Photon-Number-Resolved Spectroscopy

• Photon-number-resolved spectroscopy is a technique that measures the complete photon-number distribution P(n) across spectral, temporal, or modal coordinates.
• It employs advanced detectors like TES and SNSPDs, often combined with dispersive elements, to capture detailed statistical and temporal information of light fields.
• This method underpins research in quantum optics by distinguishing photon statistics, resolving state transitions, and enabling the computation of high-order correlation functions.

Photon-number-resolved spectroscopy (PNRS) is the direct measurement of the full photon-number distribution $P(n)$ of a light field, resolved across a spectral, temporal, or modal coordinate, rather than an indirect inference from intensity alone or from a small set of low-order correlations. In this sense, PNRS extends conventional spectroscopy from mean intensity $I(\lambda)$ to the complete discrete statistics $P(n \mid \lambda)$, from which higher-order moments and correlation functions can be computed. A distinct but related usage denotes spectra in which the resonance structure itself resolves the photon number of a quantized control field, as in photon-number-resolved Autler–Townes splitting. Contemporary realizations span transition-edge sensors (TES), superconducting nanowire single-photon detectors (SNSPDs) operated in single-pixel PNR modes, multiplexed detector networks, and integrated waveguide arrays (Klaas et al., 2018, Sauer et al., 2023, Ding et al., 2017).

1. Statistical content and observables

The central object in PNRS is the photon-number distribution $P(n)$ in a defined temporal or spatial mode. For coherent light, the distribution is Poissonian,

$P(n)=e^{-\mu}\mu^n/n!,$

with mean $\mu$. For single-mode thermal light, the distribution is geometric,

$P(n)=\frac{\bar n^n}{(1+\bar n)^{n+1}},$

and for multimode thermal light it becomes negative binomial,

$$P(n)=\binom{n+M-1}{n}\left(\frac{\bar n}{\bar n+M}\right)^n\left(\frac{M}{\bar n+M}\right)^M.$$

Because $P(n)$ contains the full factorial-moment hierarchy, PNRS can recover correlation functions of arbitrary order; in particular,

$$g^{(k)}(0)=\frac{\sum_n \prod_{i=0}^{k-1}(n-i)P_n}{\left(\sum_n nP_n\right)^k},$$

and

$I(\lambda)$0

This is why PNRS complements, rather than merely duplicates, intensity-correlation measurements: conventional $I(\lambda)$1 experiments typically access only low orders, whereas PNRS provides the full distribution from which any order can be computed (Klaas et al., 2018).

Loss enters through binomial thinning. If $I(\lambda)$2 is the incident distribution and $I(\lambda)$3 the detection efficiency, the measured distribution $I(\lambda)$4 obeys

$I(\lambda)$5

This relation is especially consequential in spectroscopy, because wavelength-dependent loss, routing loss, and detector efficiency can reshape the detected $I(\lambda)$6 unless they are either calibrated or incorporated into a forward model (Sauer et al., 2023).

The consequence is methodological as much as conceptual. A spectrum of $I(\lambda)$7 alone cannot distinguish a Poissonian field from a sub-Poissonian or super-Poissonian field with the same mean. PNRS can. In driven-dissipative condensates, weak emitters, and spectrally filtered quantum sources, this distinction is often the primary physical observable rather than a secondary diagnostic (Klaas et al., 2018).

2. Detector mechanisms and readout strategies

TES-based PNRS is calorimetric. A TES is biased in its superconducting transition, so absorption of optical energy $I(\lambda)$8 raises the sensor temperature by $I(\lambda)$9, changes the resistance, and generates a pulse whose area is proportional to absorbed energy. For a monochromatic mode, $P(n \mid \lambda)$0, so photon number is inferred from pulse area or an equivalent energy estimator. Real-time TES processing has been demonstrated with a reconfigurable FPGA that digitizes the TES output, extracts pulse features during acquisition, and classifies events on the fly. In the reported implementation, photon number was resolved up to $P(n \mid \lambda)$1 at $P(n \mid \lambda)$2 nm, with parts-per-billion discrimination for low photon numbers, <500 ns on-chip feature latency, and a data-rate reduction of approximately $P(n \mid \lambda)$3 relative to full-stream capture (Morais et al., 2020).

