Photon-Number Resolution in Quantum Detectors

• Photon-number resolution (PNR) is the capability of a detector to determine the exact number of photons in an optical pulse using methods like energy-resolving sensors, multiplexed arrays, or intrinsic timing analysis.
• PNR techniques enable precise characterization of photon statistics, which is critical for protocols in quantum communications, entanglement swapping, and quantum-enhanced metrology.
• Various architectures, including transition-edge sensors and superconducting nanowire arrays, demonstrate trade-offs in speed, efficiency, and resolution, guiding practical detector design.

Photon-number resolution (PNR) is the ability of a detector to discriminate how many photons were present in an optical pulse or mode, rather than only reporting whether at least one photon was detected. In ideal form, it corresponds to measurement outcomes $n=0,1,2,\dots$ with POVM elements $|n\rangle\langle n|$; in practice, it is realized by intrinsically energy-resolving detectors, by spatial or temporal multiplexing of ON-OFF detectors, or by intrinsic multi-photon discrimination in superconducting nanowire devices. Because many protocols rely on post-selection conditioned on specific photon-number outcomes or on precise characterization of photon-number statistics, PNR is central to entanglement swapping, teleportation, linear-optical quantum computing, Gaussian boson sampling, heralded state preparation, quantum-enhanced metrology, source characterization, and quantum communications (Sauer et al., 2023, Li et al., 2024, Zhao et al., 8 Jul 2025).

1. Operational meaning and measurement outcomes

In operational terms, PNR means mapping a detection event to an integer $n$ or to an estimate $\hat n$ that corresponds to the number of absorbed photons. Realistic devices do not perfectly resolve large $n$; they have a maximal resolvable number $N_{\max}$ with finite confidence, and different platforms realize this mapping through different observables (Jönsson et al., 2018).

One broad class is continuous-output PNR. Transition-edge sensors are described as phase-insensitive PNR detectors characterized by POVM elements diagonal in the Fock basis,

$$\Pi_n=\sum_{m=0}^{\infty}\theta_{nm}\,|m\rangle\langle m|,$$

where $\theta_{nm}$ is the conditional probability of reporting $n$ given $m$ incident photons (Li et al., 2024). A related continuous-outcome picture appears in intrinsic SNSPD PNR, where the measured variable can be arrival time, amplitude, slope, or a multidimensional feature vector derived from the pulse trace. The detector then implements a classification rule on regions of that continuous outcome space rather than directly returning a digitized energy (Schapeler et al., 14 May 2025).

A second broad class is multiplexed PNR. Here, the detector output is inferred by aggregating clicks from multiple ON-OFF detectors within a coincidence window or across many spatial, temporal, or frequency channels. In that setting, the output is not an intrinsic energy measurement; it is a statistical inference based on click patterns, coincidence windows, or binned time tags (Williamson et al., 2021, Zhao et al., 8 Jul 2025).

A third class comprises architectures that return only partially resolved outcomes. In arrival-time-based SNSPD PNR, events earlier than a final boundary can be grouped as “4+” when individual classes are no longer resolvable, so the outcome alphabet need not coincide with exact photon number for all $|n\rangle\langle n|$0 (Schapeler et al., 1 Oct 2025). This suggests that encyclopedic use of “PNR” includes exact low-$|n\rangle\langle n|$1 resolution, truncated resolution, and calibrated coarse-grainings, provided the detector output remains correlated with photon number in a single shot.

2. Statistical models and performance criteria

The starting point for most PNR analyses is the coherent-state distribution

$|n\rangle\langle n|$2

with mean photon number $|n\rangle\langle n|$3. With overall detection efficiency $|n\rangle\langle n|$4, detected counts follow a thinned distribution,

$|n\rangle\langle n|$5

which reduces to a Poisson law with mean $|n\rangle\langle n|$6 for coherent states (Sauer et al., 2023). In detector-characterization experiments this relation is used to compare measured photon-number histograms against calibrated Poisson predictions, and in tomography it provides the known input ensemble against which a detector response matrix is reconstructed (Li et al., 2024, Ding et al., 3 Apr 2025).

A general forward model writes the measured distribution as

$|n\rangle\langle n|$7

where $|n\rangle\langle n|$8 is the incident photon-number distribution and $|n\rangle\langle n|$9 is the detector response matrix or confusion matrix. This form appears in superconducting parallel nanowire detectors, in multiplexed click-detector arrays, and in amplitude-coded parallel SNSPDs (0902.4824, Jönsson et al., 2018, 2207.14538). In timestamp-based intrinsic PNR, an analogous likelihood is written over continuous kernels $n$0,

$n$1

and practical implementations often replace full maximum-likelihood estimation by decision regions and overlap integrals of fitted distributions (Sauer et al., 2023).

