAP, and outlier removal can recover an extra $N=1\ldots 5$1–3 photons of resolution (Dalbec-Constant et al., 2024). This suggests that, for calorimetric detectors, photon-number-resolving architecture includes both the sensor physics and the latent-space geometry used for classification.

3. Nanowire architectures: parallel, series, tapered, and segmented devices

Superconducting nanowire architectures replace calorimetric energy measurement with spatially distributed threshold elements whose collective response becomes amplitude-resolved. The early parallel nanowire detector (PND) realized subwavelength spatial multiplexing by wiring several NbN nanowires in parallel and reading them through a single electrical output. In this architecture, the output pulse amplitude is approximately proportional to the number of simultaneously fired branches. Experiments showed counting rate of 80 MHz and pulse duration as low as 660 ps FWHM, while pulse-height histograms exhibited no observable multiplication noise buildup (0902.4824). Telecom-wavelength PNDs demonstrated counting up to 4 photons, ultralow dark count rate, and maximum count rate $N=1\ldots 5$2; the reported system detection efficiency was $N=1\ldots 5$3 at $N=1\ldots 5$4, with dark count rate $N=1\ldots 5$5 and NEP $N=1\ldots 5$6 (0712.3080).

Waveguide-integrated nanowire PNRDs adapted the same principle to photonic integrated circuits. A GaAs/$N=1\ldots 5$7 ridge-waveguide WPNRD placed four $N=1\ldots 5$8 NbN nanowires over the optical mode, electrically in series and each shunted by a normal-metal resistor of designed value $N=1\ldots 5$9. If $\{\Pi_n\}$0 wires switch simultaneously, the readout voltage is approximately

$\{\Pi_n\}$1

allowing $\{\Pi_n\}$2–4 photon discrimination with a single output. Detection of 0–4 photons was demonstrated, and the device quantum efficiency at 1310 nm for TE polarization was $\{\Pi_n\}$3 (Sahin et al., 2013).

Later parallel SNSPD architectures emphasized efficiency and stability. A 4-pixel MoSi P-SNSPD wired pixels in parallel with series resistances to quench hotspots and suppress crosstalk, while thermal isolation prevented thermal crosstalk. The detector achieved a plateau at $\{\Pi_n\}$4 at $\{\Pi_n\}$5, with dark count rate $\{\Pi_n\}$6, timing jitter 42 ps FWHM, recovery to 50% of nominal efficiency after $\{\Pi_n\}$7, and recovery to $\{\Pi_n\}$8 by 40 ns. The reported architecture is free of latching and does not suffer from crosstalk between the pixels (2207.14538).

A different single-element nanowire strategy is the superconducting tapered nanowire detector, which turns the photon-number-dependent hotspot resistance into an analog amplitude signal through an integrated impedance-matching taper. The device uses a Klopfenstein-profile taper whose impedance varies from $\{\Pi_n\}$9 at the nanowire end to $$P_{n,m}=\langle m|\Pi_n|m\rangle=\Pr(\text{output}=n\mid \text{input}=m),$$0 at the amplifier end. It resolved up to five absorbed photons and had 16.1 ps timing jitter, $$P_{n,m}=\langle m|\Pi_n|m\rangle=\Pr(\text{output}=n\mid \text{input}=m),$$1 c.p.s. device dark count rate, $$P_{n,m}=\langle m|\Pi_n|m\rangle=\Pr(\text{output}=n\mid \text{input}=m),$$2 reset time, and $$P_{n,m}=\langle m|\Pi_n|m\rangle=\Pr(\text{output}=n\mid \text{input}=m),$$3 system detection efficiency at 1550 nm without cavity (Zhu et al., 2019).

