  Single-photon sources are of great interest because they are key elements in
different promising applications of quantum technologies. Here we demonstrate a
highly efficient tunable on-demand microwave single-photon source based on a
transmon qubit with the intrinsic emission efficiency above 98$\%$. The high
efficiency ensures a negligible pure dephasing rate and the necessary condition
for generation of indistinguishable photons. We provide an extended discussion
and analysis of the efficiency of the photon generation. To further
experimentally confirm the single-photon property of the source, correlation
functions of the emission field are also measured using linear detectors with a
GPU-enhanced signal processing technique. Our results experimentally
demonstrate that frequency tunability and negligible pure dephasing rate can be
achieved simultaneously and show that such a tunable single-photon source can
be good for various practical applications in quantum communication,
simulations and information processing in the microwave regime.
