Statistical uncertainty in background radiance measurements with single-photon detectors

Atmospheric background radiance is a critical factor in free-space Quantum Key Distribution (QKD) systems since it can significantly increase the Quantum Bit Error Rate (QBER). Accurate assessment of its impact on QKD requires measurements within the narrow spectral bandwidth and with detectors capable of sensing single photons used in QKD scenarios. These measurements serve as a valuable tool for site characterization, helping to asses the impact of background light at potential Quantum Optical Ground Station locations and to develop appropriate mitigation strategies. The single-photon detectors in the background light measurement setup need to be operated both at low detection rates at night and high detection rates by day. This is challenging because the dynamic range of sky background radiance spans roughly seven orders of magnitude. It is therefore crucial to accurately determine the statistical uncertainty of the photon rate measurements for incident background radiance of any intensity. Only then can a full error budget for the measurement results be confidently constructed. This contribution presents analytic results, obtained using Bayesian techniques, for the statistical uncertainty of rate measurements with realistic counting detectors including the effects of a finite detector dead time and a finite dark count rate. The effects of the after-pulsing probability and the detection efficiency are also discussed. In addition, these results give a formula for the measurement duration required to attain a given measurement accuracy, which can be used to optimize the ‘hemisphere scan time’ of a background radiance measurement instrument in real-time. The results are generic and can also be applied to other types of counting detectors, such as Geiger-Müller counters.