We present a generic system of three bosonic modes coupled parametrically with a time-varying coupling modulated by a combination of two pump harmonics, and we show how this system provides the minimal platform for realizing nonreciprocal couplings that can lead to gainless photon circulation, and phase-preserving or phase-sensitive directional amplification. Explicit frequency-dependent calculations within this minimal paradigm highlight the separation of amplification and directionality bandwidths, a feature generic to such schemes. We also study the influence of counterrotating interactions that can adversely affect directionality and the associated bandwidth; we find that these effects can be mitigated by suitably designing the properties of the auxiliary mode that plays the role of an engineered reservoir to the amplification mode space.

• Received 28 August 2016

DOI:https://doi.org/10.1103/PhysRevApplied.7.034031