Several non-equivalent figures of merit coexist. For multiplexed ON-OFF architectures, Jönsson and Björk define the PNR quality $n$2 as the worst-case probability that the detector correctly predicts the input photon number, and show that $n$3 is monotone non-increasing in $n$4 (Jönsson et al., 2018). For interleaved SNSPD arrays, the photon-number-resolution SNR is defined as

$n$5

where $n$6 is the $n$7-peak position and $n$8 is its FWHM (Huang et al., 2023). For intrinsic SNSPD timing methods, a resolvability criterion is proposed: $n$9 with a Gaussian-limit form

$\hat n$0

This criterion is explicitly tied to the continuous distributions typical of latency- or amplitude-based SNSPD PNR (Schapeler et al., 14 May 2025, Schapeler et al., 1 Oct 2025).

The shape of the fitted response function is itself part of the performance model. In one timing-based SNSPD study, Voigt profiles were used to model projected timing histograms, separating a Gaussian instrument response from Lorentzian broadening (Sauer et al., 2023). In another, exponentially-modified Gaussian (EMG) fits were used because Gaussian-only models underestimate the long tail and hence underestimate misidentification probabilities; for $\hat n$1, for example, $\hat n$2 with EMG versus $\hat n$3 with Gaussian fits (Schapeler et al., 1 Oct 2025). This suggests that “PNR performance” is inseparable from the statistical model used to define overlap, threshold placement, and reconstructed POVMs.

3. Detector architectures and implementation classes

Conventional routes to PNR include transition-edge sensors, multiplexing with threshold detectors, and parallel or segmented SNSPD architectures (Sauer et al., 2023). These routes differ mainly in the physical observable that carries photon-number information: absorbed energy, click occupancy, amplitude step height, or latency shift.

Transition-edge sensors remain the canonical intrinsically energy-resolving architecture. They operate as ultralow-noise microcalorimeters with near-unity efficiency and high energy resolution, but their thermal recovery tail persists for microseconds, so the sensor should ideally start from the same baseline for each event. A recent machine-learning study showed that overlapping TES traces can still be classified at 500–800 kHz, extending accurate TES PNR operation to 800 kHz while maintaining accurate photon-number assignment up to at least five photons (Li et al., 2024).

Multiplexed ON-OFF detectors realize approximate PNR by distributing photons over many bins. Spatial arrays, temporal arrays, loop architectures, and adaptive storage loops all belong to this class (Jönsson et al., 2018, Jönsson et al., 2020, Sullivan et al., 2023). Intrinsic SNSPD PNR instead exploits measurable changes in a single detector’s response time, rising-edge slope, or amplitude when more than one photon is absorbed within a short time window (Schapeler et al., 14 May 2025).

Architecture	Physical basis	Representative result
TES with machine-learning processing	Pulse amplitude, area, and full-trace classification	800 kHz operation, at least a four-fold improvement, accurate photon-number assignment up to at least five photons (Li et al., 2024)
Temporal multiplexing with two SNSPDs	16 time bins measured with binary detectors	Accurate prediction of photon numbers between $\hat n$4 to $\hat n$5; effective quantum efficiency $\hat n$6 (Jönsson et al., 2020)
Parallel/segmented SNSPD architectures	Summed amplitude from multiple pixels	4-pixel P-SNSPD with SDE $\hat n$7 and $\hat n$8–4; segmented twin-layer SNSPD with system detection efficiency of $\hat n$9 and photon-number resolution of 32 (2207.14538, Ding et al., 3 Apr 2025)
Up-conversion plus SiPM	Telecom-to-visible conversion followed by multi-pixel APD readout	Overall detection efficiency $n$0; Poissonian statistics maintained up to approximately 20 simultaneous detections (Pomarico et al., 2010)
Adaptive storage loop with a single click detector	Tunable outcoupling and Bayesian/adaptive inference	Dynamic range extended by up to an order of magnitude relative to a purely passive setup (Sullivan et al., 2023)

Parallel-nanowire implementations are historically important because they provided intrinsic amplitude-coded PNR with a single electrical output proportional to photon number. The Parallel Nanowire Detector used subwavelength-scale spatial multiplexing, achieved pulse widths as low as 660 ps FWHM, showed counting performance at 80 MHz repetition rate, and exhibited no observable multiplication-noise buildup (0807.0526, 0902.4824). A later parallel SNSPD architecture in MoSi demonstrated a system detection efficiency of $n$1, $n$2 ps FWHM jitter at the single-photon level, recovery to more than $n$3 efficiency after $n$4 ns, and full recovery in about $n$5 ns (2207.14538).

