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 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 R by more than 3 R / 2Q .