Gain, directionality and noise in microwave SQUID amplifiers:   Input-output approach
  We present a new theoretical framework to analyze microwave amplifiers based on the dc SQUID. Our analysis applies input-output theory generalized for Josephson junction devices biased in the running state. Using this approach we express the high frequency dynamics of the SQUID as a scattering between the participating modes. This enables us to elucidate the inherently nonreciprocal nature of gain as a function of bias current and input frequency. This method can, in principle, accommodate an arbitrary number of Josephson harmonics generated in the running state of the junction. We report detailed calculations taking into account the first few harmonics that provide simple semi-quantitative results showing a degradation of gain, directionality and noise of the device as a function of increasing signal frequency. We also discuss the fundamental limits on device performance and applications of this formalism to real devices. 
Archana Kamal, John Clarke, Michel Devoret
cond-mat.supr-con quant-ph