Array architectures can also improve the electronic separability of photon-number levels. In a 16-element interleaved SNSPD array, a level comparator circuit converted noisy analog pulse heights into digital steps, improved the PNR SNR by more than a factor of four, and reduced system timing jitter from about 90 ps to about 72 ps (Huang et al., 2023). At the opposite end of the complexity spectrum, a room-temperature telecom detector based on up-conversion and a SiPM achieved only moderate efficiency, but preserved coherent-state Poisson statistics up to approximately 20 simultaneous detections and operated without cryogenic nanowire hardware (Pomarico et al., 2010).

4. Intrinsic SNSPD PNR: timing, amplitude, and electrothermal dynamics

Intrinsic SNSPD PNR relies on the fact that multi-photon absorption modifies the electrothermal turn-on dynamics of the nanowire. Absorption of one or more photons creates one or more resistive hotspots, triggering redistribution of the bias current and formation of a normal domain. Multi-photon absorption accelerates hotspot growth and enlarges the initial normal domain; as modeled in prior work on turn-on dynamics, the rising slope increases with photon number, so multi-photon events cross a fixed electrical threshold earlier. The same body of work emphasizes that the rising-edge separation between adjacent photon-number levels decreases roughly as $n$6 (Sauer et al., 2023, Schapeler et al., 14 May 2025).

Several readout strategies exploit that physics. A differentiated slew-rate readout converts the $n$7-dependent rising-edge slope into amplitude. Impedance-matching tapers make the analog pulse height itself strongly sensitive to the number of simultaneous hotspots. Waveform pattern matching, linear fits to the rising-edge slope, two-threshold timing, and PCA on full traces all extract related information from the same electrothermal event (Zhu et al., 2019, Schapeler et al., 14 May 2025). These methods differ in hardware overhead, but they share a common dependence on low jitter, short optical pulses, and a stable mapping between hotspot dynamics and the measured trace.

A key demonstration used a single conventional SNSPD in the telecom C-band and ultra-high-resolution time-tagging of both the rising and falling edges of the detector pulse. At $n$8 nm, with a Single Quantum EOS Series detector specified at $n$9 system detection efficiency and $N_{\max}$0 ps RMS jitter, plus a time-tagger contribution of $N_{\max}$1 ps RMS per channel, the combined time-difference jitter was about $N_{\max}$2 ps RMS. Distinct clusters in a two-dimensional histogram of rising versus falling edge time differences were visible up to at least $N_{\max}$3; classification by optimal projection and Voigt-profile fitting showed that using both edges significantly reduced cross-talk compared to using only the rising edge. The same detector was used to measure photon-number statistics of coherent light and joint photon-number distributions of non-classical states in a type-II SPDC experiment (Sauer et al., 2023).

A different intrinsic route uses amplitude-sensitive nanowire engineering. The superconducting tapered nanowire detector resolved up to five absorbed photons at $N_{\max}$4 nm, had $N_{\max}$5 ps timing jitter, $N_{\max}$6 c.p.s. device dark count rate, $N_{\max}$7 ns reset time, and $N_{\max}$8 system detection efficiency without cavity enhancement (Zhu et al., 2019). A segmented twin-layer NbN/SiO$N_{\max}$9/NbN SNSPD on a dielectric mirror pushed this architecture further: it exhibited a single-photon system detection efficiency of about $$\Pi_n=\sum_{m=0}^{\infty}\theta_{nm}\,|m\rangle\langle m|,$$0, photon-number resolution capability up to 32, count rate about $$\Pi_n=\sum_{m=0}^{\infty}\theta_{nm}\,|m\rangle\langle m|,$$1 MHz at $$\Pi_n=\sum_{m=0}^{\infty}\theta_{nm}\,|m\rangle\langle m|,$$2 of the maximum SDE, and timing jitter as low as $$\Pi_n=\sum_{m=0}^{\infty}\theta_{nm}\,|m\rangle\langle m|,$$3 ps for $$\Pi_n=\sum_{m=0}^{\infty}\theta_{nm}\,|m\rangle\langle m|,$$4 (Ding et al., 3 Apr 2025).

