Photon number resolving detectors play a central role in quantum optics. A key challenge in resolving the number of absorbed photons in the microwave frequency range is finding a suitable material that provides not only an appropriate band structure for absorbing low-energy photons but also a means of detecting a discrete photoelectron excitation. To this end, we propose to measure the temperature gain after absorbing a photon using superconducting cadmium arsenide (Cd3As2) with a topological semimetallic surface state as the detector. The surface electrons absorb the incoming photons and then transfer the excess energy via heat to the superconducting bulks phonon modes. The temperature gain can be determined by measuring the change in the zero-bias bulk resistivity, which does not significantly affect the lattice dynamics. Moreover, the obtained temperature gain scales discretely with the number of absorbed photons, enabling a photon-number resolving function. Here, we will calculate the temperature increase as a function of the number and frequency of photons absorbed. We will also derive the timescale for the heat transfer process from the surface electrons to the bulk phonons. We will specifically show that the transfer processes are fast enough to ignore heat dissipation loss.
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