Dynamical Coulomb Blockade of Multiple Andreev Reflections

We analyze the dynamical Coulomb blockade of multiple Andreev reflections (MAR) in a superconducting quantum point contact coupled to a macroscopic impedance. We find that at very low transmission the blockade scales as $n^2$ with $n=\mathrm{Int}(2\Delta /eV)$, where $V$ is the bias voltage and $\Delta$ is the superconducting gap, as it would correspond to the occurrence of shots of charge $ne$. For higher transmission the blockade is reduced because of both the Pauli principle and the elastic renormalization of the MAR probability, and for certain voltage regions it may even become an antiblockade; i.e., the current is enhanced due to the coupling with the electromagnetic environment.

Figure 1

Mean current (upper panel) and current blockade (lower panel) for a single mode environment with ${\omega }_0=0.2\Delta$ at zero temperature for different transmission values. The current blockade curves have been displaced vertically for clarity. Notice normalization with $G_N=(2e^2/h)\tau$ and $z$, which is assumed to be a small parameter.

Figure 2

Typical diagrams contributing to the blockade in the tunnel limit for $eV\sim 2\Delta /3$. Crosses indicate the hopping events on the two branches ($+$ or $-$) of the Keldysh contour. Full lines with an arrow indicate electron or hole propagators, while double arrowed lines correspond to the anomalous ones and wavy lines denote phase correlators. Diagram (a) is an inelastic correction due to the coupling with the environment, while diagram (b) corresponds to an elastic renormalization of the MAR probability.

Figure 3

Relative blockade for $\tau =0.01$, 0.1, 0.2, and 0.3 (from top to bottom) and ${\omega }_0=0.2\Delta$. The shadowed regions indicate the maximum relative blockade (scaling as $n^2$) that is predicted for $\tau \to 0$.

Figure 4

Current blockade at small bias voltage and large transmission. Notice the sign change for intermediate bias. The dotted lines correspond to the approximate expression of Eq. (10). The inset shows the behavior of blockade for $\tau =0.9$ and different values of ${\omega }_0$.