Discontinuous Current-Phase Relations in Small One-Dimensional Josephson Junction Arrays

We study the Josephson effect in small one-dimensional (1D) Josephson junction arrays. For weak Josephson tunneling, topologically different regions in the charge-stability diagram generate distinct current-phase ($I\mathrm{\text{−}}\varphi$) relationships. We present results for a three-junction system in the vicinity of charge-degeneracy lines and triple points. We explain the generalization to larger arrays, show that discontinuities of the $I\mathrm{\text{−}}\varphi$ relation at phase $\pi$ persist and that, at maximum degeneracy, the problem can be mapped to a tight-binding model providing analytical results for arbitrary system size.

Figure 1

(a) Progression of small 1D Josephson junction arrays with increasing number of junctions $N$. (b) Detailed schematic of the three-junction JJA. Electronic transport across the junctions is due to interdot ($H_m$) and lead-dot ($H_{L,R}$) tunneling. The electrochemical potentials of the dots can be adjusted by separate gate electrodes.

Figure 2

(a) Lumped element circuit for the superconducting double quantum dot. (b) Charge-stability diagram for the superconducting double-dot system. The dashed regions mark the regimes (i)–(iii) away from all degeneracies, close to a charge-degeneracy line, and close to a triple point of the honeycomb pattern.

Figure 3

(a),(b) Leading-order processes relevant to charge transfer in the three-junction system, close to a charge-degeneracy line. Depicted are the processes starting in the initial charge state (0,1). The interdot tunneling (a) contributes to $J^{\prime }$, the dot-lead processes (b) contribute to $J$. (c) Energies $E_{\pm }(\varphi )$, Eq. (5), of the two-level system close to a charge-degeneracy line. The eigenenergies are plotted as functions of phase for the resonant case $J=J^{\prime }$. The solid (dashed) lines correspond to $h_0=0$ ($J/2$), the arrow marks the position of the low-temperature discontinuity of $I(\varphi )$ at $\pi$.

Figure 4

Current-phase relations for the three-junction JJA at zero temperature, (a) close to a charge-degeneracy line, (b) at a triple point. (a) The $I\mathrm{\text{−}}\varphi$ relation for different ratios $J^{\prime }/J$ of the interdot and dot-lead coupling. (b) $I\mathrm{\text{−}}\varphi$ relation for $E_J\equiv E_{JL}=E_{JR}$ and different ratios $E_{Jm}/E_J$.