SNSPD-based PNRS proceeds differently. In conventional single-pixel SNSPDs, photon-number information can be encoded in waveform timing and shape rather than calorimetric pulse area. Ultra-high-resolution time-tagging of a single low-jitter SNSPD shows that multi-photon absorption produces a larger initial normal region and steeper current diversion, advancing the rising threshold crossing; the falling edge also carries information. In telecom C-band operation at $P(n \mid \lambda)$4 nm, a two-dimensional histogram of trigger-to-rising-edge and trigger-to-falling-edge delays forms distinct timing clusters corresponding to photon number. With cryogenic amplification and a time-tagger of $P(n \mid \lambda)$5 ps RMS channel jitter, the primary SNSPD channel exhibited $P(n \mid \lambda)$6 ps RMS jitter, and single-shot classification was demonstrated up to $P(n \mid \lambda)$7, with clear distinguishability up to $P(n \mid \lambda)$8 by visual inspection (Sauer et al., 2023).

A related single-pixel SNSPD route uses direct threshold timing at visible and near-infrared wavelengths. NbTiN SNSPDs optimized for $P(n \mid \lambda)$9–$P(n)$0 nm have been shown to reach $P(n)$1 system detection efficiency at $P(n)$2 nm, sub-11 ps timing jitter for one photon, sub-7 ps for two photons, and photon-number resolution up to $P(n)$3 photons using conventional cryogenic electrical readout circuitry. The discrimination variable is the threshold-crossing time of the amplified waveform relative to a photodiode trigger; the relevant electrothermal scaling is reported as $P(n)$4 and $P(n)$5 (Los et al., 2024).

Another variant replaces direct electronic time discrimination with optical sampling. In an optically sampled superconducting-nanostrip PNRD, the SNSPD’s RF waveform drives a dual-output Mach–Zehnder modulator, and a synchronized ultrashort optical pulse samples the waveform at a chosen delay. Near balanced bias, the sampled amplitude becomes highly sensitive to picosecond-order differences in the SNSPD rising edge. The demonstrated system achieved a temporal resolution of $P(n)$6 ps, with an SNSPD timing jitter of $P(n)$7 ps FWHM in that setup, and enabled real-time discrimination of one- and two-photon events (Endo et al., 2024).

Finally, PNR can be engineered structurally rather than inferred from waveform dynamics. A segmented twin-layer SNSPD with spatial multiplexing reported average system detection efficiency $P(n)$8 at $P(n)$9 nm, dark count rate $P(n)=e^{-\mu}\mu^n/n!,$0 cps, photon-number resolution up to $P(n)=e^{-\mu}\mu^n/n!,$1, and count rates of $P(n)=e^{-\mu}\mu^n/n!,$2 MHz at the $P(n)=e^{-\mu}\mu^n/n!,$3 dB SDE point. In that architecture, pulse amplitude rather than timing carries the photon-number information (Ding et al., 3 Apr 2025).

3. Spectral architectures for PNRS

Detector physics alone does not define a spectrometer. PNRS becomes spectroscopic when wavelength is mapped into detection channels by a dispersive or filtering stage. A straightforward architecture places a dispersive element such as a fiber Bragg grating, arrayed waveguide grating, grating, or prism before the PNR detector, so that distinct wavelengths feed distinct ports or time-tagging channels. In the single-SNSPD timing-based scheme, this immediately enables per-channel $P(n)=e^{-\mu}\mu^n/n!,$4, per-channel $P(n)=e^{-\mu}\mu^n/n!,$5, and cross-channel $P(n)=e^{-\mu}\mu^n/n!,$6. In this configuration the detector timing does not set spectral resolution; the dispersive device does, while picosecond timing preserves per-pulse number assignment and inter-channel synchronization (Sauer et al., 2023).

TESs enable a second spectroscopic mode because they are intrinsically energy dispersive. Their pulse area is proportional to total absorbed energy even when the TES briefly saturates into the normal phase, which permits direct energy spectroscopy and photon-number resolution in the same device. The real-time TES processor described for $P(n)=e^{-\mu}\mu^n/n!,$7–$P(n)=e^{-\mu}\mu^n/n!,$8 eV operation therefore supports live histograms of energy and number-resolved spectra, immediate separation of single- from multi-photon absorption, and real-time computation of quantities such as $P(n)=e^{-\mu}\mu^n/n!,$9 (Morais et al., 2020).

