Quantum fractional resonances in superconducting circuits with an embedded Josephson junction

We present a quantum electrodynamic treatment of the generation of fractional resonances in a planar waveguide with an embedded superconducting Josephson oscillator. We analyze the dynamics of the Josephson oscillator coupled with the electromagnetic pulse which is propagating along the waveguide. The calculations are carried out entirely in the Heisenberg picture. It is shown that the quantum Josephson oscillator excited by coherent pulse field at the pump frequency, can realize frequency down-conversion and emitting sub-harmonic multiples of the fundamental (fractional harmonics). The influence of dissipation on the phenomenon of resonance capture is discussed.