The most aggressive nanowire integration in the supplied literature is the segmented twin-layer SNSPD. Here a $$P_{n,m}=\langle m|\Pi_n|m\rangle=\Pr(\text{output}=n\mid \text{input}=m),$$4 NbN active area with 80 nm line width and 160 nm pitch is segmented into $$P_{n,m}=\langle m|\Pi_n|m\rangle=\Pr(\text{output}=n\mid \text{input}=m),$$5 equal pixels connected in series. Each pixel is part of a $$P_{n,m}=\langle m|\Pi_n|m\rangle=\Pr(\text{output}=n\mid \text{input}=m),$$6 sandwich on a 13-period $$P_{n,m}=\langle m|\Pi_n|m\rangle=\Pr(\text{output}=n\mid \text{input}=m),$$7 DBR centered at 1550 nm, and each pixel is shunted by a lateral $$P_{n,m}=\langle m|\Pi_n|m\rangle=\Pr(\text{output}=n\mid \text{input}=m),$$8 Ti resistor. The system detection efficiency factorizes as

$$P_{n,m}=\langle m|\Pi_n|m\rangle=\Pr(\text{output}=n\mid \text{input}=m),$$9

with typical values at 1555 nm $n$0, $n$1, $n$2, and hence $n$3. Experimentally, the detector exhibits $n$4 SDE at dark count rate $n$5, resolves 32 distinct voltage levels with $n$6, operates at 41 MHz at 3dB-SDE rollover, and reaches timing jitter of 382 ps at $n$7, 234 ps at $n$8, and 40 ps at $n$9. Detector tomography gives $F(n)=P_{n,n}$0, $F(n)=P_{n,n}$1, $F(n)=P_{n,n}$2, and $F(n)=P_{n,n}$3 (Ding et al., 3 Apr 2025).

4. Multiplexed architectures and distributed absorption

Multiplexing remains the most general route to approximate PNR using ON–OFF elements. In integrated waveguides, this need not involve explicit beam-splitter trees. The series TES architecture described above relies on “mode transmission failure”: a detector of efficiency $F(n)=P_{n,n}$4 absorbs a fraction $F(n)=P_{n,n}$5 of the guided mode and transmits $F(n)=P_{n,n}$6 downstream, making a low-efficiency on-chip TES equivalent to a beam splitter of reflectivity $F(n)=P_{n,n}$7 followed by a perfect detector. With $F(n)=P_{n,n}$8 detectors in series, $F(n)=P_{n,n}$9, and undetected photons continue propagating for further processing or detection (Calkins et al., 2013).

The abstract multiplexed detector model uses a unitary beam-splitting network that maps a single input mode into $B_{n,m}$0 slices measured by ON–OFF detectors. In this framework, the $B_{n,m}$1-click POVM element is the symmetrized sum over all ways exactly $B_{n,m}$2 detectors click, and the estimation error in photon-number moments decreases inverse proportionally to the number of detectors: for any fixed moment order $B_{n,m}$3, $B_{n,m}$4 (Zhao et al., 8 Jul 2025). The same theory links enhanced multiplexed PNR to non-Gaussian state engineering: for two-photon subtraction with 7 dB of squeezing and an array of 20 detectors of efficiency 95%, the predicted fidelity is $B_{n,m}$5 with success probability $B_{n,m}$6 (Zhao et al., 8 Jul 2025).

Detector-tomographic comparisons of multiplexed architectures show that architecture matters beyond detector count alone. Spatial multiplexing, tree time-multiplexing, and loop time-multiplexing all realize PNR through click-pattern statistics, but more outcomes improve purity and information extraction only when the splitting network is well designed. For a flat prior and symmetric efficiencies, uniform splitting is optimal; for nonuniform priors, slightly nonuniform splitting can maximize mutual information (Schapeler et al., 2022).

Practical multiplexing has been implemented with FPGA timing electronics. One system used 7 active SNSPDs plus 1 laser-trigger channel, feeding an 8-channel TDC and an Asynchronous-Correlated-Digital-Counter implemented in a Xilinx Zynq-7020 SoC. It measured up to 8th-order single-photon correlations, reconstructed coherent-state photon statistics for average photon number up to 4, and achieved $B_{n,m}$7 after efficiency correction. Single-channel timing resolution was 22 ps r.m.s., and the full PL usage of TDC plus ACDC was about 30% with total PL power about 430 mW (Lusardi et al., 2018).

A more compact multiplexing variant uses a single click detector and a storage loop with tunable outcoupling. Here the loop traps an input pulse and taps a fraction $B_{n,m}$8 toward a single-photon detector on each round trip, while an FPGA-like controller updates $B_{n,m}$9 adaptively from prior click history. The reported dynamic-range extension factor

$C_{|n\rangle\to|n+1\rangle}=1-B_{n,n+1},$0

reaches up to $C_{|n\rangle\to|n+1\rangle}=1-B_{n,n+1},$1–10 relative to a passive loop, and the adaptive scheme can require $C_{|n\rangle\to|n+1\rangle}=1-B_{n,n+1},$2–50% fewer rounds than the passive scheme for a given mean-squared-error target (Sullivan et al., 2023).

