Microwave bifurcation of a Josephson junction: Embedding-circuit requirements

A Josephson tunnel junction which is rf driven near a dynamical bifurcation point can amplify quantum signals. However, the bifurcation point will exist robustly only if the electrodynamic environment of the junction meets certain criteria. We develop a general formalism for dealing with the nonlinear dynamics of a 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 needs 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 >> 1, and (b) the drive frequency should be low frequency detuned away from omega(R) by more than root 3 omega(R)/(2Q).