Photon-number-resolving detector

A photon-number-resolving (PNR) detector produces an output from which the number of photons absorbed in one pulse or time window can be read, rather than the binary "no photon / at least one photon" answer of an ordinary photon-counting detector. Two families exist. Intrinsic PNR detectors, chiefly the superconducting transition-edge sensor (TES), give a signal whose height is proportional to the absorbed energy, and therefore to the number of photons; tungsten TESs have reached 95% detection efficiency at 1556 nm. Multiplexed detectors split the light among many click detectors, such as the pixels of a SPAD array, several superconducting nanowires, or time bins in a fiber loop, and count how many fire.

Transition-edge sensors

A TES is a thin superconducting film held on its resistive transition, typically at around 100 mK, by a voltage bias. An absorbed photon heats the film and raises its resistance; the drop in current is read by a SQUID amplifier. A photon at 1550 nm carries 0.80 eV, and the pulse height for $n$ photons is close to $n$ times that of one photon, so a histogram of pulse heights shows separate peaks for 0, 1, 2 and more photons. Their separation degrades as the number rises, because the energy resolution is fixed while the signal saturates as the film is driven further up the transition, so in practice the number is resolved up to roughly ten photons at near-infrared wavelengths.

The price is speed and cryogenics. The thermal recovery takes of order a microsecond, the timing jitter is far larger than that of nanowires, and dilution or adiabatic demagnetization refrigeration is needed. TES detectors are therefore used where the number resolution and efficiency matter more than count rate, such as heralding and characterizing nonclassical states.

Multiplexed click detectors

If $n$ photons are spread uniformly over $N$ detector elements, the probability that each lands on a different element, so that all are counted, is

$$P_\text{distinct} = \frac{N!}{(N-n)!\,N^n}.$$

For $N = 8$, two photons are resolved with probability 0.875 and three with 0.656; with $N = 16$ the values rise to 0.938 and 0.820. The remainder are undercounted, because two photons on one element give a single click. Spatial multiplexing uses arrays of SPADs (a silicon photomultiplier is such an array read as one analog output) or interleaved nanowire segments; temporal multiplexing splits a pulse into a sequence of delayed copies in fiber and sends them to one or two detectors. Some nanowire detectors also resolve small photon numbers from the rise time or amplitude of a single pulse.

Efficiency and measured statistics

Every PNR detector is lossy, and loss is indistinguishable from fewer photons arriving. With per-photon efficiency $\eta$, an $n$-photon state is fully registered with probability $\eta^n$: for $\eta = 0.9$, one photon is detected with probability 0.9, two with 0.81 and four with 0.656. The measured distribution is the true one passed through a binomial loss channel, convolved with crosstalk, dark counts and, for multiplexed detectors, the undercounting above. Reconstructing the photon statistics of a source therefore requires a calibrated model of the detector, often obtained by detector tomography with coherent states of known mean photon number.

Where they are used

• Heralded single-photon sources: an idler detector that resolves two photons can veto multiphoton events, improving the purity of the heralded state.
• Photonic quantum computing and Gaussian boson sampling, which require the photon number in each output mode.
• Measuring the number distribution of squeezed light and other nonclassical states.
• Low-light classical metrology, where a calibrated number response extends the dynamic range of counting.

Pitfalls

Treating a multiplexed detector as ideal biases the inferred $g^{(2)}$ and photon-number distribution, especially at mean photon numbers comparable to $N$. Crosstalk between SPAD pixels adds false extra counts that mimic multiphoton events. For TESs, slow drifts in bath temperature shift the pulse heights, so the number thresholds need periodic recalibration.

Common questions

What is the difference between a single-photon detector and a photon-number-resolving detector?

A single-photon detector such as a SPAD or a single nanowire gives the same click for one photon or many arriving within its response time. A PNR detector distinguishes these cases, at least up to some maximum number.

Can an SNSPD resolve photon number?

A single meander nanowire is essentially a click detector, though some designs extract small numbers from pulse shape. Arrays of independent nanowires, read individually or through a combined output, give multiplexed number resolution.

References: A. E. Lita, A. J. Miller and S. W. Nam, "Counting near-infrared single-photons with 95% efficiency," Opt. Express 16, 3032 (2008); R. H. Hadfield, "Single-photon detectors for optical quantum information applications," Nat. Photonics 3, 696 (2009); L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).