  We demonstrate and evaluate an on-demand source of single itinerant microwave
photons. Photons are generated using a highly coherent, fixed-frequency
qubit-cavity system, and a protocol where the microwave control field is far
detuned from the photon emission frequency. By using a Josephson parametric
amplifier (JPA), we perform efficient single-quadrature detection of the state
emerging from the cavity. We characterize the imperfections of the photon
generation and detection, including detection inefficiency and state infidelity
caused by measurement backaction over a range of JPA gains from 17 to 33 dB. We
observe that both detection efficiency and undesirable backaction increase with
JPA gain. We find that the density matrix has its maximum single photon
component $\rho_{11} = 0.36 \pm 0.01$ at 29 dB JPA gain. At this gain,
backaction of the JPA creates cavity photon number fluctuations that we model
as a thermal distribution with an average photon number $\bar{n} = 0.041 \pm
0.003$.