A more radical alternative dispenses with optical mode multiplication entirely. Distributed coherent absorption places $C_{|n\rangle\to|n+1\rangle}=1-B_{n,n+1},$3 thin detector layers at standing-wave anti-nodes so that a single optical mode is absorbed uniformly across the array. For $C_{|n\rangle\to|n+1\rangle}=1-B_{n,n+1},$4 identical sublayers under counter-propagating illumination, the total absorption is

$C_{|n\rangle\to|n+1\rangle}=1-B_{n,n+1},$5

approaching unity as $C_{|n\rangle\to|n+1\rangle}=1-B_{n,n+1},$6. The correct-resolution probability scales as

$C_{|n\rangle\to|n+1\rangle}=1-B_{n,n+1},$7

and for realistic SNSPD parameters with $C_{|n\rangle\to|n+1\rangle}=1-B_{n,n+1},$8, numerical transfer-matrix simulation gave global absorption $C_{|n\rangle\to|n+1\rangle}=1-B_{n,n+1},$9 and absorption non-uniformity $88\% \pm 3\%$00 across layers (Vetlugin et al., 2022).

Linear waveguide multiplexing pursues a related goal in a traveling-wave geometry. In a single-mode waveguide carrying evanescently coupled SNSPDs, choosing detector absorption $88\% \pm 3\%$01 gives uniform absorption probabilities $88\% \pm 3\%$02, and the ideal fidelity becomes

$88\% \pm 3\%$03

With a thin-film lithium niobate platform, $88\% \pm 3\%$04, $88\% \pm 3\%$05, and a discrete array size $88\% \pm 3\%$06, the reported absolute fidelities including loss and dark counts were $88\% \pm 3\%$07, $88\% \pm 3\%$08, $88\% \pm 3\%$09, and $88\% \pm 3\%$10 (Limongi et al., 2024).

5. Waveform- and timing-domain architectures with a single detector

A recurrent misconception is that photon-number resolution requires either an intrinsically energy-resolving detector or a bank of independent pixels. Several recent architectures show that a single conventional detector can encode photon number in pulse timing or waveform shape.

One implementation uses a single telecom-band SNSPD together with ultra-high-resolution time tagging. A modelocked fiber laser at 1554 nm provides optical pulses, a fast photodiode provides a start time-stamp, and an ultra-high-resolution time-tagger records both the rising and falling edges of the SNSPD pulse relative to the photodiode trigger. The combined timing jitter per edge channel is $88\% \pm 3\%$11, and the two-dimensional histogram $88\% \pm 3\%$12 exhibits distinct clusters for $88\% \pm 3\%$13. The system detection efficiency was specified as $88\% \pm 3\%$14 for channel 1 and $88\% \pm 3\%$15 for channel 2 in two-detector experiments, dark count rate was $88\% \pm 3\%$16, and the effective maximum count rate was $88\% \pm 3\%$17 per channel (Sauer et al., 2023).

A different single-SNSPD architecture performs waveform inference rather than edge timing. Principal component analysis of digitized SNSPD traces shows that the first principal component approximately equals the time derivative of the average response trace,

$88\% \pm 3\%$18

and that almost all photon-number information is contained in the corresponding scalar projection. This reduces the full waveform to one dot product,

$88\% \pm 3\%$19

which can be implemented in streaming fashion on FPGA. The reported hardware requirements are a single 5 GSample/s A/D channel, about 3 GHz of analog bandwidth, and one FPGA multiply–accumulate block per event (Kuijf et al., 17 Nov 2025).

The significance of these results is architectural rather than merely algorithmic. Photon-number resolution emerges from a co-design of detector response, analog bandwidth, digitization rate, and classifier. In the waveform-PCA approach, the principal score is interpreted as a photon-dependent time shift; in the timing-tagger approach, the joint rise/fall-edge coordinates define a low-dimensional manifold with linear decision boundaries. Both cases replace explicit spatial multiplexing by information extraction from a single response channel (Sauer et al., 2023, Kuijf et al., 17 Nov 2025).

