  We present a new type of phase- and frequency-sensitive amplification and
attenuation in a cyclically driven three-level superconducting Josephson
system. Different from the previous linear theory of pure phase-sensitive
amplification, a new physical mechanism$-$combined action of nonlinear wave
mixing and wave interference$-$is developed and leads to not only amplification
but also attenuation. This is referred to as interference nonlinear optics. Our
results show the sudden output signal transition from large gain to deep
suppression by tuning the relative phase and in this case the system can act as
a phase-controlled amplitude modulator. We also show the continuous change from
output enhancement to attenuation by adjusting the driving-field frequency and
in this situation the system behaves as a frequency-controlled amplitude
modulator. Our study opens up a new perspective for its widespread applications
in quantum information science.
