Optimized controlled phase gate implementation and efficiency characterization for a microwave single-photon detector

Microwave single-photon detectors are critical in quantum information processing and dark matter detection. The detector based on a controlled phase (CZ) gate between microwave photons and superconducting qubits has attracted much attention due to its quantum nondemolition (QND) property. However, the practical implementation of such CZ gates and the accurate calibration of detector efficiency remain insufficiently explored. This work presents a comprehensive study on CZ gate implementation and detection efficiency characterization. Theoretically, we derive the condition for $\uppi$ phase shift in the CZ gate and model the measurement-induced dephasing at different parameters. Experimentally, we implement a 3D transmon device, and develop a precise two-stage calibration method for the CZ gate using Josephson Parametric Amplifier's phase sensitivity. We further introduce an efficiency model separating coupling efficiency and internal detection efficiency, incorporating multi-photon effects during calibration with weak coherent states. Our results provide relaxed CZ gate criteria, an accurate calibration protocol, and a loss quantification framework, advancing practical detector performance.