Three-wave mixing in Josephson travelling-wave parametric amplifiers

Abstract: This work explores possibilities of building a wide-band, quantum-limited low-noise amplifier by means of three-wave mixing (3WM) in different kinds of Josephson travelling-wave parametric amplifiers (TWPAs). We extend the theory of the continuous three-mode model to include any number of up-converted modes in the small frequency limit where the frequency dispersion is close to linear. We also extend the theory to describe a discrete chain at frequencies close to the spectral cutoff where there is no up-conversion. In both cases we find that the gain is significantly reduced compared to the prediction by the continuous three-mode model. At the high frequencies we cannot pump strongly enough to overcome the increasingly strong dispersion, while in the small frequency limit, the dispersion can be overcome but the gain is then reduced by up-conversion processes. The developed theory is in quantitative agreement with experimental observations. To recover the high gain predicted by the continuous three-mode model, we propose to engineer a TWPA with dispersive features to create a two-band dispersion relation, either by adding resonant phase matching (RPM) features, or by periodically modulating the parameters of the chain. By placing the pump frequency within the upper band, close to the cutoff frequency, while placing the signal in the lower band, we prove that there exists a sweet spot where the signal and the pump are phase matched while the up-conversion is inhibited. We solve the discrete equations for the RPM-based TWPA and show that the gain is expected to grow exponentially with the length of the TWPA.