Multiplexed architectures remain important when intrinsic PNR is unavailable or when the dynamic range must be expanded. A 16-element temporal array based on two SNSPDs splits each pulse into 16 non-overlapping time bins and estimates mean photon numbers between approximately $\mu$0 and $\mu$1 photons per pulse when averaging over $\mu$2 pulses. Its effective overall efficiency was $\mu$3, which limited high-precision single-shot PNR, but the architecture extended the resolvable mean photon number by roughly one order of magnitude relative to a single detector (Jönsson et al., 2020). An adaptive storage-loop method with a single click detector and tunable outcoupling pushes the same logic further: by selecting the per-pass outcoupling probability $\mu$4 according to an information-gain criterion, the dynamic range can in certain situations be extended by up to an order of magnitude relative to a purely passive setup (Sullivan et al., 2023).

Integrated spectroscopic platforms make these ideas scalable. A linearly multiplexed SNSPD array on a single-mode waveguide implements sequential absorption with engineered efficiencies $\mu$5 in the lossless optimum, giving ideal-system fidelity

$\mu$6

The same work gives a propagation-loss and dark-count model suitable for per-channel spectroscopy at $\mu$7, including the loss ratio $\mu$8 and the dark-count factor $\mu$9 (Limongi et al., 2024). A different non-multiplexed route uses distributed coherent absorption: a standing-wave or Salisbury-screen geometry distributes a single optical mode uniformly across an array of on-off detectors, eliminating optical mode multiplication while allowing arbitrarily high photon-number discrimination efficiency in principle as the number of detectors increases (Vetlugin et al., 2022).

4. Experimental realizations and scientific use

A canonical use of PNRS is the direct observation of state transitions in emitted light. In a microcavity exciton-polariton condensate, TES-based PNRS measured the full photon-number distribution across excitation powers from $P(n)=\frac{\bar n^n}{(1+\bar n)^{n+1}},$0 to $P(n)=\frac{\bar n^n}{(1+\bar n)^{n+1}},$1. The measured $P(n)=\frac{\bar n^n}{(1+\bar n)^{n+1}},$2 evolved from near-exponential thermal statistics to a quasi-Poissonian distribution above threshold. From the TES-derived distributions, $P(n)=\frac{\bar n^n}{(1+\bar n)^{n+1}},$3, $P(n)=\frac{\bar n^n}{(1+\bar n)^{n+1}},$4, and $P(n)=\frac{\bar n^n}{(1+\bar n)^{n+1}},$5 were extracted; $P(n)=\frac{\bar n^n}{(1+\bar n)^{n+1}},$6 dropped from near $P(n)=\frac{\bar n^n}{(1+\bar n)^{n+1}},$7 toward $P(n)=\frac{\bar n^n}{(1+\bar n)^{n+1}},$8 as the pump increased, while a displaced-thermal-state model quantified the thermal and coherent fractions. The ratio $P(n)=\frac{\bar n^n}{(1+\bar n)^{n+1}},$9 reached a plateau of about $$P(n)=\binom{n+M-1}{n}\left(\frac{\bar n}{\bar n+M}\right)^n\left(\frac{M}{\bar n+M}\right)^M.$$0 at high power (Klaas et al., 2018).

A broader demonstration of statistics-first spectroscopy was carried out with a 100-pixel superconducting nanowire PNR detector on a Si$$P(n)=\binom{n+M-1}{n}\left(\frac{\bar n}{\bar n+M}\right)^n\left(\frac{M}{\bar n+M}\right)^M.$$1N$$P(n)=\binom{n+M-1}{n}\left(\frac{\bar n}{\bar n+M}\right)^n\left(\frac{M}{\bar n+M}\right)^M.$$2 waveguide at $$P(n)=\binom{n+M-1}{n}\left(\frac{\bar n}{\bar n+M}\right)^n\left(\frac{M}{\bar n+M}\right)^M.$$3 nm. That device resolved up to $$P(n)=\binom{n+M-1}{n}\left(\frac{\bar n}{\bar n+M}\right)^n\left(\frac{M}{\bar n+M}\right)^M.$$4 photons, used $$P(n)=\binom{n+M-1}{n}\left(\frac{\bar n}{\bar n+M}\right)^n\left(\frac{M}{\bar n+M}\right)^M.$$5 ns slot spacing between pixels, and directly measured high-order correlation functions $$P(n)=\binom{n+M-1}{n}\left(\frac{\bar n}{\bar n+M}\right)^n\left(\frac{M}{\bar n+M}\right)^M.$$6 up to $$P(n)=\binom{n+M-1}{n}\left(\frac{\bar n}{\bar n+M}\right)^n\left(\frac{M}{\bar n+M}\right)^M.$$7. For an amplified-spontaneous-emission source spectrally filtered to $$P(n)=\binom{n+M-1}{n}\left(\frac{\bar n}{\bar n+M}\right)^n\left(\frac{M}{\bar n+M}\right)^M.$$8 MHz and temporally gated by an electro-optic modulator, the detector observed the transition from Bose–Einstein to Poisson statistics as the time–bandwidth product $$P(n)=\binom{n+M-1}{n}\left(\frac{\bar n}{\bar n+M}\right)^n\left(\frac{M}{\bar n+M}\right)^M.$$9 was increased. At $P(n)$0, the measured $P(n)$1 was $P(n)$2, close to the single-mode thermal value $P(n)$3. The same platform also showed photon-subtraction-induced photon-number enhancement and quantum-limited discrimination between coherent and thermal states (Cheng et al., 2022).

