# Quantum Calibration of Photon-Number-Resolving Detectors Based on Multi-Pixel Photon Counters  
[PERSON]  
1 State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China1  
[PERSON]  
1 State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China1  
[PERSON]  
1 State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China1  
[PERSON]  
1 State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China1  
[PERSON]  
1 State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China1  
repetition rate [30]. In the visible regime, the multi-pixel photon counter (MPPC) is usually used as a convenient PNRD based on spatial multiplexing [31]. MPPC has many advantages such as excellent photon-number-resolving capability, great response speed, high time resolution, and wide spectral response range. The compactness of MPPC enables it to be packaged into different devices for different demands and be applied to many research fields [32]. MPPC can also be combined with frequency up-conversion systems to achieve the infrared PNRD [6; 33; 34].  
To characterize the detection performance of MPPCs, the traditional parameters include detection efficiency, dark count rate, after pulsing probability, crosstalk probability, etc. Among them, the detection efficiency is determined by the materials and the coating on the detector. And at different wavelengths, the detection efficiency varies. In the past, partial calibrations or elaborate models of a detector were given based on these traditional parameters, which can no longer meet the requirements of describing its quantum characteristics completely and accurately. Since the quantum detector is employed in more and more complex situations, [PERSON] et al. first realized the tomography of quantum detectors in 2009 [35]. PNRD was first fully characterized by the so-called quantum detector tomography (QDT). In quantum mechanics, POVM elements can be used to describe quantum detectors. In previous work [36], the quantum state tomography (QST) has been demonstrated with MPPC, in which the quantum property of the detector was considered as a known fact. The quantum state of the incident photons was reconstructed according to the QST, but in QDT, the quantum property of the detector is unknown and the incident photon states are well-known. In 2013, the time-multiplexed superconducting nanowire single photon detector has already been well characterized by QDT at 1550 nm [37]. Comparing with the previous model description methods [38], we can characterize the quantum properties of MPPC based on spatial-multiplexing technique precisely and comprehensively through the reconstructed POVM elements at two different wavelengths. It will facilitate the application of precisely calibrated MPPC in the field of quantum information. Under very few preconditions, QDT can realize the \"black box\" calibration without model construction and calibration of specific parameters such as detection efficiency. The characterized detectors could also be used to prepare various non-Gaussian states [26].  
Here, we characterized the photon-number-resolving capability of MPPC by reconstructing the POVM elements using the method of QDT. We obtained its quantum features at the wavelengths of 791 nm and 523 nm, respectively. Quantum calibration of this kind of detector at different wavelengths provides an experimental and theoretical basis for the applications of MPPC in quantum optics and quantum sensing.