Intrinsic timing PNR is strongly affected by optical pulse shape and duration. A practical study using a commercial SNSPD at $$\Pi_n=\sum_{m=0}^{\infty}\theta_{nm}\,|m\rangle\langle m|,$$5 nm found that Gaussian spectral filtering yields much cleaner arrival-time histograms than a bandpass filter of equal bandwidth, that $$\Pi_n=\sum_{m=0}^{\infty}\theta_{nm}\,|m\rangle\langle m|,$$6 ps gives clean separation, that separation begins to degrade at $$\Pi_n=\sum_{m=0}^{\infty}\theta_{nm}\,|m\rangle\langle m|,$$7 ps, and that at $$\Pi_n=\sum_{m=0}^{\infty}\theta_{nm}\,|m\rangle\langle m|,$$8 ps the $$\Pi_n=\sum_{m=0}^{\infty}\theta_{nm}\,|m\rangle\langle m|,$$9 and $\theta_{nm}$0 peaks overlap strongly. In that system, EMG fits described the data over four orders of magnitude in count rate, reconstructed POVMs showed sharp photon-number-diagonal features, and the resolvability criterion yielded $\theta_{nm}$1 (Schapeler et al., 1 Oct 2025). By contrast, NbTiN SNSPDs optimized for $\theta_{nm}$2–$\theta_{nm}$3 nm demonstrated over $\theta_{nm}$4 system detection efficiency at $\theta_{nm}$5 nm, sub-11 ps timing jitter for one photon, sub-7 ps for two photons, and resolution up to 7 photons using conventional cryogenic readout circuitry (Los et al., 2024).

Kinetic inductance introduces a direct speed–resolution trade-off. In waveguide-integrated SNSPDs, increasing the nanowire kinetic inductance from $\theta_{nm}$6 nH to $\theta_{nm}$7 nH improved PNR quality by $\theta_{nm}$8, $\theta_{nm}$9, and $n$0 over the first three photon numbers, because larger $n$1 increased the latency separation between photon-number peaks. The same change reduced the detector’s count rate from $n$2 Mcps to $n$3 Mcps because the recovery time increased linearly with $n$4 (Jaha et al., 2024). This suggests that intrinsic SNSPD PNR is best understood as a circuit-engineering problem coupled to hotspot physics rather than as a purely material property.

5. Multiplexed PNR, scaling laws, and adaptive schemes

Multiplexed PNR begins from a binary detector model and distributes the input across many modes so that the total number of clicks approximates photon number. For a balanced array of $n$5 ideal ON-OFF detectors, the probability of obtaining exactly $n$6 clicks given $n$7 photons is

$n$8

where $n$9 are Stirling numbers of the second kind and $m$0 (Zhao et al., 8 Jul 2025). In a temporal array measured with two SNSPDs, coherent light with mean $m$1 per pulse gives a single-bin click probability

$m$2

and the maximum-likelihood estimate of $m$3 follows by inverting the binomial mean across $m$4 bins (Jönsson et al., 2020).

These models make the scaling limits explicit. Jönsson and Björk showed that the required quantum efficiency is very high in order to achieve even moderate photon resolution with multiplexed click detectors, and that the number of click detectors required grows quadratically with the maximal number of photons resolvable. For $m$5, their simulations gave $m$6 elements to resolve 5 photons with $m$7, and $m$8 elements to resolve 10 photons with $m$9 (Jönsson et al., 2018). A more recent theoretical framework proves that, for multiplexed PNR detectors, the estimation error in terms of photon-number moments decreases inverse proportionally to the number of detectors, i.e. as $|n\rangle\langle n|$00, under the paper’s assumptions (Zhao et al., 8 Jul 2025).

Experimentally, multiplexing is valuable even when single-shot fidelity is limited. A 16-element temporal array built from fiber couplers and two SNSPDs accurately predicted photon numbers between $|n\rangle\langle n|$01 and $|n\rangle\langle n|$02 for the same number of input pulses used to estimate the click statistics, but its effective quantum efficiency of $|n\rangle\langle n|$03 prevented high-precision single-shot PNR (Jönsson et al., 2020). Adaptive storage-loop PNR addresses a related limitation by changing the outcoupling fraction round by round. In that architecture, the click probability in bin $|n\rangle\langle n|$04 is

$|n\rangle\langle n|$05

and the control law chooses $|n\rangle\langle n|$06 to keep each bin informative. Simulations showed that adaptive feedback can extend dynamic range by up to an order of magnitude relative to a passive setup and can produce sub-shot-noise photon-number variance under favorable $|n\rangle\langle n|$07 and $|n\rangle\langle n|$08 (Sullivan et al., 2023).