Single-pixel SNSPD PNRS has also been used in explicitly non-classical experiments. In the telecom C band, trigger-referenced timing clusters from a single low-jitter SNSPD were used to measure the photon-number statistics of coherent light with tunable mean photon number and to reconstruct joint photon-number distributions for $P(n)$4 N00N states generated by type-II SPDC in a Hong–Ou–Mandel interferometer. The SPDC/HOM experiment used two SNSPD channels, with the second channel having estimated detection probability $P(n)$5 and $P(n)$6 ps RMS jitter; the overall system efficiency was approximately $P(n)$7, inferred from singles versus pair detection rates (Sauer et al., 2023).

Optically sampled SNSPD-PNRDs have been used as heralding detectors for non-classical state generation. In a photon-subtraction experiment on squeezed vacuum, exact number conditioning improved non-classicality relative to an ON–OFF herald. For single-photon subtraction in the higher-pump configuration, the Wigner value at the origin changed from $P(n)$8 for ON–OFF heralding to $P(n)$9 for PNR heralding. Two-photon subtraction was also realized, producing the expected even-parity structure (Endo et al., 2024).

5. Photon-number-resolved spectral structure in light–matter systems

In a distinct theoretical tradition, photon-number-resolved spectroscopy refers not to detector readout but to spectral lines whose positions encode the photon number of a quantized control field. For a $$g^{(k)}(0)=\frac{\sum_n \prod_{i=0}^{k-1}(n-i)P_n}{\left(\sum_n nP_n\right)^k},$$0-type three-level system with states $$g^{(k)}(0)=\frac{\sum_n \prod_{i=0}^{k-1}(n-i)P_n}{\left(\sum_n nP_n\right)^k},$$1, $$g^{(k)}(0)=\frac{\sum_n \prod_{i=0}^{k-1}(n-i)P_n}{\left(\sum_n nP_n\right)^k},$$2, and $$g^{(k)}(0)=\frac{\sum_n \prod_{i=0}^{k-1}(n-i)P_n}{\left(\sum_n nP_n\right)^k},$$3, a quantized cavity control field couples $$g^{(k)}(0)=\frac{\sum_n \prod_{i=0}^{k-1}(n-i)P_n}{\left(\sum_n nP_n\right)^k},$$4 while a classical probe addresses $$g^{(k)}(0)=\frac{\sum_n \prod_{i=0}^{k-1}(n-i)P_n}{\left(\sum_n nP_n\right)^k},$$5. The dressed-state energies are

$$g^{(k)}(0)=\frac{\sum_n \prod_{i=0}^{k-1}(n-i)P_n}{\left(\sum_n nP_n\right)^k},$$6

which imply probe resonance frequencies

$$g^{(k)}(0)=\frac{\sum_n \prod_{i=0}^{k-1}(n-i)P_n}{\left(\sum_n nP_n\right)^k},$$7

On cavity resonance, $$g^{(k)}(0)=\frac{\sum_n \prod_{i=0}^{k-1}(n-i)P_n}{\left(\sum_n nP_n\right)^k},$$8, these reduce to

$$g^{(k)}(0)=\frac{\sum_n \prod_{i=0}^{k-1}(n-i)P_n}{\left(\sum_n nP_n\right)^k},$$9

The $I(\lambda)$00 scaling is the spectral signature of photon number (Ding et al., 2017).

This formulation also distinguishes vacuum induced transparency (VIT) from vacuum induced Autler–Townes splitting (ATS). Under resonant control, the threshold is

$I(\lambda)$01

For $I(\lambda)$02, the system is in the VIT regime: resonances overlap at line center and no photon-number-resolved spectrum appears even for finite mean photon number in the control field. For $I(\lambda)$03, the system enters the vacuum-induced ATS regime, in which the resonances split and photon-number-resolved peaks can be resolved provided their separation exceeds the linewidth. The paper states a practical resolvability criterion as

$I(\lambda)$04

This use of the term shows that PNRS can denote either detector-enabled reconstruction of $I(\lambda)$05 or number-resolved spectral response of a strongly coupled quantum system (Ding et al., 2017).

