  We describe the back action of microwave-photon detection via a Josephson
photomultiplier (JPM), a superconducting qubit coupled strongly to a
high-quality microwave cavity. The back action operator depends qualitatively
on the duration of the measurement interval, resembling the regular photon
annihilation operator at short interaction times and approaching a variant of
the photon subtraction operator at long times. The optimal operating conditions
of the JPM differ from those considered optimal for processing and storing of
quantum information, in that a short $T_2$ of the JPM suppresses the cavity
dephasing incurred during measurement. Understanding this back action opens the
possibility to perform multiple JPM measurements on the same state, hence
performing efficient state tomography.
