Minimal models for nonreciprocal amplification using biharmonic drives
  We present a generic system of three harmonic modes coupled parametrically with a time-varying coupling modulated by a combination of two pump harmonics, and show how this system provides the minimal platform to realize 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, universal in such schemes. We also study the influence of counter-rotating interactions that can adversely affect directionality and 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. 
A. Kamal and A. Metelmann
cond-mat.mes-hall quant-ph