PNR detector tomography using an analytical unequal-channel model and a hybrid reconstruction approach

Though photon-number-resolving (PNR) detectors based on multiplexed threshold elements are key tools for quantum optics and quantum information, yet accurate calibration of their response in the presence of imperfections remains challenging. Here we develop a model for a four-channel PNR detector designed as a superconducting nanowire single-photon detector (SNSPD) array, allowing for unequal channel transmission coefficients and dark counts. Closed-form expressions for the click probabilities and transfer-matrix elements are obtained for coherent-state probes, and a numerical study quantifies the range of channel imbalance where the commonly used equal-channel, single-parameter approximation fails. Applying this framework to experimental data from a four-channel PNR SNSPD, we show that direct model-based fits do not reproduce the measured click statistics probably because of inter-channel cross-talks. To overcome this, we implement a hybrid quantum detector tomography scheme that combines a model-based fit to the mean click number with a data-driven reconstruction of the low-photon subspace. The resulting transfer matrix reproduces the experimental click statistics with average fidelity above 99.9% and provides a robust description of the detector in the presence of cross-talks and other non-idealities.