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Laser Physics Letters

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

G. V. Avosopiants, Konstantin G. Katamadze, N. A. Borshchevskaya, S. Svyatodukh, Roman V. Ozhegov, Bogdanov Yu, S. P. Kulik

Published Dec 1, 2025• 0 citations
Publisher

Abstract

Abstract 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.

DOI
10.1088/1612-202x/ae27aa

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Abstract 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.
We demonstrate quantum detector tomography of a commercial 2×2 array of superconducting nanowire single photon detectors. We show that detector-specific figures of merit including efficiency, dark-count and cross-talk probabilities can be directly extracted, without recourse to the underlying detector physics. These figures of merit are directly identified from just four elements of the reconstructed positive operator valued measure (POVM) of the device. We show that the values for efficiency and dark-count probability extracted by detector tomography show excellent agreement with independent measurements of these quantities, and we provide an intuitive operational definition for cross-talk probability. Finally, we show that parameters required for the reconstruction must be carefully chosen to avoid oversmoothing the data.
We develop a combinatorial and Bayesian framework to quantify the information gained by increasing the local photon-number resolution of multiplexed detection systems based on superconducting nanowire single-photon detectors (SNSPDs). Motivated by recent studies that attribute photon-number resolving capabilities to a single SNSPD overcoming the typical on/off detection mechanism, we generalize the existing formulation for the binary devices by deriving the probability response tensor of multiplexed systems based on SNSPDs with an arbitrary number of resolvable states. We include finite efficiency through binomial loss convolution and, separately, assess the impact of dark counts. From this response tensor we derive a generalized N-photon detector fidelity, a Bayesian photon-number reconstruction, and the corresponding reconstruction fidelity. We particularize this formulation to study 2-, 3- and 4-state multiplexed systems with different numbers of detectors and show that increasing local photon-number resolution systematically improves both detector fidelity and Bayesian photon-number reconstruction. This substantially relaxes the quadratic scaling between detector number and resolvable photon-number required by purely binary multiplexed arrays. These results provide a quantitative value-of-information assessment for multi-state photon-number-resolving detector arrays.
Photon-number-resolving (PNR) single-photon detectors are an enabling technology in many areas, such as photonic quantum computing, nonclassical light-source characterization, and quantum imaging.Here, we demonstrate high-efficiency PNR detectors using a parallel superconducting nanowire singlephoton-detector (P-SNSPD) architecture that does not suffer from crosstalk between the pixels and that is free of latching.The behavior of the detector is modeled and used to predict the possible outcomes given a certain number of incoming photons.We apply our model to a four-pixel P-SNSPD with a system detection efficiency of 92.5%.We also demonstrate how this detector allows reconstructing the photonnumber statistics of a coherent source of light, which paves the way towards the characterization of the photon statistics of other types of light source using a single detector.
Superconducting nanowire single-photon detectors (SNSPDs) have abstracted widespread attention in the fields of quantum information and quantum photonics due to their high detection efficiency and low dark count rates (DCR). However, single-pixel SNSPDs exist the notable limitations in detection speed and photon number resolution (PNR) capabilities, which pose challenges for practical applications. In this study, we propose a design of a series four-pixel SNSPD that incorporates a SPICE circuit simulation model to achieve PNR and spatial resolution of the four photons in an ideal state, and then determine the optical cavity structure of the detector at the highest light absorption of up to 97% through optical simulation. This design provides a theoretical foundation for the subsequent fabrication of large-area SNSPD with PNR and spatial resolution, significantly improving the flexibility and adaptability of SNSPDs to meet diverse detection requirements. Furthermore, our work also lays a research foundation for multi-photon detection in quantum communication and quantum computing.
Abstract Photon number resolving (PNR) capability is very important in many optical applications, including quantum information processing, fluorescence detection, and few-photon-level ranging and imaging. Superconducting nanowire single-photon detectors (SNSPDs) with a multipixel interleaved architecture give the array an excellent spatial PNR capability. However, the signal-to-noise ratio (SNR) of the PNR (SNRPNR) of the array will be degraded with increasing the element number due to the electronic noise in the readout circuit, which limits the PNR resolution as well as the maximum PNR number. In this study, a 16-element interleaved SNSPD array was fabricated, and the PNR capability of the array was investigated and analyzed. By introducing a level comparator circuit (LCC), the SNRPNR of the detector array was improved over a factor of four. In addition, we performed a statistical analysis of the photon number on this SNSPD array with LCC, showing that the LCC method effectively enhances the PNR resolution. Besides, the system timing jitter of the detector was reduced from 90 ps to 72 ps due to the improved electrical SNR.
We present an experimental realization of a 16 element, temporal-array, photon-number-resolving (PNR) detector, which is a multiplexed single-photon detector that splits an input signal over multiple time bins, and the time bins are detected using two superconducting nanowire single-photon detectors (SNSPD). A theoretical investigation of the PNR capabilities of the detector is performed and it is concluded that, compared to a single-photon detector, our array detector can resolve one order of magnitude higher mean photon numbers, given the same number of input pulses to measure. This claim is experimentally verified and we show that the detector can accurately predict photon numbers between ${10}^{\ensuremath{-}3}$ and 102{10}^{2}102. Our present detector is incapable of single-shot photon-number measurements with high precision since its effective quantum efficiency is 4949%49. Using SNSPDs with a higher quantum efficiency the PNR performance will improve, but the photon-number resolution will still be limited by the array size.