6. Trade-offs, applications, and design implications

The literature makes clear that no single architecture dominates all figures of merit. The comparison compiled for the segmented 32-pixel detector places TES devices at $88\% \pm 3\%$20–95% SDE at 1550 nm, resolution up to $88\% \pm 3\%$21 photons, count rate $88\% \pm 3\%$22, timing jitter $88\% \pm 3\%$23–100 ns, cooling at $88\% \pm 3\%$24, and SQUID-based readout. Prior multi-pixel SNSPDs are summarized there as reaching up to 24 pixels, resolution $88\% \pm 3\%$25, SDE $88\% \pm 3\%$26–90%, count rate $88\% \pm 3\%$27, and jitter $88\% \pm 3\%$28–100 ps, while time-multiplexed schemes reach up to $88\% \pm 3\%$29 photons at the cost of $88\% \pm 3\%$30–3 dB coupling loss per loop and count rates $88\% \pm 3\%$31 (Ding et al., 3 Apr 2025). These are not merely engineering details; they define the operational envelope of each PNR architecture.

Photon-number-resolving architecture is also not restricted to superconducting platforms. A SiPM-based detection chain using a 14 bit, 1 Gs/s acquisition system with FPGA-based baseline subtraction, digital deconvolution, and charge integration was characterized in the mesoscopic intensity regime. The reported system achieves true photon-number resolution up to $88\% \pm 3\%$32 photons per pulse at 1 MHz, with maximum resolvable photon number $88\% \pm 3\%$33 of $88\% \pm 3\%$34, $88\% \pm 3\%$35, and $88\% \pm 3\%$36 for the 50 CS, 25 PS, and 25 CS models, respectively; for coherent input, $88\% \pm 3\%$37 over $88\% \pm 3\%$38 at 1 MHz (Pozzoli et al., 19 May 2026). This suggests a distinct design regime in which PNR is optimized for mesoscopic quantum communication rather than single-photon heralding.

Applications track these architectural differences. Integrated waveguide TESs explicitly target on-chip state heralding, photon subtraction, large interferometric networks with embedded read-out, boson sampling, and cluster-state generation (Calkins et al., 2013). FPGA-multiplexed SNSPD systems reconstruct coherent-state statistics and high-order correlations (Lusardi et al., 2018), while parallel SNSPDs similarly reconstruct photon-number statistics of a coherent source using a single detector package (2207.14538). Timing-resolved single-SNSPD PNR has been used to measure coherent-state statistics and photon-number correlations of non-classical states, including $88\% \pm 3\%$39 N00N-state measurements (Sauer et al., 2023). In theoretical state engineering, multiplexed PNR detectors provide an advantage in cat-state breeding protocols, with the 20-detector, 95%-efficiency example above predicting fidelity $88\% \pm 3\%$40 and success probability $88\% \pm 3\%$41 (Zhao et al., 8 Jul 2025).

Two broader design lessons recur. First, more bins, pixels, or outcomes do not automatically yield better PNR: the splitting or absorption distribution must be uniform or optimized, whether in a beam-splitter tree, a loop detector, or a linear waveguide array (Schapeler et al., 2022, Limongi et al., 2024). Second, optical hardware and signal-processing hardware are becoming inseparable components of the architecture. TES nonlinear embeddings, FPGA real-time area extraction, principal-component readout of SNSPD traces, and adaptive outcoupling control all show that photon-number resolution is increasingly a hybrid optical-electrical-inferential system problem (Dalbec-Constant et al., 2024, Morais et al., 2020, Kuijf et al., 17 Nov 2025, Sullivan et al., 2023).

A plausible implication is that future PNR architectures will continue to blur the boundary between detector physics and readout inference. One reported expectation is that combining the high-efficiency twin-layer nanowire design with further multiplexing and edge-rising-slope analysis could push the resolution beyond 50 photons while retaining $88\% \pm 3\%$42 SDE and tens of MHz count rates (Ding et al., 3 Apr 2025). Another reported direction is that advanced calibrations could extend SiPM dynamic range beyond 50 photons (Pozzoli et al., 19 May 2026). The common trend is not convergence on a single device type, but increasing specialization of architectures to distinct operating points in efficiency, dynamic range, latency, wavelength, and integration density.