Detecting an itinerant microwave photon with high efficiency is an
outstanding problem in microwave photonics and its applications. We present a
scheme to detect an itinerant microwave photon in a transmission line via the
nonlinearity provided by a transmon in a driven microwave resonator. With a
single transmon we achieve 84% distinguishability between zero and one
microwave photons and 90% distinguishability with two cascaded transmons by
performing continuous measurements on the output field of the resonator. We
also show how the measurement diminishes coherence in the photon number basis
thereby illustrating a fundamental principle of quantum measurement: the higher
the measurement efficiency, the faster is the decoherence