6. Error sources, calibration, and design limits

PNRS is limited by a combination of loss, timing uncertainty, saturation, and model mismatch. In timing-based SNSPD schemes, the instrument response function is effectively the convolution of detector jitter, amplifier bandwidth distortion, and time-tagger jitter. In the telecom single-SNSPD implementation, marginal timing distributions along the optimized projection axis were fit by sums of Voigt profiles, and cross-talk was quantified by overlap integrals between neighboring bins. The use of both rising and falling edge timings relative to the trigger significantly reduced cross talk compared with using the rising edge alone (Sauer et al., 2023).

Optical pulse shape can be equally decisive. A dedicated study of arrival-time-based photon-number assignment with SNSPDs at $I(\lambda)$06 nm found that Gaussian temporal pulse shapes yield cleaner arrival-time histograms than bandpass-filtered pulses of equal bandwidth. For Gaussian-shaped spectra, $I(\lambda)$07 nm produced a calculated temporal FWHM of approximately $I(\lambda)$08 ps and the cleanest separation; $I(\lambda)$09 nm gave approximately $I(\lambda)$10 ps and visible degradation; $I(\lambda)$11 nm gave approximately $I(\lambda)$12 ps and significantly reduced discrimination. That work also showed that exponentially modified Gaussian distributions are required for realistic error estimates: with approximately $I(\lambda)$13 ps optical pulses, the EMG model gave $I(\lambda)$14 and $I(\lambda)$15 for $I(\lambda)$16, and $I(\lambda)$17 and $I(\lambda)$18 for $I(\lambda)$19. A Gaussian-only model underestimated these errors (Schapeler et al., 1 Oct 2025).

Multiplexed spectrometers face different constraints. In temporal arrays, efficiency and bin count jointly limit single-shot performance: with $I(\lambda)$20 bins and overall efficiency $I(\lambda)$21, robust single-shot discrimination of more than about one photon was not feasible, even though mean photon numbers over five decades could be estimated accurately by multi-pulse statistics (Jönsson et al., 2020). In linearly multiplexed waveguide arrays, propagation loss and total nanowire length degrade fidelity through the loss ratio $I(\lambda)$22 and the dark-count factor $I(\lambda)$23 (Limongi et al., 2024). In adaptive storage loops, performance depends on per-bin dark-count probability $I(\lambda)$24, detector dead time, and whether the control update can be completed within one loop round trip (Sullivan et al., 2023).

Across detector classes, the principal trade-off is between dynamic range, efficiency, temporal resolution, and operational speed. TESs provide intrinsically energy-dispersive PNRS and resolve up to $I(\lambda)$25 in real time in the cited implementation, but their thermal decays of $I(\lambda)$26–$I(\lambda)$27 constrain practical count rates to the kHz regime (Morais et al., 2020). Fast SNSPD platforms provide picosecond timing and high repetition rates, but their dynamic range is typically set by timing-cluster overlap, amplitude-cluster overlap, or the number of engineered pixels (Sauer et al., 2023, Ding et al., 3 Apr 2025).

The near-term trajectory in the literature is therefore mixed rather than singular. Proposed improvements include shorter reset times and tailored cryogenic amplifiers for timing-based SNSPD PNR, multi-threshold evaluation and signal pre-conditioning for more robust classification, multi-pixel or hybrid architectures to expand dynamic range, and on-chip spectrometers that map wavelength to output ports while preserving per-port PNR (Sauer et al., 2023). High-efficiency segmented SNSPDs already point toward $I(\lambda)$28-level amplitude readout at $I(\lambda)$29 nm with $I(\lambda)$30 average system detection efficiency, while distributed coherent absorption and coherently interacting nanoscale detectors suggest routes to large-$I(\lambda)$31 PNRS without conventional optical mode multiplication (Ding et al., 3 Apr 2025, Vetlugin et al., 2022, Young et al., 2019). A plausible implication is that PNRS will continue to divide into two technologically distinct regimes: energy-dispersive microcalorimetry for direct spectral readout, and ultrafast number-discriminating nanowire platforms for externally dispersed or time-resolved quantum spectroscopy.