Adiabatic Pumping in a Superconductor-Normal-Superconductor Weak Link

We present a formalism to study adiabatic pumping through a superconductor-normal-superconductor weak link. At zero temperature, the pumped charge is related to the Berry phase accumulated, in a pumping cycle, by the Andreev bound states. We analyze in detail the case when the normal region is short compared to the superconducting coherence length. The pumped charge turns out to be an even function of the superconducting phase difference. Hence, it can be distinguished from the charge transferred due to the standard Josephson effect.

Figure 1

Schematic setup for the SNS pump. The weak link is composed of a scattering region contacted to the $S$ terminals (in orange) through two fictitious $N$ leads (in light blue). The transport properties of the scattering region (characterized by the matrix $S_0$) are cycled in time via two external gates which control the parameters $X_1$ and $X_2$. The superconducting order parameters of the $S$ terminals on the left-hand side and on the right-hand side have equal magnitude ${\Delta }_0$, but differ by a phase $\phi$.

Figure 2

Pumped charge, in units of $2e$, as a function of superconducting phase difference $\phi$, computed for a sinusoidal weak-pumping cycle. Different lines refer to different values of average transmission: solid line $\overline{g}=0.5$, dashed line $\overline{g}=0.7$, and dot-dashed line $\overline{g}=0.9$. For the pumping cycle we have chosen $\Delta g=0.1$, $\Delta \alpha =2\pi$, ${\varphi }_0=\pi /2$, and $\hslash \omega =0.1{\Delta }_0(1-\sqrt{1-\overline{g}/2})$. In the inset the Josephson charge is plotted as a function of $\phi$ for the same parameters of the main panel.