Evaluating the performance of photon-number-resolving detectors

We analyze the performance of photon-number-resolving (PNR) detectors and introduce a figure of merit for the accuracy of such detectors. This figure of merit is the (worst-case) probability that the photon-number-resolving detector correctly predicts the input photon number. Simulations of various PNR detectors based on multiplexed single-photon “click detectors” is performed. We conclude that the required quantum efficiency is very high in order to achieve even moderate (up to a handful) photon resolution, we derive the required quantum efficiency as a function of the the maximal photon number one wants to resolve, and we show that the number of click detectors required grows quadratically with the maximal number of photons resolvable.

Figure 1

Schematic image of a spatial array consisting of four detector elements with quantum efficiency $\eta$. An input signal equally distributed over the four detector elements, e.g., with a 1-to-4 fiber coupler.

Figure 2

Simulation result of the PNR quality for spatial arrays with negligible dark count rates. In panels (a) and (c), the PNR quality using the set of all probability distributions is presented for an array consisting of 8 and 32 elements, respectively. In panels (b) and (d), the PNR quality using the set of Poisson distributions with mean $\mu \le n$ is presented for an array consisting of 8 and 32 elements, respectively. Both array sizes show improvement in the quality when restricted to the set of Poisson distributions. However, reaching a quality larger than 0.5 for many photons still requires a very high quantum efficiency.

Figure 3

The number of detector elements $M$ needed get a PNR quality of $Q_n\ge 0.5$ in a spatial array with quantum efficiency $\eta =1$ and negligible dark counts as a function of the number of resolved photons. The simulated result is in agreement with the approximation in Eq. (13) for sufficiently large $M$.

Figure 4

PNR quality for a 16-element array with a quantum efficiency of $\eta =0.95$ as a function of the dark-count probability of each detector element. The general trend shows that the PNR quality decreases with increased dark counts; however, in some instances it is possible to have an increasing PNR quality.

Figure 5

Schematic image of a temporally multiplexed detector. The fibers between the 50:50 couplers are chosen so the detectors have time to recover between pulses and so a split signal can not recombine. Between coupler $k$ and $k+1$ the lower fiber has a short (but arbitrary) length $l$ and the upper fiber has length $2^{k-1}L+l$, where $L$ corresponds to the smallest length rendering two subsequent pulses that the detector can resolve.

Figure 6

Schematic image of a loop-multiplexed detector. Incoming light enters the multimode fiber end to the left and enters a splitting circuit in the middle. The probability for photons to exit to the detector is $T$ independent of which of the two input fibers was used by the photon. The remaining probability is that the photon enters the loop, where it has an ${\eta }_l$ probability to survive per loop.

Figure 7

PNR quality of a loop-multiplexed detector without dark counts. The probability that a photon survives the loop is ${\eta }_l=0.97$ and at most $l=32$ loops are allowed. (a) The PNR quality for all distributions is presented. Only two photons can be detected with a quality higher than 0.5. (b) The PNR quality for all Poisson distributions with mean $\mu \le n$. Compared to panel (a) the quality has improved significantly, yet it is still only possible to detect up to four photons even with a perfect detector.