  The interaction between propagating microwave fields and Cooper-pair
tunneling across a DC voltage-biased Josephson junction can be highly
nonlinear. We show theoretically that this nonlinearity can be used to convert
an incoming single microwave photon into an outgoing $n$-photon Fock state in a
different mode. In this process, the electrostatic energy released in a
Cooper-pair tunneling event is transferred to the outgoing Fock state,
providing energy gain. The created multi-photon Fock state is frequency
entangled and highly bunched. The conversion can be made reflectionless
(impedance-matched) so that all incoming photons are converted to $n$-photon
states. With realistic parameters multiplication ratios $n > 2$ can be reached.
By two consecutive multiplications, the outgoing Fock-state number can get
sufficiently large to accurately discriminate it from vacuum with linear
post-amplification and power measurement. Therefore, this amplification scheme
can be used as single-photon detector without dead time.
