  By properly driving a qubit-resonator system in the strong dispersive regime,
we implement an "impedance-matched" $\Lambda$ system in the dressed states,
where a resonant single photon deterministically induces a Raman transition and
excites the qubit. Combining this effect and a fast dispersive readout of the
qubit, we realize a detector of itinerant microwave photons. We theoretically
analyze the single-photon response of the $\Lambda$ system and evaluate its
performance as a detector. We achieve a high detection efficiency close to
unity without relying on precise temporal control of the input pulse shape and
under a conservative estimate of the system parameters. The detector can also
be reset quickly by applying microwave pulses, which allows a short dead time
and a high repetition rate.
