RF bifurcation of a Josephson junction: microwave embedding circuit requirements

A Josephson tunnel junction which is RF-driven near a dynamical bifurcation point can amplify quantum signals. The bifurcation point will exist robustly only if the electrodynamic environment of the junction meets certain criteria. In this article we develop a general formalism for dealing with the non-linear dynamics of Josephson junction embedded in an arbitrary microwave circuit. We find sufficient conditions for the existence of the bifurcation regime: a) the embedding impedance of the junction need to present a resonance at a particular frequency $\omega_{R}$, with the quality factor $Q$ of the resonance and the participation ratio $p$ of the junction satisfying $Qp\gg 1$, b) the drive frequency should be low frequency detuned away from $\omega_{R}$ by more than $\sqrt{3}\omega_{R}/(2Q)$.