Shock wave in series connected Josephson transmission line: Theoretical foundations and effects of resistive elements

TL;DR

The study analytically characterizes shock waves in Josephson transmission lines under ohmic dissipation, revealing that shock velocity $U$ is set by the states on either side and is independent of dissipation in the dissipationless limit. It shows that dissipation broadens shocks when resistors shunt the JJs or are placed in series with ground capacitors, while dissipation in series with the JJ preserves a sharp front and the same $U$. The authors develop a simple-wave approximation that extends to dissipative JTLs and provide a Newtonian-analogy formulation to understand shock stability and profiles, including weak-damping expansions and time-averaged energy methods. The work has practical implications for JTL-based amplifiers and motivates future exploration of quantum effects on shock dynamics in these nonlinear transmission lines.

Abstract

We analytically study shock wave in the Josephson transmission line (JTL) in the presence of ohmic dissipation. When ohmic resistors shunt the Josephson junctions (JJ) or are introduced in series with the ground capacitors the shock is broadened. When ohmic resistors are in series with the JJ, the shock remains sharp, same as it was in the absence of dissipation. In all the cases considered, ohmic resistors don't influence the shock propagation velocity. We study an alternative to the shock wave - an expansion fan - in the framework of the simple wave approximation for the dissipationless JTL and formulate the generalization of the approximation for the JTL with ohmic dissipation.

Figures (13)

• Figure 1: Dissipationless JTL.
• Figure 2: Small amplitude disturbance propagation velocity $\overline{u}(I)$ according to (\ref{['ccc']}) (blue solid line) and shock propagation velocity $\overline{U}(I_1,I_2)$ according to (\ref{['velocity']}) (red dashed line) for $I_L=4I_c$ and $I_2=.5I_c$.
• Figure 3: A snapshot of the shock wave given by (\ref{['b']}) ($\tau=25$).
• Figure 4: Equation (\ref{['b2']}) (solid blue line) and (\ref{['bro4']}) (dashed red line) for $n=12$.
• Figure 5: JTL with ohmic resistors shunting the JJ and in series with the ground capacitor.