On Parametric Amplification In Discrete Josephson Transmission Line

TL;DR

The work extends parametric amplification in Josephson transmission lines from a continuum to a discrete JTL by deriving pump–signal–idler coupling equations and thresholds in a lossy, 1D array. It develops both three‑wave and four‑wave mixing formalisms, each with a local pump‑dominated threshold that accounts for loss, dispersion, and the discrete structure, providing explicit inequalities for amplification. In the small‑wave‑vector limit, it yields exact amplitude relations and conserved quantities that illuminate energy exchange between modes, while in the general case it presents threshold criteria incorporating detuning and dissipation. The results offer practical guidance for realizing broadband superconducting traveling‑wave parametric amplifiers in discrete circuits, highlighting how momentum mismatch and finite‑length effects shape the amplification threshold.

Abstract

We consider the discrete series-connected lossy Josephson transmission line, constructed from Josephson junctions, capacitors and resistors (one-dimensional array of Josephson junctions). We derive equations describing pump, signal, and idler interaction in the system and calculate the thresholds for the parametric amplification.

Figures (3)

• Figure 1: Josephson transmission line composed of superconducting grains
• Figure 2: The solution (\ref{['good2']}) plotted in the variables $A_{\alpha}^{(r)}$. We have chosen $\mu_i/\mu_s=1.2$ and $\theta=.1$ for $\phi=0$.
• Figure 3: Numerical solution of Eqs. (\ref{['pa']}), (\ref{['pb']}) for $\theta$ (blue solid line) and $\varphi$ (red dashed line). We have chosen $\mu_s=.2$, $\mu_i=.24$, $\theta(0)=.1$ and $\varphi(0)=0$ (no idle signal at $\tilde{Z}=0$).