Effect of disorder on coherent quantum phase slips in Josephson junction chains

We study coherent quantum phase slips in a Josephson junction chain, including two types of quenched disorder: random spatial modulation of the junction areas and random induced background charges. Usually, the quantum phase-slip amplitude is sensitive to the normal-mode structure of superconducting phase oscillations in the ring (Mooij-Schön modes, which are all localized by the area disorder). However, we show that the modes' contribution to the disorder-induced phase-slip action fluctuations is small, and the fluctuations of the action on different junctions are mainly determined by the local junction parameters. We study the statistics of the total quantum phase-slip amplitude on the chain and show that it can be non-Gaussian for not sufficiently long chains.

Figure 1

A schematic representation of a superconducting ring threaded by a magnetic flux $\mathrm{\Phi }$ and containing $N$ Josephson junctions with a capacitance $C$ between the neighboring islands and a capacitance $C_g$ to the ground. $E_J$ is the Josephson energy. ${\varphi }_n$ is the condensate phase of the $n\mathrm{th}$ superconducting island.

Figure 2

A schematic representation of flux dependence for the ground-state energy (top panel) and persistent current (bottom panel). The gray dotted lines correspond to the purely classical approximation [i.e., neglecting the kinetic terms in the action (1)] for the static configurations ${\varphi }_n=2\pi n/N,{\varphi }_n=0$, and ${\varphi }_n=-2\pi n/N$ (the left, middle, and right parabolas in the top panel, respectively). The red solid lines correspond to the exact ground state including the effect of the quantum tunneling.

Figure 3

Distribution $f\left(A\right)$ in the absence of induced charges, calculated for $\sigma =4$ and different $N$ by direct numerical sampling (blue dots), using the weakest-junction approximation (21) (red dashed lines), the saddle-point approximation (19) (orange dotted lines), and the lognormal fit (solid green lines).

Figure 4

Cumulative probability distribution of $\delta S_n$ and estimates of the two smallest $\delta S_n$ for a typical sample.

Figure 5

Distribution $f\left(\right|A\left|\right)$ with random induced charges, calculated for $\sigma =4$ and different $N$ by direct numerical sampling (blue dots) and using the weakest-junction approximation (21) (red dashed lines).

Figure 6

Two factors of the integrand in $I\left(z\right)$ (B5), $exp(-\frac{x^2}{2{\sigma }^2})$ (dashed blue line) and $1-exp\left(-z{e}^{-x}\right)$ (dashed green line), and their product (solid red line).