Multiplexing also extends beyond beamsplitters and delay lines. A proposed PNR detector based on a cascade of waveguide-coupled $|n\rangle\langle n|$09-type emitters uses deterministic single-photon subtraction followed by ON-OFF detection. For a two-photon subtraction case, 7 dB of squeezing, and an array of 20 detectors of efficiency $|n\rangle\langle n|$10, the calculation predicts fidelity $|n\rangle\langle n|$11 with success probability $|n\rangle\langle n|$12, and the paper argues that megahertz-rate cat-state generation is achievable using an on-chip array of tens of ON-OFF detectors (Pasharavesh et al., 11 Jul 2025, Zhao et al., 8 Jul 2025).

6. Applications, interpretive issues, and outlook

Across architectures, PNR is used both as a detector primitive and as a characterization tool. In coherent-state characterization it measures truncated Poisson distributions and detector POVMs; in non-classical-state experiments it gives joint photon-number distributions $|n\rangle\langle n|$13, conditional moments, covariance, and bunching or anti-bunching signatures (Sauer et al., 2023, Li et al., 2024). Timing-based SNSPD PNR has already been used to observe photon-pair bunching at a single output port of a Hong–Ou–Mandel interferometer, and segmented or parallel architectures are explicitly motivated by linear-optical quantum computing, Gaussian boson sampling, quantum communications, heralded source characterization, $|n\rangle\langle n|$14 measurements, and precision metrology (Zhu et al., 2019, Ding et al., 3 Apr 2025).

Calibration remains an open technical issue. Coherent states are a robust reference, but coherent-state-based characterization cannot by itself reveal absolute loss; accurate attenuator calibration or a reference threshold detector is required for absolute detector tomography (Sauer et al., 2023). Timing-based intrinsic PNR additionally requires a precise optical timing reference, re-optimization of clustering boundaries to accommodate slow drifts, and a statistical model that captures non-Gaussian broadening (Sauer et al., 2023, Schapeler et al., 1 Oct 2025). In TES systems, overlap-aware supervised learning and unsupervised clustering reduce the penalty imposed by slow thermal recovery, but retraining or online adaptation may be needed if pulse shapes drift (Li et al., 2024).

The interpretation of multiplexed PNR has also been debated. One analysis argues that, for realistic detector parameters and commonly used coincidence windows, accidental coincidences dominate, so low-SNR multiplexed PNR outputs can be described by a classical wave model based on amplitude-threshold detection and coincidence aggregation. That work proposes that reliable PNR consistent with corpuscular light requires $|n\rangle\langle n|$15, $|n\rangle\langle n|$16, and operation away from saturation (Williamson et al., 2021). Other works, by contrast, focus on calibrated response matrices, POVMs, confusion matrices, and task-specific fidelity rather than on using PNR as stand-alone evidence for the discrete nature of light (Jönsson et al., 2018, Li et al., 2024). A plausible implication is that “PNR” now denotes a family of calibrated measurement schemes whose validity depends on the detector model, the regime of operation, and the figure of merit being reported.

Current development directions are consistent across platforms: further jitter reduction, optimized amplification and thresholding, more than two threshold levels, advanced statistical classifiers such as Bayesian or EM inference, on-chip timing references, integrated electronics, and combinations of intrinsic PNR with modest multiplexing to extend dynamic range (Sauer et al., 2023, Schapeler et al., 14 May 2025). In SNSPDs, Gaussian pulse shaping, EMG-based histogram modeling, larger or engineered kinetic inductance, and richer use of full-trace information all improve separability, but often at the cost of count rate (Schapeler et al., 1 Oct 2025, Jaha et al., 2024). In multiplexed systems, larger $|n\rangle\langle n|$17, higher efficiency, and lower dark counts directly tighten the $|n\rangle\langle n|$18 approximation error and expand the domain in which click statistics track photon-number moments faithfully (Zhao et al., 8 Jul 2025). The field therefore continues to evolve along two parallel lines: better intrinsic detectors and better inference on imperfect detectors.