A noiseless, photon-counting detector, which resolves the energy of each photon, could radically change astronomy, biophysics, and quantum optics. Superconducting detectors promise an intrinsic resolving power at visible wavelengths of R = E/δE ≈ 100 due to their low excitation energy. We study superconducting energy-resolving microwave kinetic inductance detectors (MKIDs), which hold particular promise for larger cameras. A visible and near-infrared photon absorbed in the superconductor

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... a few thousand quasiparticles through several stages of electron-phonon interaction. Here we demonstrate experimentally that the resolving power of MKIDs at visible to near-infrared wavelengths is limited by the loss of hot phonons during this process. We measure the resolving power of our aluminum-based detector as a function of photon energy using four lasers with wavelengths between 1545-402 nm. For detectors on thick SiN/Si and sapphire substrates the resolving power is limited to 10-21 for the respective wavelengths, consistent with the loss of hot phonons. When we suspend the sensitive part of the detector on a 110-nm-thick SiN membrane, the measured resolving power improves to 19-52, respectively. The improvement is equivalent to a factor 8 ± 2 stronger phonon trapping on the membrane, which is consistent with a geometrical phonon propagation model for these hot phonons. We discuss a route towards the Fano limit by phonon engineering.