Theory of Josephson photomultipliers: Optimal working conditions and back action

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 of performing multiple JPM measurements on the same state, hence performing efficient state tomography.