  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.

Quantum Physics